Pseudo-dual pairs and branching of Discrete Series

TL;DR

The paper investigates Discrete Series representations of semisimple Lie groups with admissible branching to symmetric subgroups, introducing pseudo-dual pairs and explicit branching laws, with applications to computational algorithms like Atlas.

Contribution

It introduces the concept of pseudo-dual pairs for symmetric subgroups and develops explicit branching laws, connecting to symmetry breaking and computational methods.

Findings

Explicit examples of branching laws provided.

Links established to symmetry breaking and holographic operators.

Method suitable for computer algorithms like Atlas.

Abstract

For a semisimple Lie group $G$, we study Discrete Series representations with admissible branching to a symmetric subgroup $H$. This is done using a canonical associated symmetric subgroup $H_0$, forming a pseudo-dual pair with $H$, and a corresponding branching law for this group with respect to its maximal compact subgroup. This is in analogy with either Blattner's or Kostant-Heckmann multiplicity formulas, and has some resemblance to Frobenius reciprocity. We give several explicit examples and links to Kobayashi-Pevzner theory of symmetry breaking and holographic operators. Our method is well adapted to computer algorithms, such as for example the Atlas program.

Equations

$$\dim Ho{m}_H(\sigma ,{\pi }_{{|}_H})=\dim Ho{m}_{\tilde{K}}(F_{\sigma },\tilde{\pi }).$$

$$‡$$

$$L^2(G{\times }_{\tau }W)$$

$$(f,g{)}_{V_{\lambda }}={\int }_G(f(x),g(x){)}_{W}dx,$$

$$H^2(G,\tau )=\{f\in L^2(G)\otimes W:f(gk)=\tau (k{)}^{-1}f(g)$$

$$K_{\lambda }:G\times G\to En{d}_{\mathbb{C}}(W)$$

$$(P_{\lambda }(f)(x),w{)}_W={\int }_G(f(y),K_{\lambda }(y,x{)}^{\star }w{)}_{W}dy,f\in L^2(G{\times }_{\tau }W).$$

$$\mathcal{U}({\mathfrak{h}}_0)W:=L_{\mathcal{U}({\mathfrak{h}}_0)}{H}^2(G,\tau )[V_{\lambda +{\rho }_n^{\lambda }}^K]\equiv {\pi }_{\lambda }(\mathcal{U}({\mathfrak{h}}_0))(V_{\lambda }^G[V_{\lambda +{\rho }_n^{\lambda }}^K]).$$

$$H^2(G,\tau )[V_{\lambda +{\rho }_n^{\lambda }}^K]=\{K_{\lambda }(\cdot ,e{)}^{\star }w:w\in W\}\equiv W.$$

$$W={\oplus }_{1\le j\le r}W[({\sigma }_j,Z_j)]={\oplus }_{1\le j\le r}W[{\sigma }_j].$$

$${\mathbf{H}}^2(H_0,\tau ):=\sum _j\dim Ho{m}_L(\tau ,{\sigma }_j)H^2(H_0,{\sigma }_j).$$

$$\sum _j{L}^2(H_0{\times }_{{\sigma }_j}W[{\sigma }_j]{)}_{H_0-disc}\equiv L^2(H_0{\times }_{\tau }W{)}_{H_0-disc}.$$

$$K_T(e,\cdot )z\in H^2(G,\tau )[V_{\mu }^H][V_{\mu +{\rho }_n^H}^L]$$

$$r_0:H^2(G,\tau )\to L^2(H_0{\times }_{\tau }W),r_0(f)(h_0)=f(h_0),h_0\in H_0,$$

$$T\stackrel{r_0^D}{⟼}(V_{\mu +{\rho }_n^{\mu }}^L\ni z\mapsto r_0(D(K_T(e,\cdot )z))\in {\mathbf{H}}^2(H_0,\tau ))$$

$$K_T(hl,gk)=\tau (k^{-1})K_T(h,g)\sigma (l),k\in K,l\in L,g\in G,h\in H.$$

$$K_T(h{h}_1,hx)=K_T(h_1,x),h,h_1\in H,x\in G.$$

$$\mathcal{U}(\mathfrak{h})W:=L_{\mathcal{U}(\mathfrak{h})}{H}^2(G,\tau )[V_{\lambda +{\rho }_n^{\lambda }}^K]\equiv {\pi }_{\lambda }(\mathcal{U}(\mathfrak{h}))(V_{\lambda }^G[V_{\lambda +{\rho }_n^{\lambda }}^K]).$$

$$\dim Ho{m}_{\mathfrak{h},H\cap K}(V_{\mu }^H,V_{\lambda }^G)=\dim Ho{m}_{\tilde{K}}(F_{\mu ,\chi },({\Gamma }_{H\cap L_1}^{\tilde{K}}{)}^{p_0+q_0}(N(\Lambda ))),$$

$$m^{G,H}(\lambda ,\mu )=(-1{)}^{\frac{1}{2}\dim (H/H\cap L_1)}\sum _{i=1}^d\sum _{s\in W_{\tilde{K}}}ϵ(s)p({\Lambda }_i+{\rho }_{\tilde{K}}+s{s}_{H\cap K}\mu ),$$

$$m^{G,H}(\lambda ,\mu )=\pm \sum _{w\in W_K}ϵ(w)p_{S_w^H}(\mu -q_{\mathfrak{u}}(w\lambda )).$$

$$S_w^H:=S_w^L\backslash \Phi (\mathfrak{h}/\mathfrak{l},\mathfrak{u}),\mathrm{where},S_w^L:=q_{\mathfrak{u}}(w({\Psi }_{\lambda }{)}_n)\cup \Delta (\mathfrak{k}/\mathfrak{l},\mathfrak{u}).$$

$$r_m:H^2(G,\tau )\to L^2(H{\times }_{S^m(Ad)⊠\tau }(S^m(\mathfrak{p}\cap \mathfrak{q}{)}^{\star }\otimes W)).$$

$$m^{G,H}(\lambda ,\mu )=\underset{m\to \infty }{\lim }\dim {\varphi }_{{\pi }_{\lambda },{\pi }_{\mu }^H,m}((H^2(G,\tau )\cap \mathcal{C}(G,\tau ))[V_{\mu +{\rho }_n^H}]).$$

$$re{s}_{L\cap K_2}({\pi }_{{\Lambda }_2})=re{s}_{L\cap K_2}({\pi }_{{\Lambda }_2}^{K_2})=\sum _{{\nu }_2\in (\mathfrak{u}\cap {\mathfrak{k}}_2{)}^{\star }}{m}^{K_2,L\cap K_2}({\Lambda }_2,{\nu }_2){\pi }_{{\nu }_2}^{L\cap K_2},$$

$$re{s}_L({\pi }_{{\Lambda }_1}^{K_1}⊠{\pi }_{{\Lambda }_2}^{K_2})=\sum _{{\nu }_2\in Spe{c}_{L\cap K_2}({\pi }_{{\Lambda }_2}^{K_2})}{m}^{K_2,L\cap K_2}({\Lambda }_2,{\nu }_2){\pi }_{{\Lambda }_1}^{K_1}⊠{\pi }_{{\nu }_2}^{L\cap K_2},$$

$$re{s}_L({\pi }_{({\eta }_1,{\eta }_2)}^{H_0})=\sum _{({\theta }_1,{\theta }_2)\in Spe{c}_L({\pi }_{({\eta }_1,{\eta }_2)}^{H_0})}{m}^{H_0,L}(({\eta }_1,{\eta }_2),({\theta }_1,{\theta }_2)){\pi }_{({\theta }_1,{\theta }_2)}^L.$$

$$re{s}_H({\pi }_{\lambda })=re{s}_H({\pi }_{\lambda }^G)=\sum _{\mu \in Spe{c}_H({\pi }_{\lambda })}{m}^{G,H}(\lambda ,\mu ){\pi }_{\mu }^H.$$

$$m^{G,H}(\lambda ,\mu )=\dim Ho{m}_H(H^2(H,V_{\mu +{\rho }_n^H}^L),H^2(G,V_{\lambda +{\rho }_n^{\lambda }}^K))$$

$$\begin{array}{rl}& ic(H^2(H_0,{\pi }_{{\Lambda }_1}^{K_1}⊠{\pi }_{{\nu }_2}^{L\cap K_2}))=({\Lambda }_1,{\nu }_2)-{\rho }_n^{H_0}\\ & =({\lambda }_1+({\rho }_n^{\lambda }{)}_1-({\rho }_n^{H_0}{)}_1,{\nu }_2-({\rho }_n^{H_0}{)}_2)=({\lambda }_1+({\rho }_n^H{)}_1,{\nu }_2-({\rho }_n^{H_0}{)}_2)\\ & =({\lambda }_1,{\nu }_2-({\rho }_n^H{)}_2-({\rho }_n^{H_0}{)}_2)+{\rho }_n^H=({\lambda }_1,{\nu }_2-q_{\mathfrak{u}}({\rho }_n^{\lambda }{)}_2)+{\rho }_n^H.\end{array}$$

Full text

Partially supported by Aarhus University (Denmark), CONICET (Argentina)

We are delighted to make this paper part of a tribute to Toshiyuki Kobayashi,

for his indefatigable dedication to representation theory

and Lie theory, and his many new ideas here.

111Keywords: Admissible restriction, Branching laws, Discrete Series, Multiplicity formulae.222MSC2020: Primary 22E45; Secondary 20G05, 17B10.

Abstract For a semisimple Lie group $G$, we study Discrete Series representations with admissible branching to a symmetric subgroup $H$. This is done using a canonical associated symmetric subgroup $H_{0}$, forming a pseudo-dual pair with $H$, and a corresponding branching law for this group with respect to its maximal compact subgroup. This is in analogy with either Blattner’s or Kostant-Heckman multiplicity formulas, and has some resemblance to Frobenius reciprocity. We give several explicit examples and links to Kobayashi-Pevzner theory of symmetry breaking and holographic operators. Our method is well adapted to computer algorithms, such as for example the Atlas program.

1. 1 Introduction

2. 2 Preliminaries and some notation

3. 3 Duality Theorem, explicit isomorphism

4. 3.1 Statement and proof of the duality result

5. 3.1.1

6. 3.1.2

7. 3.1.3

8. 3.1.4

9. 3.1.5

10. 3.1.6

11. 3.1.7

12. 3.2 Kernel of the restriction map

13. 3.3 The map $r_{0}^{D}$ is injective

14. 3.4 Previous work on duality formula and Harish-Chandra parameters

15. 3.4.1 Computing Harish-Chandra parameters from Theorem 3.1

16. 3.4.2 Gross-Wallach multiplicity formula

17. 3.4.3 Duflo-Vargas multiplicity formula, [5]

18. 3.4.4 Harris-He-Olafsson multiplicity formula, [10]

19. 3.5 Completion of the Proof of Theorem 3.1, the map $r_{0}^{D}$ is surjective

20. 4 Duality Theorem, proof of dimension equality

21. 4.1 Dimension equality Theorem, statement

22. 4.1.1 Conclusion proof of Theorem 4.2

23. 4.1.2 Existence of $D$

24. 4.1.3 Representations $\pi_{\lambda}$ so that $res_{L}(\tau)$ is irreducible

25. 4.1.4 Analysis of $\mathcal{U}(\mathfrak{h}_{0})W$, $\mathcal{L}_{\lambda}$, existence of $D$, case $\tau_{|_{L}}$ is irreducible

26. 4.1.5 Analysis of $\mathcal{U}(\mathfrak{h}_{0})W$, $\mathcal{L}_{\lambda}$, existence of $D$, for general $(\tau,W)$

27. 4.2 Explicit inverse map to $r_{0}^{D}$

28. 4.2.1 Case $(\tau,W)$ restricted to $L$ is irreducible

29. 4.2.2 Value of $b=d=c$ when $res_{L}(\tau)$ is irreducible

30. 4.2.3 Analysis of $r_{0}^{D}$ for arbitrary $(\tau,W),(\sigma,Z)$

31. 4.2.4 Eigenvalues of $r_{0}^{\star}r_{0}$

32. 5 Examples

33. 5.1 Multiplicity free representations

34. 5.1.1 Holomorphic representations

35. 5.1.2 Quaternionic real forms, quaternionic representations

36. 5.1.3 More examples of multiplicity free restriction

37. 5.2 Explicit examples

38. 5.2.1 Quaternionic representations for $Sp(1,b)$

39. 5.2.2 Explicit example I

40. 5.2.3 Explicit example II

41. 5.2.4 Explicit example III

42. 5.2.5 Comments on admissible restriction of quaternionic representations

43. 5.2.6 Explicit example IV

44. 5.3 Existence of Discrete Series whose lowest $K$-type restricted to $K_{1}(\Psi)$ is irreducible

45. 5.3.1

46. 6 Symmetry breaking operators and normal derivatives

47. 6.0.1 Converse to Proposition 6.1

48. 6.0.2 Comments on the interplay among the subspaces, $\mathcal{L}_{\lambda}$, $\mathcal{U}(\mathfrak{h}_{0})W$, $H^{2}(G,\tau)_{K-fin}$ and symmetry breaking operators

49. 6.0.3 A functional equation for symmetry breaking operators

50. 7 Tables

51. 8 Partial list of symbols and definitions

1. Introduction

For a semisimple Lie group $G$, an irreducible representation $(\pi,V)$ of $G$ and closed reductive subgroup $H\subset G$ the problem of decomposing the restriction of $\pi$ to $H$ has received attention ever since number theory or physics and other branches of mathematics required a solution. In this paper, we are concerned with the important particular case of branching representations of the Discrete Series, i.e. those $\pi$ arising as closed irreducible subspaces of the left regular representation in $L^{2}(G)$, and breaking the symmetry by a reductive subgroup $H$. Here much work has been done. Notable is the paper of Gross-Wallach, [9], and the work of Toshiyuki Kobayashi and his school. For further references on the subject, we refer to the overview work of Toshiyuki Kobayashi and references therein. The aim of this note is to compute the decomposition of the restriction of an $H$-admissible representation $\pi$ to a symmetric subgroup $H$ (see 3.4.2), in [9] it is derive a duality Theorem for Discrete Series representation. Their duality is based on the dual subgroup $G^{d}$ (this is the dual subgroup which enters the duality introduced by Flensted-Jensen in his study of discrete series for affine symmetric spaces [7]) and, roughly speaking, their formula looks like

[TABLE]

Here, $\pi$ is a irreducible square integrable representation of $G$, $\sigma$ is a irreducible representation of $H$, $F_{\sigma}$ is a irreducible representation of a maximal compact subgroup $\widetilde{K}$ of $G^{d}$, and $\tilde{\pi}$ is a finite sum of fundamental representations of $G^{d}$ attached to $\pi$. In [23], B. Speh and the first author considered a different duality Theorem for restriction to a symmetric subgroup, let $H_{0}$ the associated subgroup to $H$ (see 3.1), hence, $L:=H_{0}\cap H$ is a maximal compact subgroup of both $H,H_{0}$. Then,

[TABLE]

Here, $\pi$ is certain irreducible representation of $G$, $\sigma$ is a irreducible representation of $H$, $\sigma_{0}$ is the lowest $L$-type of $\sigma$ and $\widetilde{\Pi}$ is a finite sum of irreducible representation of $H_{0}$ attached to $\pi$. The purpose of this paper is, for a $H$-admissible Discrete Series $\pi$ for $G$, to show a formula as the above and to provide an explicit isomorphism between the two vector spaces involved in the equality. This is embodied in Theorem 3.1.

Theorem 3.1 reduces the branching law in two steps (1) For the maximal compact subgroup $K$ of $G$ and the lowest $K$-type of $\pi$, branching this under $L$ (maximal compact in $H$ and also in $H_{0}$) (2) branching a Discrete Series of $H_{0}$ with respect to $L$, i.e. finding its $L$-types with multiplicity. Both of these steps can be implemented in algorithms, as they are available for example in the computer program Atlas, http://atlas.math.umd.edu.

We would like to point out that T. Kobayashi, T. Kobayashi-Pevzner and Nakahama have shown a duality formula as $({\ddagger})$ for holomorphic Discrete Series representation $\pi$. In order to achieve their result, they have shown an explicit isomorphism between the two vector spaces in the formula. Further, with respect to analyze $res_{H}(\pi)$, Kobayashi-Oshima have shown a way to compute the irreducible components of $res_{H}(\pi)$ in the language of Zuckerman modules $A_{\mathfrak{q}}(\lambda)$ [18][19].

As a consequence of the involved material, we obtain a necessary and sufficient condition for a symmetry breaking operators to be represented via normal derivatives. This is presented in Proposition 6.1.

Another consequence is Proposition 4.9. That is, for the closure of the linear span of the totality of $H_{0}$-translates (resp. $H$-translates) of the isotypic component associated to the lowest $K$-type of $\pi$, we exhibit its explicit decomposition as a finite sum of Discrete Series representations of $H_{0}$ (resp. $H$).

Our proof is based on the fact that Discrete Series representations are realized in reproducing kernel Hilbert spaces. As a consequence, in Lemma 3.6, we obtain a general result on the structure of the kernel of a certain restriction map. The proof also relies on the work of Hecht-Schmid [11], and a result of Schmid in [27].

It follows from the work of Kobayashi-Oshima, also, from Tables 1,2,3, that whenever a Discrete Series for $G$ has an admissible restriction to a symmetric subgroup, then, the infinitesimal character of the representation is dominant with respect to either a Borel de Siebenthal system of positive roots or to a system of positive roots so that it has two noncompact simple roots, each one, has multiplicity one in the highest root. Under the $H$-admissible hypothesis, the infinitesimal character of each of the irreducible components of $\widetilde{\Pi}$ in formula $({\ddagger})$, has the same property as the infinitesimal character of $\pi$. Thus, for most $H$-admissible Discrete series, to compute the right hand side of $({\ddagger})$, we may appeal to the work of the first author and Wolf [26]. Their results let us compute the highest weight of each irreducible factor in the restriction of $\pi$ to $K_{1}(\Psi)$. Next, we apply [5, Theorem 5] for the general case.

We may speculate that a formula like $({\ddagger})$ might be true for $\pi$ whose underlying Harish-Chandra module is equivalent to a unitarizable Zuckerman module. In this case, the definition of $\sigma_{0}$ would be the subspace spanned by the lowest $L$-type of $\sigma$ and $\widetilde{\Pi}$ would be a Zuckerman module attached to the lowest $K$-type of $\pi$.

The paper is organized as follows. In Section 2, we introduce facts about Discrete Series representation and notation. In Section 3, we state the main Theorem and begin its proof. As a tool, we obtain information on the kernel of the restriction map.

In Section 4, we complete the proof of the main Theorem. As a by-product, we obtain information on the kernel of the restriction map, under admissibility hypothesis. We present examples and applications of the Main Theorem in section 5. This includes lists of multiplicity free restriction of representations, many of the multiplicity free representations are non holomorphic Discrete Series representations. We also dealt with quaternionic and generalized quaternionic representations.

In Section 6, we analyze when symmetry breaking operators are represented by means of normal derivatives. Section 7 presents the list of $H$-admissible Discrete Series and related information.

Acknowledgements: The authors would like to thank T. Kobayashi for much insight and inspiration on the problems considered here. Also, we thank Michel Duflo, Birgit Speh, Yosihiki Oshima and Jan Frahm for conversations on the subject. Part of the research in this paper was carried out within the online research community on Representation Theory and Noncommutative Geometry sponsored by the American Institute of Mathematics. Also, some of the results in this note were the subject of a talk in the "Conference in honour of Prof. Toshiyuki Kobayashi" to celebrate his sixtieth birthday, the authors thank the organizers for the facilities to present and participate in such a wonderful meeting via zoom. Finally, we thank the referees for their truly expert and careful advice in improving the paper.

2. Preliminaries and some notation

Let $G$ be an arbitrary, matrix, connected semisimple Lie group. Henceforth, we fix a maximal compact subgroup $K$ for $G$ and a maximal torus $T$ for $K.$ Harish-Chandra showed that $G$ admits square integrable irreducible representations if and only if $T$ is a Cartan subgroup of $G.$ For this paper, we always assume $T$ is a Cartan subgroup of $G.$ Under these hypothesis, Harish-Chandra showed that the set of equivalence classes of irreducible square integrable representations is parameterized by a lattice in $i\mathfrak{t}^{\star}.$ In order to state our results we need to make explicit this parametrization and set up some notation. As usual, the Lie algebra of a Lie group is denoted by the corresponding lower case German letter. To avoid notation, the complexification of the Lie algebra of a Lie group is also denoted by the corresponding German letter without any subscript. $V^{\star}$ denotes the dual space to a vector space $V.$ Let $\theta$ be the Cartan involution which corresponds to the subgroup $K,$ the associated Cartan decomposition is denoted by $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}.$ Let $\Phi(\mathfrak{g},\mathfrak{t})$ denote the root system attached to the Cartan subalgebra $\mathfrak{t}.$ Hence, $\Phi(\mathfrak{g},\mathfrak{t})=\Phi_{c}\cup\Phi_{n}=\Phi_{c}(\mathfrak{g},\mathfrak{t})\cup\Phi_{n}(\mathfrak{g},\mathfrak{t})$ splits up as the union the set of compact roots and the set of noncompact roots. From now on, we fix a system of positive roots $\Delta$ for $\Phi_{c}.$ For this paper, either the highest weight or the infinitesimal character of an irreducible representation of $K$ is dominant with respect to $\Delta.$ The Killing form gives rise to an inner product $(...,...)$ in $i\mathfrak{t}^{\star}.$ As usual, let $\rho=\rho_{G}$ denote half of the sum of the roots for some system of positive roots for $\Phi(\mathfrak{g},\mathfrak{t}).$ A Harish-Chandra parameter for $G$ is $\lambda\in i\mathfrak{t}^{\star}$ such that $(\lambda,\alpha)\not=0,$ for every $\alpha\in\Phi(\mathfrak{g},\mathfrak{t}),$ and so that $\lambda+\rho$ lifts to a character of $T.$ To each Harish-Chandra parameter $\lambda$, Harish-Chandra, associates a unique irreducible square integrable representation $(\pi_{\lambda}^{G},V_{\lambda}^{G})$ of $G$ of infinitesimal character $\lambda.$ Moreover, he showed the map $\lambda\rightarrow(\pi_{\lambda}^{G},V_{\lambda}^{G})$ is a bijection from the set of Harish-Chandra parameters dominant with respect to $\Delta$ onto the set of equivalence classes of irreducible square integrable representations for $G$ (see [32, Chap 6]). For short, we will refer to an irreducible square integrable representation as a Discrete Series representation.

Each Harish-Chandra parameter $\lambda$ gives rise to a system of positive roots

xxxxxxxxxxxxxxx$\Psi_{\lambda}=\Psi_{G,\lambda}=\{\alpha\in\Phi(\mathfrak{g},\mathfrak{t}):(\lambda,\alpha)>0\}.$

From now on, we assume that Harish-Chandra parameter for $G$ are dominant with respect to $\Delta.$ Whence, $\Delta\subset\Psi_{\lambda}.$ We write $\rho_{n}^{\lambda}=\rho_{n}=\frac{1}{2}\sum_{\beta\in\Psi_{\lambda}\cap\Phi_{n}}\beta$, $(\Psi_{\lambda})_{n}:=\Psi_{\lambda}\cap\Phi_{n}$. We define $\rho_{c}=\frac{1}{2}\sum_{\alpha\in\Delta}\alpha$.

We denote by $(\tau,W):=(\pi_{\lambda+\rho_{n}}^{K},V_{\lambda+\rho_{n}}^{K})$ the lowest $K-$type of $\pi_{\lambda}:=\pi_{\lambda}^{G}.$ The highest weight of $(\pi_{\lambda+\rho_{n}}^{K},V_{\lambda+\rho_{n}}^{K})$ is $\lambda+\rho_{n}-\rho_{c}.$ We recall a Theorem of Vogan’s thesis [31][6] which states that $(\tau,W)$ determines $(\pi_{\lambda},V_{\lambda}^{G})$ up to unitary equivalence. We recall the set of square integrable sections of the vector bundle determined by the principal bundle $K\rightarrow G\rightarrow G/K$ and the representation $(\tau,W)$ of $K$ is isomorphic to the space

[TABLE]

Here, the action of $G$ is by left translation $L_{x},x\in G.$ The inner product on $L^{2}(G)\otimes W$ is given by

[TABLE]

where $(...,...)_{W}$ is a $K-$invariant inner product on $W.$ Subsequently, $L_{D}$ (resp. $R_{D})$ denotes the left infinitesimal (resp. right infinitesimal) action on functions from $G$ of an element $D$ in universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ for the Lie algebra $\mathfrak{g}$. As usual, $\Omega_{G}$ denotes the Casimir operator for $\mathfrak{g}.$ Following Hotta-Parthasarathy [13], Enright-Wallach [6], Atiyah-Schmid [1], we realize $V_{\lambda}:=V_{\lambda}^{G}$ as the space

[TABLE]

We also recall, $R_{\Omega_{G}}=L_{\Omega_{G}}$ is an elliptic $G-$invariant operator on the vector bundle $W\rightarrow G\times_{\tau}W\rightarrow G/K$ and hence, $\,H^{2}(G,\tau)$ consists of smooth sections, moreover point evaluation $e_{x}$ defined by $\,H^{2}(G,\tau)\ni f\mapsto f(x)\in W$ is continuous for each $x\in G$ (cf. [25, Appendix A4]). Therefore, the orthogonal projector $P_{\lambda}$ onto $\,H^{2}(G,\tau)$ is an integral map (integral operator) represented by the smooth matrix kernel or reproducing kernel [25, Appendix A1, Appendix A4, Appendix A6]

[TABLE]

which satisfies $K_{\lambda}(\cdot,x)^{\star}w$ belongs to $\,H^{2}(G,\tau)$ for each $x\in G,w\in W$ and

[TABLE]

For a closed reductive subgroup $H$, after conjugation by an inner automorphism of $G$ we may and will assume $L:=K\cap H$ is a maximal compact subgroup for $H.$ That is, $H$ is $\theta-$stable. In this paper for irreducible square integrable representations $(\pi_{\lambda},V_{\lambda})$ for $G$ we would like to analyze its restriction to $H.$ In particular, we study the irreducible $H-$subrepresentations for $\pi_{\lambda}$. A known fact is that any irreducible $H-$subrepresentation of $V_{\lambda}$ is a square integrable representation for $H$, for a proof (cf. [9]). Thus, owing to the result of Harish-Chandra on the existence of square integrable representations, from now on, we may and will assume $H$* admits a compact Cartan subgroup*. After conjugation, we may assume $U:=H\cap T$ is a maximal torus in $L=H\cap K.$ From now on, we set a square integrable representation $V_{\mu}^{H}\equiv H^{2}(H,\sigma)\subset L^{2}(H\times_{\sigma}Z)$ of lowest $L-$type $(\pi_{\mu+\rho_{n}^{\mu}}^{L},V_{\mu+\rho_{n}^{\mu}}^{L})\equiv:(\sigma,Z)$.

For a representation $M$ and irreducible representation $N$, $M[N]$ denotes the isotypic component of $N$, that is, $M[N]$ is the linear span of the irreducible subrepresentations of $M$ equivalent to $N$. If topology is involved $M[N]$ is the closure of the linear span.

For a $H$-admissible representation $\pi$, $Spec_{H}(\pi)$, denotes the set of Harish-Chandra parameters of the irreducible $H$-subrepresentations of $\pi$.

3. Duality Theorem, explicit isomorphism

3.1. Statement and proof of the duality result

The unexplained notation is as in section 2, our hypotheses are $(G,H=(G^{\sigma})_{0})$ is a symmetric pair and $(\pi_{\lambda},V_{\lambda}^{G})$ is a $H$-admissible, square integrable irreducible representation for $G$. $K=G^{\theta}$ is a maximal compact subgroup of $G$, $H_{0}:=(G^{\sigma\theta})_{0}$ and $K$ is so that $L=H\cap K=H_{0}\cap K$ is a maximal compact subgroup of both $H$ and $H_{0}$. By definition, $H_{0}$ is the associated subgroup to $H$.

In this section, under our hypothesis, for a $H$-irreducible factor $V_{\mu}^{H}$ for $res_{H}(\pi_{\lambda})$, we show an explicit isomorphism from the space

xxxx $Hom_{H}(V_{\mu}^{H},V_{\lambda}^{G})$ onto $Hom_{L}(V_{\mu+\rho_{n}^{\mu}}^{L},\,\pi_{\lambda}(\mathcal{U}(\mathfrak{h}_{0}))V_{\lambda}^{G}[V_{\lambda+\rho_{n}}^{K}])$.

We also analyze the restriction map $r_{0}:H^{2}(G,\tau)\rightarrow L^{2}(H_{0}\times_{\tau}W)$.

To follow, we present the necessary definitions and facts involved in the main statement.

3.1.1.

We consider the linear subspace $\mathcal{L}_{\lambda}$ spanned by the lowest $L$-type subspace of each irreducible $H$-factor of $res_{H}((L,H^{2}(G,\tau)))$. That is,

$\mathcal{L}_{\lambda}$ is the linear span of $\cup_{\mu\in Spec_{H}(\pi_{\lambda})}H^{2}(G,\tau)[V_{\mu}^{H}][V_{\mu+\rho_{n}^{\mu}}^{L}]$.

We recall that our hypothesis yields that the subspace of $L$-finite vectors in $V_{\lambda}^{G}$ is equal to the subspace of $K$-finite vectors [16, Prop. 1.6 ]. Whence, we have $\mathcal{L}_{\lambda}$ is a subspace of the space of $K$-finite vectors in $H^{2}(G,\tau)$.

3.1.2.

We also need the subspace

[TABLE]

We write $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ for the closure of $\mathcal{U}(\mathfrak{h}_{0})W$. Hence, $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is the closure of the left translates by the algebra $\mathcal{U}(\mathfrak{h}_{0})$ of the subspace of $K$-finite vectors

[TABLE]

Thus, $\mathcal{U}(\mathfrak{h}_{0})W$ consists of analytic vectors for $\pi_{\lambda}$. Therefore, $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is invariant under left translations by $H_{0}$. In Proposition 4.9 we present the decomposition of $\mathcal{U}(\mathfrak{h}_{0})W$ as a sum of irreducible representations for $H_{0}$.

We point out

The $L$-module $\mathcal{L}_{\lambda}$ is equivalent to the underlying $L$ -module in $\mathcal{U}(\mathfrak{h}_{0})W$.

This has been proven in [30, (4.5)]. For completeness we present a proof in Proposition 4.9.

Under the extra assumption $res_{L}(\tau)$ is irreducible, we have $\mathcal{U}(\mathfrak{h}_{0})W$ is a irreducible $(\mathfrak{h}_{0},L)$-module, and, in this case, the lowest $L$-type of $\mathcal{U}(\mathfrak{h}_{0})W$ is $(res_{L}(\tau),W)$. That is, $\mathcal{U}(\mathfrak{h}_{0})W$ is equivalent to the underlying Harish-Chandra module for $H^{2}(H_{0},res_{L}(\tau))$. The Harish-Chandra parameter $\eta_{0}\in i\mathfrak{u}^{\star}$ for $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is computed in 3.4.1.

For scalar holomorphic Discrete Series, the classification of the symmetric pairs $(G,H)$ such that the equality $\mathcal{U}(\mathfrak{h}_{0})W=\mathcal{L}_{\lambda}$ holds, is:

$(\mathfrak{su}(m,n),\mathfrak{su}(m,l)+\mathfrak{su}(n-l)+\mathfrak{u}(1)),$ $(\mathfrak{so}(2m,2),\mathfrak{u}(m,1)),$

$(\mathfrak{so}^{\star}(2n),\mathfrak{u}(1,n-1)),$ $(\mathfrak{so}^{\star}(2n),\mathfrak{so}(2)+\mathfrak{so}^{\star}(2n-2)),$ $(\mathfrak{e}_{6(-14)},\mathfrak{so}(2,8)+\mathfrak{so}(2)).$ [30, (4.6)]. Thus, there exists scalar holomorphic Discrete Series with $\mathcal{U}(\mathfrak{h}_{0})W\not=\mathcal{L}_{\lambda}$.

3.1.3.

To follow, we set some more notation. We fix a representative for $(\tau,W)$. We write

$(res_{L}(\tau),W)=\sum_{1\leq j\leq r}q_{j}(\sigma_{j},Z_{j})$, $q_{j}=\dim Hom_{L}(Z_{j},res_{L}(W))$

and the decomposition in isotypic components

[TABLE]

From now on, we fix respective representatives for $(\sigma_{j},Z_{j})$ with $Z_{j}\subset W[(\sigma_{j},Z_{j})]$.

Henceforth, we denote by

[TABLE]

We think the later module as a linear subspace of

[TABLE]

Hence, $\mathbf{H}^{2}(H_{0},\tau)\subset L^{2}(H_{0}\times_{\tau}W)_{H_{0}-disc}$. We note that when $res_{L}(\tau)$ is irreducible, then $\mathbf{H}^{2}(H_{0},\tau)=H^{2}(H_{0},res_{L}(\tau))$.

3.1.4.

Owing to both spaces $H^{2}(H,\sigma),H^{2}(G,\tau)$ are reproducing kernel spaces, we represent each $T\in Hom_{H}(H^{2}(H,\sigma),H^{2}(G,\tau))$ by a kernel $K_{T}:H\times G\rightarrow Hom_{\mathbb{C}}(Z,W)$ so that $K_{T}(\cdot,x)^{\star}w\in H^{2}(H,\sigma)$ and $(T(g)(x),w)_{W}=\int_{H}(g(h),K_{T}(h,x)^{\star}w)_{Z}dh$. Here, $x\in G,w\in W,g\in H^{2}(H,\sigma)$. In [25], it is shown: $K_{T}$ is a smooth function, $K_{T}(h,\cdot)z=K_{T^{\star}}(\cdot,h)^{\star}z\in H^{2}(G,\tau)$ and

[TABLE]

is a $L$-finite vector in $H^{2}(G,\tau)$.

3.1.5.

Finally, we recall the restriction map

[TABLE]

is $(L^{2},L^{2})$-continuous [24].

The main result of this section is,

Theorem 3.1**.**

*We assume $(G,H)$ is a symmetric pair and $res_{H}(\pi_{\lambda})$ is admissible. We fix a irreducible factor $V_{\mu}^{H}$ for $res_{H}(\pi_{\lambda})$. Then, the following statements hold.

i) The map $r_{0}:H^{2}(G,\tau)\rightarrow L^{2}(H_{0}\times_{\tau}W)$ restricted to $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ yields a isomorphism between $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ onto $\mathbf{H}^{2}(H_{0},\tau)$.

ii) For each fixed intertwining $L$-equivalence*

xxxxxx* $D:\mathcal{L}_{\lambda}[V_{\mu+\rho_{n}^{\mu}}^{L}]=H^{2}(G,\tau)[V_{\mu}^{H}][V_{\mu+\rho_{n}^{\mu}}^{L}]\rightarrow(\mathcal{U}(\mathfrak{h}_{0})W)[V_{\mu+\rho_{n}^{\mu}}^{L}]$,

the map*

xxx* $r_{0}^{D}:Hom_{H}(H^{2}(H,\sigma),H^{2}(G,\tau))\rightarrow Hom_{L}(V_{\mu+\rho_{n}^{\mu}}^{L},\mathbf{H}^{2}(H_{0},\tau))$

defined by*

[TABLE]

is a linear isomorphism.

Remark 3.2*.*

When the natural inclusion $H/L\rightarrow G/K$ is a holomorphic map, T. Kobayashi, M. Pevzner and Y. Oshima in [17],[20] have shown a similar dual multiplicity result after replacing the underlying Harish-Chandra module in $\mathbf{H}^{2}(H_{0},\tau)$ by its representation as a Verma module. Also, in the holomorphic setting, Jakobsen-Vergne in [14] has shown the isomorphism $H^{2}(G,\tau)\equiv\sum_{r\geq 0}H^{2}(H,\tau_{|_{L}}\otimes S^{(r)}((\mathfrak{h}_{0}\cap\mathfrak{p}^{+}))^{\star})$. On the papers, [22] [21], we find applications of the result of Kobayashi for their work on decomposing holomorphic Discrete Series. H. Sekiguchi [28] has obtained a similar result of branching laws for singular holomorphic representations.

Remark 3.3*.*

The proof of Theorem 3.1 requires to show the map $r_{0}^{D}$ is well defined as well as several structure Lemma’s. Once we verify the map is well defined, we will show injectivity, Corollary 3.9, Proposition 4.2 and linear algebra will give the surjectivity. In Proposition 4.9, we show $i)$, and, in the same Proposition we give a proof of the existence of the map $D$ as well as its bijectivity, actually this result has been shown in [30]. However, we sketch a proof in this note. The surjectivity also depends heavily on a result in [30], for completeness we give a proof. We may say that our proof of Theorem 1 is rather long and intricate, involving both linear algebra for finding the multiplicities, and analysis of the kernels of the intertwining operators in question to set up the equivalence of the $H$-morphisms and the $L$-morphisms. The structure of the branching and corresponding symmetry breaking is however very convenient to apply in concrete situations, and we give several illustrations.

We explicit the inverse map to the bijection $r_{0}^{D}$ in subsection 4.2.

Remark 3.4*.*

When $\mathcal{L}_{\lambda}=\mathcal{U}(\mathfrak{h}_{0})W$ we may take $D$ equal to the identity map.

Remark 3.5*.*

A mirror statement to Theorem 3.1 for symmetry breaking operators is as follows: $Hom_{H}(H^{2}(G,\tau),H^{2}(H,\sigma))$ is isomorphic to $Hom_{L}(Z,\mathbf{H}^{2}(H_{0},\tau))$ via the map $S\mapsto(z\mapsto(H_{0}\ni x\mapsto r_{0}^{D}(S^{\star})(z)(x)=r_{0}(D(K_{S}(\cdot,x)^{\star})(z))\in W))$.

3.1.6.

We verify $r_{0}(D(K_{T}(e,\cdot)z))(\cdot)$ belongs to $L^{2}(H_{0}\times_{\tau}W)_{H_{0}-disc}$.

Indeed, owing to our hypothesis, a result [5] (see [4, Proposition 2.4]) implies $\pi_{\lambda}$ is $L$-admissible. Hence, [15, Theorem 1.2] implies $\pi_{\lambda}$ is $H_{0}$-admissible. Also, [16, Proposition 1.6] shows the subspace of $L$-finite vectors in $H^{2}(G,\tau)$ is equal to the subspace of $K$-finite vectors and $res_{\mathcal{U}(\mathfrak{h}_{0})}(H^{2}(G,\tau)_{K-fin})$ is a admissible, completely algebraically decomposable representation. Thus, the subspace $H^{2}(G,\tau)[W]\equiv W$ is contained in a finite sum of irreducible $\mathcal{U}(\mathfrak{h}_{0})$-factors. Hence, $\mathcal{U}(\mathfrak{h}_{0})W$ is a finite sum of irreducible $\mathcal{U}(\mathfrak{h}_{0})$ factors. In [9], we find a proof that each irreducible summand for $res_{H_{0}}(\pi_{\lambda})$ is a square integrable representations for $H_{0}$, hence, the equivariance and continuity of $r_{0}$ yields $r_{0}(\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W))$ is contained in $L^{2}(H_{0}\times_{\tau}W)_{H_{0}-disc}$. 3.1.4 shows $K_{T}(e,\cdot)\in V_{\lambda}[V_{\mu}^{H}][V_{\mu+\rho_{n}^{H}}^{L}]$, hence, $D(K_{T}(e,\cdot)z)(\cdot)\in\mathcal{U}(\mathfrak{h}_{0})W$, and the claim follows.

3.1.7.

The map $Z\ni z\mapsto r_{0}(D(K_{T}(e,\cdot)z))(\cdot)\in L^{2}(H_{0}\times_{\tau}W)$ is a $L$-map.

For this, we recall the equalities

[TABLE]

[TABLE]

Therefore, $K_{T}(e,hl_{1})\sigma(l_{2})z=\tau(l_{1}^{-1})K_{T}(l_{2},h)z=\tau(l_{1}^{-1})K_{T}(e,l_{2}^{-1}h)z$ for $l_{1},l_{2}\in L,h\in H_{0}$ and we have shown the claim.

We have enough information to verify the injectivity we have claimed in $i)$ as well as the injectivity of the map $r_{0}^{D}$. For these, we show a fact valid for a arbitrary reductive pair $(G,H)$ and arbitrary Discrete Series representation.

3.2. Kernel of the restriction map

In this paragraph we show a fact valid for any reductive pair $(G,H)$ and arbitrary representation $\pi_{\lambda}$. The objects involved here are the restriction map $r$ from $H^{2}(G,\tau)$ into $L^{2}(H\times_{\tau}W)$ and the subspace

[TABLE]

We write $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$ for the closure of $\mathcal{U}(\mathfrak{h})W$. The subspace $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$ is the closure of the left translates by the algebra $\mathcal{U}(\mathfrak{h})$ of the subspace of $K$-finite vectors

xxxxxxxxxxxxx $\{K_{\lambda}(\cdot,e)^{\star}w:w\in W\}=H^{2}(G,\tau)[W]$.

Thus, $\mathcal{U}(\mathfrak{h})W$ consists of analytic vectors for $\pi_{\lambda}$. Hence, $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$ is invariant by left translations by $H$. Therefore the subspace

xx $L_{H}(H^{2}(G,\tau)[W])=\{K_{\lambda}(\cdot,h)^{\star}w=L_{h}(K_{\lambda}(\cdot,e)^{\star}w):w\in W,h\in H\}$

is contained in $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$. Actually,

xxxxxxx $\mathrm{Cl}(L_{H}(H^{2}(G,\tau)[W]))=\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$.

The other inclusion follows from that $\mathrm{Cl}(L_{H}(H^{2}(G,\tau)[W]))$ is invariant by left translation by $H$ and $\{K_{\lambda}(\cdot,e)^{\star}w:w\in W\}$ is contained in the subspace of smooth vectors in $\mathrm{Cl}(L_{H}(H^{2}(G,\tau)[W]))$.

The result pointed out in the title of this paragraph is:

Lemma 3.6**.**

Let $(G,H)$ be a arbitrary reductive pair and an arbitrary representation $(\pi_{\lambda},H^{2}(G,\tau))$. Then, $Ker(r)$ is equal to the orthogonal subspace to $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$.

Proof.

Since, [24], $r:H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\tau}W)$ is a continuous map, we have $Ker(r)$ is a closed subspace of $H^{2}(G,\tau)$. Next, for $f\in H^{2}(G,\tau)$, it holds the identity

xxx $(f(x),w)_{W}=\int_{G}(f(y),K_{\lambda}(y,x)^{\star}w)_{W}dy,\forall x\in G,\forall w\in W$.

Thus, $r(f)=0$ if and only if $f$ is orthogonal to the subspace spanned by $\{K_{\lambda}(\cdot,h)^{\star}w:w\in W,h\in H\}$.

Hence, $\mathrm{Cl}(Ker(r))=(\mathrm{Cl}(L_{H}(H^{2}(G,\tau)[W])))^{\perp}$. Applying the considerations after the definition of $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$ we obtain $Ker(r)^{\perp}=\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$. Thus $Ker(r)=(Ker(r)^{\perp})^{\perp}=\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$. ∎

Corollary 3.7**.**

Any irreducible $H$-discrete factor $M$ for $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$ contains a $L$-type in $res_{L}(\tau)$. That is, $M[res_{L}(\tau)]\not=\{0\}.$

The corollary follows from that $r$ restricted to $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$ is injective and that Frobenius reciprocity for $L^{2}(H\times_{\tau}W)$ holds.

3.3. The map $r_{0}^{D}$ is injective

As a consequence of the general fact shown in the previous subsection, we obtain the injectivity in $i)$ and the map $r_{0}^{D}$ is injective.

Corollary 3.8**.**

Let $(G,H)$ be a symmetric pair and $H_{0}=G^{\sigma\theta}$. Then, the restriction map $r_{0}:H^{2}(G,\tau)\rightarrow L^{2}(H_{0}\times_{\tau}W)$ restricted to the subspace $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is injective.

Corollary 3.9**.**

Let $(G,H)$ be a symmetric pair, $H_{0}=G^{\sigma\theta}$ and we assume $res_{H}(\pi_{\lambda})$ is $H$-admissible. Then, the map $r_{0}^{D}$ is injective.

In fact, for $T\in Hom_{H}(H^{2}(H,\sigma),H^{2}(G,\tau))$, if $r_{0}(D(K_{T}(e,\cdot)z))=0\,\forall z\in Z$, then, since $D(K_{T}(e,x)z)\in\mathcal{U}(\mathfrak{h}_{0})W$, the previous corollary implies $D(K_{T}(e,x)z)=0\,\forall z,x\in G$. Since, $K_{T}(e,\cdot)z\in V_{\lambda}^{G}[V_{\mu}^{H}][V_{\mu+\rho_{n}^{H}}^{L}]$, and $D$ is injective we obtain $K_{T}(e,x)z=0\,\forall z,\forall x$. Lastly, we recall equality $K_{T}(h,x)=K_{T}(e,h^{-1}x)$. Whence we have verified the corollary.

Before we show the surjectivity for the map $r_{0}^{D}$ we would like to comment on other works on the topic of this note.

3.4. Previous work on duality formula and Harish-Chandra parameters

The setting for this subsection is: $(G,H)$ is a symmetric pair and $(\pi_{\lambda},V_{\lambda}^{G})$ is a irreducible square integrable representation of $G$ and $H$-admissible. As before, we fix $K,L=H\cap K,T,U=H\cap T$. The following Theorem has been shown by [9], a different proof is in [18].

Theorem 3.10** (Gross-Wallach, T. Kobayashi-Y. Oshima).**

We assume $(G,H)$ is symmetric pair, $\pi_{\lambda}^{G}$-is $H$-admissible, then

a) $res_{H}(\pi_{\lambda}^{G})$ is the Hilbert sum of inequivalent Square integrable representations for $H$, $\pi_{\mu_{j}}^{H},j=1,2,\dots$, with respective finite multiplicity $0<m_{j}<\infty$.

b) The Harish-Chandra parameters of the totality of discrete factors for $res_{H}(\pi_{\lambda}^{G})$ belong to a "unique" Weyl Chamber in $i\mathfrak{u}^{\star}$.

That is, xx $V_{\lambda}^{G}=\oplus_{1\leq j<\infty}V_{\lambda}^{G}[V_{\mu_{j}}^{H}]\equiv\oplus_{j}\,\,Hom_{H}(V_{\mu_{j}}^{H},V_{\lambda}^{G})\otimes V_{\mu_{j}}^{H}$,

xxxx* $\dim Hom_{H}(V_{\mu_{j}}^{H},V_{\lambda}^{G})=m_{j}$, xx $\pi_{\mu_{j}}^{H}\neq\pi_{\mu_{i}}^{H}$ iff $i\not=j$,

and there exists a system of positive roots $\Psi_{H,\lambda}\subset\Phi(\mathfrak{h},\mathfrak{u}),$ such that for all $j$, $(\alpha,\mu_{j})>0$   for all $\alpha\in\Psi_{H,\lambda}$.*

In [30][18] (see Tables 1,2,3) we find the list of pairs $(\mathfrak{g},\mathfrak{h})$, as well as systems of positive roots $\Psi_{G}\subset\Phi(\mathfrak{g},\mathfrak{t}),\Psi_{H,\lambda}\subset\Phi(\mathfrak{h},\mathfrak{u})$ such that,

• $\lambda$ dominant with respect to $\Psi_{G}$ implies $res_{H}(\pi_{\lambda}^{G})$ is admissible.

• For all $\mu_{j}$ in $a)$ we have $(\mu_{j},\Psi_{H,\lambda})>0$.

• When $U=T$, we have $\Psi_{H,\lambda}=\Psi_{\lambda}\cap\Phi(\mathfrak{h},\mathfrak{t}).$

Since $(G,H_{0})$ is a symmetric pair, Theorem 3.10 as well as its comments apply to $(G,H_{0})$ and $\pi_{\lambda}$. Here, when $U=T$, $\Psi_{H_{0},\lambda}=\Psi_{\lambda}\cap\Phi(\mathfrak{h}_{0},\mathfrak{t})$.

From the tables in [30] it follows that any of the system $\Psi_{\lambda}$, $\Psi_{H,\lambda},\Psi_{H_{0},\lambda}$ has, at most, two noncompact simple roots, and the sum of the respective multiplicity of each noncompact simple root in the highest root is less or equal than two.

3.4.1. Computing Harish-Chandra parameters from Theorem 3.1

As usual, $\rho_{n}=\frac{1}{2}\sum_{\beta\in\Psi_{\lambda}\cap\Phi_{n}}\beta$, $\rho_{n}^{H}=\frac{1}{2}\sum_{\beta\in\Psi_{H,\lambda}\cap\Phi_{n}}\beta$, $\rho_{K}=\frac{1}{2}\sum_{\alpha\in\Psi_{\lambda}\cap\Phi_{c}}\alpha$,

$\rho_{L}=\frac{1}{2}\sum_{\alpha\in\Psi_{H,\lambda}\cap\Phi_{c}}\alpha$. We write $res_{L}(\tau)=res_{L}(V_{\lambda+\rho_{n}}^{K})=\oplus_{1\leq j\leq r}\,q_{j}\,\pi_{\nu_{j}}^{L}=\sum_{j}q_{j}\sigma_{j}$, with $\nu_{j}$ dominant with respect to $\Psi_{H,\lambda}\cap\Phi_{c}$. we recall $\nu_{j}$ is the infinitesimal character (Harish-Chandra parameter) of $\pi_{\nu_{j}}^{L}$. Then, the Harish-Chandra parameter for $H^{2}(H_{0},\pi_{\nu_{j}}^{L})$ is $\eta_{j}=\nu_{j}-\rho_{n}^{H_{0}}$.

According to [27, Lemma 2.22](see Remark 4.8), the infinitesimal character of a $L$-type of $H^{2}(H_{0},\pi_{\nu_{j}}^{L})$ is equal to $\nu_{j}+B=\eta_{j}+\rho_{n}^{H_{0}}+B$ where $B$ is a sum of roots in $\Psi_{H_{0},\lambda}\cap\Phi_{n}$.

The isomorphism $r_{0}^{D}$ in Theorem 3.1, let us conclude:

For each subrepresentation $V_{\mu_{s}}^{H}$ of $res_{H}(\pi_{\lambda})$, we have $\mu_{s}+\rho_{n}^{H}$ is a $L$-type of

xxxxxxx $\mathbf{H}^{2}(H_{0},\tau)\equiv\oplus_{j}\,q_{j}\,H^{2}(H_{0},\pi_{\nu_{j}}^{L})\equiv\oplus_{j}\,\underbrace{V_{\eta_{j}}^{H_{0}}\oplus\cdots\oplus V_{\eta_{j}}^{H_{0}}}_{q_{j}}$,

and the multiplicity of $V_{\mu_{s}}^{H}$ is equal to the multiplicity of $V_{\mu_{s}+\rho_{n}^{H}}^{L}$ in $\mathbf{H}^{2}(H_{0},\tau)$.

3.4.2. Gross-Wallach multiplicity formula

To follow we describe the duality Theorem due to [9]. $(G,H)$ is a symmetric pair. For this paragraph, in order to avoid subindexes we write $\mathfrak{g}=Lie(G),\mathfrak{h}=Lie(H)$ etc. We recall $\mathfrak{h}_{0}=\mathfrak{g}^{\sigma\theta}$. We have the decompositions $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}=\mathfrak{h}+\mathfrak{q}=\mathfrak{h}_{0}+\mathfrak{p}\cap\mathfrak{h}+\mathfrak{q}\cap\mathfrak{k}$. The dual real Lie algebra to $\mathfrak{g}$ is $\mathfrak{g}^{d}=\mathfrak{h}_{0}+i(\mathfrak{p}\cap\mathfrak{h}+\mathfrak{q}\cap\mathfrak{k})$, the algebra $\mathfrak{g}^{d}$ is a real form for $\mathfrak{g}_{\mathbb{C}}$. A maximal compactly embedded subalgebra for $\mathfrak{g}^{d}$ is $\widetilde{\mathfrak{k}}=\mathfrak{h}\cap\mathfrak{k}+i(\mathfrak{h}\cap\mathfrak{p})$. Let $\pi_{\lambda}$ be a $H$-admissible Discrete Series for $G$. One of the main results of [9] attach to $\pi_{\lambda}$ a finite sum of the underlying Harish-Chandra module of finitely many fundamental representations for $G^{d}$, $(\Gamma_{H\cap L}^{\widetilde{K}})^{p_{0}+q_{0}}(N(\Lambda))$, so that for each subrepresentation $V_{\mu}^{H}$ of $V_{\lambda}^{G}$ we compute the multiplicity $m^{G,H}(\lambda,\mu)$ of $V_{\mu}^{H}$ by means of Blattner’s formula [11] applied to $(\Gamma_{H\cap L_{1}}^{\widetilde{K}})^{p_{0}+q_{0}}(N(\Lambda))$ . In more detail, since $Lie(H)_{\mathbb{C}}=Lie(\widetilde{K})_{\mathbb{C}}$, and the center of $H$ is equal to the center of $\widetilde{K}$, for the infinitesimal character $\mu$ and the central character $\chi$ of $V_{\mu}^{H}$, we may associate a finite dimensional irreducible representation $F_{\mu,\chi}$ for $\widetilde{K}$. Then, they show

[TABLE]

[TABLE]

where $\tau=F^{\Lambda}=\sum_{i}M^{\Lambda_{i}}$ as a sum of irreducible $H\cap L_{1}$-module and $p$ is the partition function associated to $\Phi(\mathfrak{u}_{1}/\mathfrak{u}_{1}\cap\mathfrak{h}_{\mathbb{C}},\mathfrak{u})$, here, $\mathfrak{u}_{1}$ is the nilpotent radical of certain parabolic subalgebra $\mathfrak{q}=\mathfrak{l}_{1}+\mathfrak{u}_{1}$ used to define the $A_{\mathfrak{q}}(\lambda)$-presentation for $\pi_{\lambda}$. Explicit example IV presents the result of [9] for the pair $(SO(2m,2n),SO(2m,2n-1))$.

3.4.3. Duflo-Vargas multiplicity formula, [5]

We keep notation and hypothesis as in the previous paragraph. Then,

[TABLE]

Here, $q_{\mathfrak{u}}:\mathfrak{t}^{\star}\rightarrow\mathfrak{u}^{\star}$ is the restriction map. $p_{S_{w}^{H}}$ is the partition function associated to the multiset

[TABLE]

We recall for a strict multiset of elements in vector space $V$ the partition function attached to $S$, roughly speaking, is the function that counts the number of ways of expressing each vector as a nonnegative integral linear combinations of elements of $S$. For a precise definition see [5] or the proof of Lemma 4.4.

3.4.4. Harris-He-Olafsson multiplicity formula, [10]

Notation and hypothesis as in the previous paragraphs. Let

[TABLE]

the normal derivative map defined in [24]. Let $\Theta_{\pi_{\mu}^{H}}$ denote the Harish-Chandra character of $\pi_{\mu}^{H}$. For $f$ a tempered function in $H^{2}(G,\tau)$, they define $\phi_{\pi_{\lambda},\pi_{\mu}^{H},m}(f)=\Theta_{\pi_{\mu}^{H}}\star r_{m}(f)$ . They show:

[TABLE]

3.5. Completion of the Proof of Theorem 3.1, the map $r_{0}^{D}$ is surjective

Item $i)$ in Theorem 3.1 is shown in Proposition 4.9 $c)$. The existence of the map $D$ is shown in Proposition 4.9 $e)$.

To show the surjectivity of $r_{0}^{D}$ we appeal to Theorem 4.2, [30, Theorem 1], where we show the initial space and the target space are equidimensional, linear algebra concludes de proof of Theorem 3.1. Thus, we conclude the proof of Theorem 3.1 as soon as we complete the proof of Theorem 4.2 and Proposition 4.9.

4. Duality Theorem, proof of dimension equality

The purpose of this subsection is to sketch a proof of the equality of dimensions in the duality formula presented in Theorem 3.1 as well as some consequences. Part of the notation has already been introduced in the previous section. Sometimes notation will be explained after it has been used. Unexplained notation is as in [5], [25], [30].

4.1. Dimension equality Theorem, statement

The setting is as follows, $(G,H)$ is a symmetric pair,$(\pi_{\lambda},V_{\lambda}^{G})=(L,H^{2}(G,\tau))$ a $H$-admissible irreducible square integrable representation. Then, the Harish-Chandra parameter $\lambda$ gives rise to systems of positive roots $\Psi_{\lambda}$ in $\Phi(\mathfrak{g},\mathfrak{t})$ and by mean of $\Psi_{\lambda}$, in [5] is defined a nontrivial normal connected subgroup $K_{1}(\Psi_{\lambda})=:K_{1}$ of $K$, and, it is shown that the $H$-admissibility yields $K_{1}\subset H$333This also follows from the tables in [18]. Thus, $\mathfrak{k}=\mathfrak{k}_{1}\oplus\mathfrak{k}_{2}$, $\mathfrak{l}=\mathfrak{k}_{1}\oplus\mathfrak{l}\cap\mathfrak{k}_{2}$ (as ideals), and $\mathfrak{t}=\mathfrak{t}\cap\mathfrak{k}_{1}+\mathfrak{t}\cap\mathfrak{k}_{2}$, $\mathfrak{u}:=\mathfrak{t}\cap\mathfrak{l}=\mathfrak{u}\cap\mathfrak{k}_{1}+\mathfrak{u}\cap\mathfrak{k}_{2}$ is a Cartan subalgebra of $\mathfrak{l}$. Let $q_{\mathfrak{u}}$ denote restriction map from $\mathfrak{t}^{\star}$ onto $\mathfrak{u}^{\star}$. Let $K_{2}$ denote the analytic subgroup corresponding to $\mathfrak{k}_{2}$. We recall $H_{0}:=(G^{\sigma\theta})_{0}$, $L=K\cap H=K\cap H_{0}$. We have $K=K_{1}K_{2}$, $L=K_{1}(K_{2}\cap L)$. We set $\Delta:=\Psi_{\lambda}\cap\Phi(\mathfrak{k},\mathfrak{t})$. Applying Theorem 3.10 to both $H$ and $H_{0}$ we obtain respective systems of positive roots $\Psi_{H,\lambda}$ in $\Phi(\mathfrak{h},\mathfrak{u})$, $\Psi_{H_{0},\lambda}$ in $\Phi(\mathfrak{h}_{0},\mathfrak{u})$. For a list of six-tuples $(G,H,\Psi_{\lambda},$ $\Psi_{H,\lambda},\Psi_{H_{0},\lambda},K_{1})$ we refer to [30, Table 1, Table 2, Table 3]. Always, $\Psi_{H,\lambda}\cap\Phi_{c}(\mathfrak{l},\mathfrak{u})=\Psi_{H_{0},\lambda}\cap\Phi_{c}(\mathfrak{l},\mathfrak{u})$. As usual, either $\Phi_{n}(\mathfrak{g},\mathfrak{t})$ or $\Phi_{n}$ denotes the subset of noncompact roots in $\Phi(\mathfrak{g},\mathfrak{t})$, $\rho_{n}^{\lambda}$ (resp. $\rho_{n}^{H},\rho_{n}^{H_{0}}$) denotes one half of the sum of the elements in $\Psi_{\lambda}\cap\Phi_{n}(\mathfrak{g},\mathfrak{t})$ (resp. $\Phi_{n}\cap\Psi_{H,\lambda},\Phi_{n}\cap\Psi_{H_{0},\lambda}$). When $\mathfrak{u}=\mathfrak{t},\rho_{n}^{\lambda}=\rho_{n}^{H}+\rho_{n}^{H_{0}}$. From now on, the infinitesimal character of an irreducible representation of $K$ (resp $L$) is dominant with respect to $\Delta$ (resp. $\Psi_{H,\lambda}\cap\Phi(\mathfrak{l},\mathfrak{u}))$. The lowest $K$-type $(\tau,W)$ of $\pi_{\lambda}$ decomposes $\pi_{\lambda+\rho_{n}^{\lambda}}^{K}=\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\Lambda_{2}}^{K_{2}}$, with $\pi_{\Lambda_{s}}^{K_{s}}$ an irreducible representation for $K_{s},s=1,2$. We express $\gamma=(\gamma_{1},\gamma_{2})\in\mathfrak{t}^{\star}=\mathfrak{t}_{1}^{\star}+\mathfrak{t}_{2}^{\star}$. Hence, [11][3], $\Lambda_{1}=\lambda_{1}+(\rho_{n}^{\lambda})_{1},\Lambda_{2}=\lambda_{2}+(\rho_{n}^{\lambda})_{2}$. Sometimes $(\rho_{n}^{\lambda})_{2}\not=0$. This happens only for $\mathfrak{su}(m,n)$ and some particular systems $\Psi_{\lambda}$ (see proof of Lemma 5). Harish-Chandra parameters for the irreducible factors of either $res_{H}(\pi_{\lambda})$ (resp. $res_{H_{0}}(\pi_{\lambda})$)) will always be dominant with respect to $\Psi_{H,\lambda}\cap\Phi(\mathfrak{l},\mathfrak{u})$ (resp. $\Psi_{H_{0},\lambda}\cap\Phi(\mathfrak{l},\mathfrak{u}))$.

For short, we write $\pi_{\Lambda_{2}}:=\pi_{\Lambda_{2}}^{K_{2}}$. We write

[TABLE]

as a sum of irreducible representations of $L\cap K_{2}$.

The set of $\nu_{2}$ so that $m^{K_{2},L\cap K_{2}}(\Lambda_{2},\nu_{2})\not=0$ is denoted by $Spec_{L\cap K_{2}}(\pi_{\Lambda_{2}}^{K_{2}})$. Thus,

[TABLE]

as a sum of irreducible representations of $L$. Besides, for a Harish-Chandra parameter $\eta=(\eta_{1},\eta_{2})$ for $H_{0}$, we write

[TABLE]

The restriction of $\pi_{\lambda}$ to $H$ is expressed by (see 3.10)

[TABLE]

In the above formulaes, $m^{\cdot,\cdot}(\cdot,\cdot)$ are non negative integers and represent multiplicities; for $\nu_{2}\in Spec_{L\cap K_{2}}(\pi_{\Lambda_{2}}^{K_{2}})$, $\nu_{2}$ is dominant with respect to $\Psi_{H,\lambda}\cap\Phi(\mathfrak{k}_{2},\mathfrak{u}\cap\mathfrak{k}_{2})$, and $(\Lambda_{1},\nu_{2})$ is $\Psi_{H_{0},\lambda}$-dominant (see [30]); in the third formulae, $(\eta_{1},\eta_{2})$ is dominant with respect to $\Psi_{H_{0},\lambda}$ and $(\theta_{1},\theta_{2})$ is dominant with respect to $\Psi_{H_{0},\lambda}\cap\Phi_{c}(\mathfrak{h}_{0},\mathfrak{u})$; in the fourth formula, $\mu$ is dominant with respect to $\Psi_{H,\lambda}$. Sometimes, for $\mu\in Spec_{H}(\pi_{\lambda}^{G})$, we replace $\rho_{n}^{\mu}$ by $\rho_{n}^{H}$.

We make a change of notation:

xxxxxxxxxxx$\sigma_{j}=\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}}$ and $q_{j}=m^{K_{2},L\cap K_{2}}(\Lambda_{2},\nu_{2})$.

Then, in order to show either the existence of the map $D$ or the surjectivity of the map $r_{0}^{D}$, we need to show:

Theorem 4.1**.**

[TABLE]

A complete proof of the result is in [30]. However, for sake of completeness and clarity we would like to sketch a proof. We also present some consequences of the Theorem.

Next, we compute the infinitesimal character, $ic(...)$, equivalently the Harish-Chandra parameter, for $H^{2}(H_{0},\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}})$ and restate the previous Theorem.

[TABLE]

To follow, we state Theorem 4.1 regardless of the realization of the involved Discrete Series.

Theorem 4.2**.**

Duality, dimension formula. The hypothesis is $(G,H)$ is a symmetric pair and $\pi_{\lambda}$ is a $H$-admissible representation. Then,

[TABLE]

Remark: In the proof of Lemma 5 we obtain that $(\rho_{n}^{\lambda})_{2}\not=0$ forces $\mathfrak{g}=\mathfrak{su}(m,n)$, $\mathfrak{u}=\mathfrak{t}$ and $(\rho_{n}^{\lambda})_{2}$ is orthogonal to $(\mathfrak{z}_{\mathfrak{k}})^{\perp}$. That is, $(\rho_{n}^{\lambda})_{2}\in\mathfrak{z}_{\mathfrak{k}}^{\star}$, i.e., for every symmetric pair $(G,H)$ we have $(\rho_{n}^{\lambda})_{2}$ determines a central character of $\mathfrak{k}$. It is also verified that since $\Lambda_{2}=(\rho_{n}^{\lambda})_{2}+\lambda_{2}$, we have $Spec_{L\cap K_{2}}(\pi_{\Lambda_{2}}^{K_{2}})=q_{\mathfrak{u}}(\rho_{n}^{\lambda})_{2}+Spec_{L\cap K_{2}}(\pi_{\lambda_{2}}^{K_{2}})$ and for $Spec_{L\cap K_{2}}(\pi_{\Lambda_{2}}^{K_{2}})\ni\nu_{2}=q_{\mathfrak{u}}(\rho_{n}^{\lambda})_{2}+\nu_{2}^{\prime},\nu_{2}^{\prime}\in Spec_{L\cap K_{2}}(\pi_{\lambda_{2}}^{K_{2}})$ it holds the equality $m^{K_{2},L\cap K_{2}}(\Lambda_{2},\nu_{2})=m^{K_{2},L\cap K_{2}}(\lambda_{2},\nu_{2}^{\prime})$. Finally, Lemma 4.5 yields the formula in Theorem 4.2 is equivalent to the formula

[TABLE]

This formula is the one we show in Lemma 4.4.

The following diagram helps to understand the equalities in Theorem 4.2 and in the next three Lemmas.

\textstyle{Spec_{H}(V_{(\lambda_{1},\lambda_{2})}^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu\mapsto\mu}$$\scriptstyle{\nu\mapsto\nu+\rho_{n}^{H}}$$\textstyle{\cup_{\nu_{2}^{\prime}\in Spec_{L\cap K_{2}}(\pi_{\lambda_{2}}^{K_{2}})}Spec_{L}(V_{(\lambda_{1},\nu_{2}^{\prime})}^{H_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu\mapsto\nu+\rho_{n}^{H}}$$\textstyle{Spec_{L}(\mathbf{H}^{2}(H_{0},\tau))=\cup_{\nu_{2}^{\prime}\in Spec_{L\cap K_{2}}(\pi_{\lambda_{2}}^{K_{2}})}Spec_{L}(V_{(\lambda_{1},\nu_{2}^{\prime})+\rho_{n}^{H}}^{H_{0}})}

A consequence of Theorem 4.2, Lemma 4.5 and Lemma 4.4 is:

Corollary 4.3**.**

[TABLE]

xxxxxx* $Spec_{H}(\pi_{\lambda})=\cup_{\nu_{2}^{\prime}\in Spec_{L\cap K_{2}}(\pi_{\lambda_{2}}^{K_{2}})}Spec_{L}(V_{(\lambda_{1},\nu_{2}^{\prime})}^{H_{0}}).$*

As we pointed out, Theorem 4.2 follows after we verify the next three Lemmas.

Lemma 4.4**.**

The hypothesis is $(G,H)$ is a symmetric space, $\pi_{\lambda}$ is $H$-admissible. Then

[TABLE]

Proof of Lemma 4.4.

The hypothesis $(G,H)$ is a symmetric pair and $\pi_{\lambda}$ is $H$-admissible, let us to apply notation and facts in [5], [30] as well as in [12] [3] [9] [18]. The proof is based on an idea in [3] of piling up multiplicities by means of Dirac delta distributions. That is, let $\delta_{\nu}$ denote the Dirac delta distribution at $\nu\in i\mathfrak{u}^{\star}$. Under our hypothesis, the function $m^{G,H}(\lambda,\mu)$ has polynomial growth in $\mu$, whence, the series $\sum_{\mu}m^{G,H}(\lambda,\mu)\,\,\delta_{\mu}$ converges in the space of distributions in $i\mathfrak{u}^{\star}$. Since Harish-Chandra parameter is regular, we may and will extend the function $m^{G,H}(\lambda,\cdot)$ to a $W_{L}$-skew symmetric function by the rule $m^{G,H}(\lambda,w\mu)=\epsilon(w)m^{G,H}(\lambda,\mu),w\in W_{L}$. Thus, the series $\sum_{\mu\in HC-param(H)}m^{G,H}(\lambda,\mu)\delta_{\mu}$ converges in the space of distributions in $i\mathfrak{u}^{\star}$. Next, for $0\not=\gamma\in i\mathfrak{u}^{\star}$ we consider the discrete Heaviside distribution $y_{\gamma}:=\sum_{n\geq 0}\delta_{\frac{\gamma}{2}+n\gamma}$, and for a strict, finite, multiset $S=\{\gamma_{1},\dots,\gamma_{r}\}$ of elements in $i\mathfrak{u}^{\star}$, we set

[TABLE]

Here, $\star$ is the convolution product in the space of distributions on $i\mathfrak{u}^{\star}$. $p_{S}$ is called the partition function attached to the set $S$. Then, in [5] there is presented the equality

[TABLE]

Here, $W_{S}$ is the Weyl group of the compact connected Lie group $S$; for a $ad(\mathfrak{u})$-invariant linear subspace $R$ of $\mathfrak{g}_{\mathbb{C}}$, $\Phi(R,\mathfrak{u})$ denotes the multiset of elements in $\Phi(\mathfrak{g},\mathfrak{u})$ such that its root space is contained in $R$, and $S_{w}^{H}=[q_{\mathfrak{u}}(w(\Psi_{\lambda})_{n})\cup\Delta(\mathfrak{k}/\mathfrak{l},\mathfrak{u})]\backslash\Phi(\mathfrak{h}/\mathfrak{l},\mathfrak{u})$.

Since, $K=K_{1}K_{2}$, $W_{K}=W_{K_{1}}\times W_{K_{2}}$, we write $W_{K}\ni w=st,s\in W_{K_{1}},t\in W_{K_{2}}$. We recall the hypothesis yields $K_{1}\subset L$. It readily follows: $s\Phi(\mathfrak{h}/\mathfrak{l},\mathfrak{u})=\Phi(\mathfrak{h}/\mathfrak{l},\mathfrak{u})$, $s\Delta(\mathfrak{k}/\mathfrak{l},\mathfrak{u})=\Delta(\mathfrak{k}/\mathfrak{l},\mathfrak{u}),t(\Psi_{\lambda})_{n}=(\Psi_{\lambda})_{n},t\eta_{1}=\eta_{1},s\eta_{2}=\eta_{2}\,\text{for}\,\eta_{j}\in\mathfrak{k}_{j}\cap\mathfrak{u}$, $sq_{\mathfrak{u}}(\cdot)=q_{\mathfrak{u}}(s\cdot)$. Hence,

[TABLE]

Thus,

[TABLE]

Following [12], we write the restriction of $\pi_{\lambda_{2}}^{K_{2}}$ to $L\cap K_{2}$ in the language of Dirac, Heaviside distributions in $i\mathfrak{u}^{\star}$, whence

[TABLE]

In the previous formula, we will apply $\Delta(\mathfrak{k}_{2}/(\mathfrak{k}_{2}\cap\mathfrak{l}),\mathfrak{u})=\Delta(\mathfrak{k}/\mathfrak{l},\mathfrak{u})$.

We also write in the same language the restriction to $L$ of a Discrete Series $\pi_{(\lambda_{1},\nu_{2}^{\prime})}^{H_{0}}$ for $H_{0}$. This is.

[TABLE]

Putting together the previous equalities, we obtain

[TABLE]

Since, the family $\{\delta_{\nu}\}_{\nu\in i\mathfrak{u}^{\star}}$ is linearly independent, we have shown Lemma 4.4. ∎

In order to conclude the proof of the dimension equality we state and prove a translation invariant property of multiplicity.

Lemma 4.5**.**

*For a dominant integral $\mu\in i\mathfrak{u}^{\star}$ it holds:

xxxxxxxxxxxxxx $m^{H_{0},L}((\lambda_{1},\nu_{2}^{\prime})+\rho_{n}^{H},\mu+\rho_{n}^{H})=m^{H_{0},L}((\lambda_{1},\nu_{2}^{\prime}),\mu)$.*

Proof.

We recall that the hypothesis of the Lemma 4.5 is: $(G,H)$ is a symmetric pair and $\pi_{\lambda}$ is $H$-admissible. The proof of Lemma 4.5 is an application of Blattner’s multiplicity formula, facts from [11] and observations from [30, Table 1,2,3]. In the next paragraphs we only consider systems $\Psi_{\lambda}$ so that $res_{H}(\pi_{\lambda})$ is admissible. We check the following statements by means of case by case analysis and the tables in [9] and [30].

OBS0. Every quaternionic system of positive roots that we are dealing with, satisfies the Borel de Siebenthal property, except for the algebra $\mathfrak{su}(2,2n)$ and the systems $\Psi_{1}$ (see 4.6). Its Dynkin diagram is \textstyle{{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bullet}} . Bullet represents non compact roots, circle compact.

OBS1. Always the systems $\Psi_{H,\lambda},\Psi_{H_{0},\lambda}$ have the same compact simple roots.

OBS2. When $\Psi_{\lambda}$ satisfies the Borel de Siebenthal property, it follows that both systems $\Psi_{H,\lambda},\Psi_{H_{0},\lambda}$ satisfy the Borel de Siebenthal property. Except for444We are indebted to the referee for this observation: i) $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2)+\mathfrak{so}(2n-2)$ and $\Psi_{H,\lambda}$; ii) $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2n-2)+\mathfrak{so}(2)$ and $\Psi_{H_{0},\lambda}$, for details see the proof of Remark 4.6.

OBS3. $\Psi_{\lambda}$ satisfies the Borel de Siebenthal property except for two families of algebras: a) the algebra $\mathfrak{su}(m,n)$ and the systems $\Psi_{a},a=1,\cdots,m-1$, $\tilde{\Psi}_{b},b=1,\cdots,n-1$, the corresponding systems $\Psi_{H_{0},\lambda},\Psi_{H,\lambda}$ do not satisfy the Borel de Siebenthal property. They have two noncompact simple roots; b) For the algebra $\mathfrak{so}(2m,2)$ each system $\Psi_{\pm}$ does not satisfy the Borel de Siebenthal property, however, each associated system $\Psi_{SO(2m,1),\lambda},\Psi_{H_{0},\lambda}$ satisfies the Borel de Siebenthal property.

OBS4. For the pair $(\mathfrak{su}(2,2n),\mathfrak{sp}(1,n))$. $\Psi_{1}$ does not satisfy the Borel de Siebenthal property. Here, $\Psi_{H,\lambda}=\Psi_{H_{0},\lambda}$ and they have Borel de Siebenthal property.

OBS5. Summing up. Both systems $\Psi_{H,\lambda},\Psi_{H_{0},\lambda}$ satisfy the Borel de Siebenthal property except for: i) $(\mathfrak{su}(m,n),\mathfrak{su}(m,k)+\mathfrak{su}(n-k)+\mathfrak{u}(1))$, $(\mathfrak{su}(m,n),\mathfrak{su}(k,n)+\mathfrak{su}(m-k)+\mathfrak{u}(1))$ and the systems $\Psi_{a},a=1,\cdots,m-1$, $\tilde{\Psi}_{b},b=1,\cdots,n-1$; ii) $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2)+\mathfrak{so}(2n-2)$ and $\Psi_{H,\lambda}$; iii) $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2n-2)+\mathfrak{so}(2)$ and $\Psi_{H_{0},\lambda}$, for details see the proof of Remark 4.6.

To continue, we explicit Blattner’s formula according to our setting, we recall fact’s from [11] and finish the proof of Lemma 4.5 under the assumption $\Psi_{H_{0},\lambda}$ satisfies the property of Borel de Siebenthal. Later on, we consider other systems.

Blattner’s multiplicity formula applied to the $L$-type $V_{\mu+\rho_{n}^{H}}^{L}$ of $V_{(\lambda_{1},\nu_{2}^{\prime})+\rho_{n}^{H}}^{H_{0}}$ yields

[TABLE]

Here, $Q_{0}$ is the partition function associated to the set $\Phi_{n}(\mathfrak{h}_{0})\cap\Psi_{H_{0},\lambda}$.

We recall a fact that allows to simplify the formula of above under our setting.

Fact 1: [11, Statement 4.31]. For a system $\Psi_{H_{0},\lambda}$ having the Borel de Siebenthal property, it is shown that in the above sum, if the summand attached to $s\in W_{L}$ contributes nontrivially, then $s$ belongs to the subgroup $W_{U}(\Psi_{H_{0},\lambda})$ spanned by the reflections about the compact simple roots in $\Psi_{H_{0},\lambda}$.

From OBS1 we have $W_{U}(\Psi_{H_{0},\lambda})=W_{U}(\Psi_{H,\lambda})$. Owing that either $\Psi_{H_{0},\lambda}$ or $\Psi_{H,\lambda}$ has the Borel de Siebenthal property we apply [11, Lemma 3.3], whence $W_{U}(\Psi_{H,\lambda})=\{s\in W_{L}:s(\Psi_{H,\lambda}\cap\Phi_{n}(\mathfrak{h},\mathfrak{u}))=\Psi_{H,\lambda}\cap\Phi_{n}(\mathfrak{h},\mathfrak{u})\}$. Thus, for $s\in W_{U}(\Psi_{H_{0},\lambda})$ we have $s\rho_{n}^{H}=\rho_{n}^{H}$. We apply the equality $s\rho_{n}^{H}=\rho_{n}^{H}$ in 4.1 and we obtain

[TABLE]

Blattner’s formula and the previous observations gives us that the right hand side of the above equality is

cccccccccccccccc$\dim Hom_{L}(V_{\mu}^{L},V_{(\lambda_{1},\nu_{2}^{\prime})}^{H_{0}})=m^{H_{0},L}((\lambda_{1},\nu_{2}^{\prime}),\mu)$,

whence, we have shown Lemma 4.5 when $\Psi_{H_{0},\lambda}$ has the Borel de Siebenthal property.

In order to complete the proof of Lemma 4.5, owing to OBS5, we are left to consider the pairs $(\mathfrak{su}(m,n),\mathfrak{su}(m,k)+\mathfrak{su}(n-k)+\mathfrak{u}(1))$ as well as $(\mathfrak{su}(m,n),\mathfrak{su}(k,n)+\mathfrak{su}(m-k)+\mathfrak{u}(1))$ and the systems $\Psi_{a},a=1,\cdots,m-1$, $\tilde{\Psi}_{b},b=1,\cdots,n-1$; $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2)+\mathfrak{so}(2n-2)$ and $\Psi_{H,\lambda}$; $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2n-2)+\mathfrak{so}(2)$ and $\Psi_{H_{0},\lambda}$, for details see the proof of Remark 4.6.

The previous reasoning says we are left to extend Fact 1, [11, Statement (4.31)], for the pairs $(\mathfrak{su}(m,n),\mathfrak{su}(m,k)+\mathfrak{su}(n-k)+\mathfrak{u}(1))$ (resp. $(\mathfrak{su}(m,n),\mathfrak{su}(k,n)+\mathfrak{su}(m-k)+\mathfrak{u}(1))$) and the systems $(\Psi_{a})_{a=1,\cdots,m-1}$ (resp. $(\tilde{\Psi}_{b})_{b=1,\cdots,n-1}$); $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2n-2)+\mathfrak{so}(2)$ and $\Psi_{H_{0},\lambda}$, for details see the proof of Remark 4.6. Under this setting we first verify:

Remark 4.6*.*

If $w\in W_{L}$ and $Q_{0}(w\mu-(\lambda+\rho_{n}^{H_{0}}))\not=0$, then $w\in W_{U}(\Psi_{H_{0},\lambda})$.

To show Remark 4.6 we follow [11]. To begin with we consider the pairs associated to $\mathfrak{g}=\mathfrak{su}(m,n)$. We fix as Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{su}(m,n)$ the set of diagonal matrices in $\mathfrak{su}(m,n).$ For certain orthogonal basis $\epsilon_{1},\dots,\epsilon_{m},\delta_{1},\dots,\delta_{n}$ of the dual vector space to the subspace of diagonal matrices in $\mathfrak{gl}(m+n,\mathbb{C}),$ we may, and will choose $\Delta=\{\epsilon_{r}-\epsilon_{s},\delta_{p}-\delta_{q},1\leq r<s\leq m,1\leq p<q\leq n\},$ the set of noncompact roots is $\Phi_{n}=\{\pm(\epsilon_{r}-\delta_{q})\}.$ We recall the positive roots systems for $\Phi(\mathfrak{g},\mathfrak{t})$ containing $\Delta$ are in a bijective correspondence with the totality of lexicographic orders for the basis $\epsilon_{1},\dots,\epsilon_{m},\delta_{1},\dots,\delta_{n}$ which contains the "suborder" $\epsilon_{1}>\dots>\epsilon_{m},\,\,\delta_{1}>\dots>\delta_{n}.$ The two holomorphic systems correspond to the orders $\epsilon_{1}>\dots>\epsilon_{m}>\delta_{1}>\dots>\delta_{n};\,\,\delta_{1}>\dots>\delta_{n}>\epsilon_{1}>\dots>\epsilon_{m}.$ We fix $1\leq a\leq m-1,$ and let $\Psi_{a}$ denote the set of positive roots associated to the order $\epsilon_{1}>\dots>\epsilon_{a}>\delta_{1}>\dots>\delta_{n}>\epsilon_{a+1}>\dots>\epsilon_{m}$. We fix $1\leq b\leq n-1$ and let $\tilde{\Psi}_{b}$ denote the set of positive roots associated to the order $\delta_{1}>\dots>\delta_{b}>\epsilon_{1}>\dots>\epsilon_{m}>\delta_{b+1}>\dots>\delta_{n}$. Since, $\mathfrak{h}=\mathfrak{su}(m,k)+\mathfrak{u}(n-k)$, $\mathfrak{h}_{0}=\mathfrak{su}(m,n-k)+\mathfrak{u}(k)$. The root systems for $(\mathfrak{h},\mathfrak{t})$ and $(\mathfrak{h}_{0},\mathfrak{t})$ respectively are:

[TABLE]

[TABLE]

The system $\Psi_{H,\lambda}=\Psi_{\lambda}\cap\Phi(\mathfrak{h},\mathfrak{t}),$ $\Psi_{H_{0},\lambda}=\Psi_{\lambda}\cap\Phi(\mathfrak{h}_{0},\mathfrak{t})$ which correspond to $\Psi_{a}$ are the system associated to the respective lexicographic orders

[TABLE]

[TABLE]

Without loss of generality, and in order to simplify notation, we may and will assume $\mathfrak{h}_{0}=\mathfrak{su}(m,n)$, $\Psi_{H_{0},\lambda}=\Psi_{a}$ and we show Remark 4.6 for $\mathfrak{su}(m,n)$ and $\Psi_{a}$. $Q$ denotes the partition function for $\Psi_{a}\cap\Phi_{n}$.

The subroot system spanned by the compact simple roots in $\Psi_{a}$ is

$\Phi_{U}=\{\epsilon_{i}-\epsilon_{j},1\leq i\leq a,1\leq j\leq a\,\text{or}\,a+1\leq i\leq m,a+1\leq j\leq m\}\cup\{\delta_{i}-\delta_{j},1\leq i\not=j\leq n\}.$

$\Psi_{a}\cap\Phi_{c}\backslash\Phi_{U}=\{\epsilon_{i}-\epsilon_{j},1\leq i\leq a,a+1\leq j\leq m\}$.

$\Psi_{a}\cap\Phi_{n}=\{\epsilon_{i}-\delta_{j},\delta_{j}-\epsilon_{r},1\leq i\leq a,a+1\leq r\leq m,1\leq j\leq n\}$.

$2\rho_{n}^{H_{0}}=n(\epsilon_{1}+\dots+\epsilon_{a})-n(\epsilon_{a+1}+\dots+\epsilon_{m})-(a-(m-a))(\delta_{1}+\dots+\delta_{n})$.

A finite sum of non compact roots in $\Psi_{a}$ is equal a to

$B=\sum_{1\leq j\leq a}A_{j}\epsilon_{j}-\sum_{a+1\leq i\leq m}B_{i}\epsilon_{i}+\sum_{r}C_{r}\delta_{r}$ with $A_{j},B_{i}$ non negative numbers.

Let $w\in W_{L}$ so that $Q(w\mu-(\lambda+\rho_{n}^{H_{0}}))\not=0$. Hence, $\mu=w^{-1}(\lambda+\rho_{n}^{H_{0}}+B)$, with $B$ a sum of roots in $\Psi_{a}\cap\Phi_{n}$. Thus, $w^{-1}$ is the unique element in $W_{L}$ that takes $\lambda+\rho_{n}^{H_{0}}+B$ to the Weyl chamber determined by $\Psi_{a}\cap\Phi_{c}$.

Let $w_{1}\in W_{U}(\Psi_{a})$ so that $w_{1}(\lambda+\rho_{n}+B)$ is $\Psi_{a}\cap\Phi_{U}$-dominant. Next we verify $w_{1}(\lambda+\rho_{n}+B)$ is $\Psi_{a}\cap\Phi_{c}$-dominant. For this, we fix $\alpha\in\Psi_{a}\cap\Phi_{c}\backslash\Phi_{U}$ and check $(w_{1}(\lambda+\rho_{n}+B),\alpha)>0$. $\alpha=\epsilon_{i}-\epsilon_{j},i\leq a<j$, and $w_{1}\in W_{U}(\Psi_{a})$, hence, $w_{1}^{-1}(\alpha)=\epsilon_{r}-\epsilon_{s},r\leq a<s$ belongs to $\Psi_{a}$. Thus, $(w_{1}(\lambda+\rho_{n}+B),\alpha)=(\lambda+\rho_{n}+B,w_{1}^{-1}\alpha)=(\lambda,w_{1}^{-1}\alpha)+(\rho_{n},w_{1}^{-1}\alpha)+(B,w_{1}^{-1}\alpha)=(\lambda,w_{1}^{-1}\alpha)+n-(-n)+A_{r}+B_{s}$, the first summand is positive because $\lambda$ is $\Psi_{a}$-dominant, the third and fourth are nonnegative. Therefore, $w^{-1}=w_{1}$ and we have shown Remark 4.6 for $\mathfrak{g}=\mathfrak{su}(m,n)$.

To follow we fix $\mathfrak{g}=\mathfrak{so}(2m,2n),n>1,\mathfrak{h}=\mathfrak{so}(2m,2n-2)+\mathfrak{so}(2),\mathfrak{h}_{0}=\mathfrak{so}(2m,2)+\mathfrak{so}(2n-2)$, then, for certain orthogonal basis $\epsilon_{1},\dots,\epsilon_{m},\delta_{1},\dots,\delta_{n}$ of the dual vector space to $\mathfrak{t}$, the system of positive roots $\Psi_{\lambda}=\{\epsilon_{i}\pm\epsilon_{j},\delta_{r}\pm\delta_{s},\epsilon_{a}\pm\delta_{b},1\leq i<j\leq m,1\leq r<s\leq n,1\leq a\leq m,1\leq b\leq n\}$ is so that $K_{1}(\Psi_{\lambda})=SO(2m)\subset H$. Since $n>1$ this is a Borel de Siebenthal system.

$\Psi_{H,\lambda}=\{\epsilon_{i}\pm\epsilon_{j},\delta_{r}\pm\delta_{s},\epsilon_{a}\pm\delta_{b},1\leq i<j\leq m,1\leq r<s\leq n-1,1\leq a\leq m,1\leq b\leq n-1\}$.

For $n>2$, $\Psi_{H,\lambda}$ is Borel de Siebenthal, for $n=2$, $\Psi_{H,\lambda}$ is not Borel de Siebenthal.

$\Psi_{H_{0},\lambda}=\{\epsilon_{i}\pm\epsilon_{j},\delta_{r}\pm\delta_{s},\epsilon_{a}\pm\delta_{n},1\leq i<j\leq m,1\leq r<s\leq n-1,1\leq a\leq m\}$. Since $n>1$, $\Psi_{H_{0},\lambda}$ is not a Borel de Siebenthal system. The simple roots for $\Psi_{H_{0},\lambda}$ are $\epsilon_{1}-\epsilon_{2},\cdots,\epsilon_{m-1}-\epsilon_{m},\epsilon_{m}\pm\delta_{n},\delta_{1}-\delta_{2},\cdots,\delta_{n-3}-\delta_{n-2},\delta_{n-2}\pm\delta_{n-1}$.

$\Phi_{U}=\{\epsilon_{i}-\epsilon_{j},\delta_{r}\pm\delta_{s},1\leq i<j\leq m,1\leq r<s\leq n-1\}$.

$W_{L}=span\{S_{\epsilon_{i}\pm\epsilon_{j}},S_{\delta_{r}\pm\delta_{s}},1\leq i<j\leq m,1\leq r<s\leq n-1\}$.

$W_{U}(\Psi_{H_{0},\lambda})=span\{S_{\epsilon_{i}-\epsilon_{j}},S_{\delta_{r}\pm\delta_{s}},1\leq i<j\leq m,1\leq r<s\leq n-1\}$.

$\Psi_{\lambda}\cap\Phi_{c}\backslash\Phi_{U}=\{\epsilon_{i}+\epsilon_{j},1\leq i<j\leq m\}.$

$Q_{0}$ is the partition function associated to $\Psi_{H_{0},\lambda}\cap\Phi_{n}=\{\epsilon_{i}\pm\delta_{n},1\leq i\leq m\}$.

$\rho_{n}^{H_{0}}=\epsilon_{1}+\dots+\epsilon_{m}$.

A finite sum of noncompact roots $B$ in $\Psi_{H_{0},\lambda}$ is equal to

$B=\sum_{j=1}^{j=m}(a_{j}+b_{j})\epsilon_{j}+(\sum_{1\leq j\leq m}(a_{j}-b_{j}))\delta_{n},a_{j}\geq 0,b_{j}\geq 0\,\forall j$.

Let $w\in W_{L}$ so that $Q_{0}(w\mu-(\lambda+\rho_{n}))\not=0$. Hence, $\mu=w^{-1}(\lambda+\rho_{n}+B)$, with $B$ a sum of roots in $\Psi_{\lambda}\cap\Phi_{n}$. Thus, $w^{-1}$ is the unique element in $W_{L}$ that takes $\lambda+\rho_{n}+B$ to the Weyl chamber determined by $\Psi_{\lambda}\cap\Phi_{c}$.

Let $w_{1}\in W_{U}(\Psi_{H_{0},\lambda})$ so that $w_{1}(\lambda+\rho_{n}+B)$ is $\Psi_{H_{0},\lambda}\cap\Phi_{U}$-dominant. Next we verify $w_{1}(\lambda+\rho_{n}+B)$ is $\Psi_{H_{0},\lambda}\cap\Phi_{c}$-dominant. For this, we fix $\alpha\in\Psi_{H_{0},\lambda}\cap\Phi_{c}\backslash\Phi_{U}$ and check $(w_{1}(\lambda+\rho_{n}+B),\alpha)>0$. $\alpha=\epsilon_{i}+\epsilon_{j}$, and $w_{1}\in W_{U}(\Psi_{H_{0},\lambda})$, hence, $w_{1}^{-1}(\alpha)=\epsilon_{r}+\epsilon_{s}$ belongs to $\Psi_{H_{0},\lambda}$. Thus, $(w_{1}(\lambda+\rho_{n}+B),\alpha)=(\lambda+\rho_{n}+B,w_{1}^{-1}\alpha)=(\lambda,w_{1}^{-1}\alpha)+(\rho_{n},w_{1}^{-1}\alpha)+(B,w_{1}^{-1}\alpha)$, the first summand is positive because $\lambda$ is $\Psi_{H_{0},\lambda}$-dominant, the second and third are nonnegative. Therefore, $w^{-1}=w_{1}$ and we have shown Remark 4.6 for $\mathfrak{g}=\mathfrak{so}(2m,2n)$.

Whence, we have concluded the proof of Lemma 4.5. ∎

Lemma 4.7**.**

555We thank the referee for improving this Lemma

*We recall $\Lambda_{2}=\lambda_{2}+(\rho_{n}^{\lambda})_{2}$. We claim:

xxxxxxxxxxx $m^{K_{2},L\cap K_{2}}(\Lambda_{2},\nu_{2}^{\prime}+q_{\mathfrak{u}}(\rho_{n}^{\lambda})_{2})=m^{K_{2},L\cap K_{2}}(\lambda_{2},\nu_{2}^{\prime})$.

xxxxxxxxxxx $Spec_{L\cap K_{2}}(\pi_{\Lambda_{2}}^{K_{2}})=Spec_{L\cap K_{2}}(\pi_{\lambda_{2}}^{K_{2}})+q_{\mathfrak{u}}(\rho_{n}^{\lambda})_{2}$.*

In fact, when $\Psi_{\lambda}$ is holomorphic, $\rho_{n}^{\lambda}$ is in $\mathfrak{z}_{\mathfrak{k}}=\mathfrak{k}_{1}$ hence $(\rho_{n}^{\lambda})_{2}=0$. In [30] it is shown that when $K$ is semisimple $(\rho_{n}^{\lambda})_{2}=0$. Actually, this is so, owing that the simple roots for $\Psi_{\lambda}\cap\Phi(\mathfrak{k}_{2},\mathfrak{t}_{2})$ are simple roots for $\Psi_{\lambda}$ and that $\rho_{n}^{\lambda}$ is orthogonal to every compact simple root for $\Psi_{\lambda}$. For general $\mathfrak{g}$, the previous considerations together with that $(\rho_{n}^{\lambda})_{2}$ is orthogonal to $\mathfrak{k}_{1}$ yields that $(\rho_{n}^{\lambda})_{2}$ belongs to the dual of the center of $\mathfrak{l}\cap\mathfrak{k}_{2}$. From Tables 1,2,3 we deduce we are left to analyze $(\rho_{n}^{\lambda})_{2}$ for $\mathfrak{su}(m,n),\mathfrak{so}(2m,2)$. For $\mathfrak{so}(2m,2)$ we follow the notation in 4.1.3, then $\mathfrak{t}_{1}^{\star}=span(\epsilon_{1},\dots,\epsilon_{m}),\mathfrak{t}_{2}^{\star}=span(\delta_{1})$ and $\rho_{n}^{\Psi_{\pm}}=c(\epsilon_{1}+\dots+\epsilon_{m})\in\mathfrak{t}_{1}^{\star}$. For $\mathfrak{su}(m,n)$ we follow the notation in Lemma 4.5. It readily follows that for $1\leq a<m$, $\rho_{n}^{\Psi_{a}}=\frac{n}{m}((m-a)(\epsilon_{1}+\dots+\epsilon_{a})-a(\epsilon_{a+1}+\dots+\epsilon_{m}))+\frac{2a-m}{2m}((n(\epsilon_{1}+\dots+\epsilon_{m})-m(\delta_{1}+\dots+\delta_{n}))$. The first summand is in $(\mathfrak{t}\cap\mathfrak{su}(m))^{\star}$, the second summand belongs to $\mathfrak{z}_{\mathfrak{k}}^{\star}$, thus, $(\rho_{n}^{\Psi_{a}})_{2}=0$ if and only if $2a=m$. Hence, for $(\mathfrak{su}(2,2m),\mathfrak{sp}(1,m))$, we have $(\rho_{n}^{\Psi_{1}})_{2}=0$. For $(\mathfrak{su}(m,n),\mathfrak{su}(m,k)+\mathfrak{su}(n-k)+\mathfrak{z}_{\mathfrak{l}})$, always, $(\rho_{n}^{\Psi_{a}})_{2}$ determines a character of the center of $\mathfrak{k}$. In this case, $\lambda_{2}=\Lambda_{2}$ except for $(\mathfrak{su}(m,n),\mathfrak{su}(m,k)+\mathfrak{su}(n-k)+\mathfrak{z}_{\mathfrak{l}}),\Psi_{a}$ and $a\not=2m$, hence, $\pi_{\Lambda_{2}}^{K_{2}}$ is equal to $\pi_{\lambda_{2}}^{K_{2}}$ times a central character of $K$. Thus, the equality holds.

As a corollary we obtain: $(\rho_{n}^{\lambda})_{2}\not=0$ forces $\mathfrak{g}=\mathfrak{su}(m,n),\mathfrak{h}=\mathfrak{s}(\mathfrak{u}(m,k)+\mathfrak{u}(n-k)),m>1,n>1,1\leq k\leq n-1$, $\mathfrak{u}=\mathfrak{t}$. Thus, we always have $q_{\mathfrak{u}}((\rho_{n}^{\lambda})_{2})=(\rho_{n}^{\lambda})_{2}$.

4.1.1. Conclusion proof of Theorem 4.2

We just put together Lemma 4.5, Lemma 4.4 and Lemma 5, hence, we obtain the equalities we were searching for. This concludes the proof of Theorem 4.2.

4.1.2. Existence of $D$

To follow we show the existence of the isomorphism $D$ in Theorem 4.2 $ii)$ and derive the decomposition into irreducible factors of the semisimple $\mathfrak{h}_{0}$-module $\mathcal{U}(\mathfrak{h}_{0})W$. On the mean time, we also consider some particular cases of Theorem 4.2. Before, we proceed we comment on the structure of the representation $\tau$.

4.1.3. Representations $\pi_{\lambda}$ so that $res_{L}(\tau)$ is irreducible

Under our $H$-admissibility hypothesis of $\pi_{\lambda}$ we analyze the cases so that the representation $res_{L}(\tau)$ is irreducible. The next structure statements are verified in [30]. To begin with, we recall the decomposition $K=K_{1}Z_{K}K_{2}$, (this is not a direct product!, $Z_{K}$ connected center of $K$) and the direct product $K=K_{1}K_{2}$, we also recall that actually, either $K_{1}$ or $K_{2}$ depend on $\Psi_{\lambda}$. When $\pi_{\lambda}$ is a holomorphic representation $K_{1}=Z_{K}$ and $\mathfrak{k}_{2}=[\mathfrak{k},\mathfrak{k}]$; when, $Z_{K}$ is nontrivial and $\pi_{\lambda}$ is not a holomorphic representation we have $Z_{K}\subset K_{2}$; for $\mathfrak{g}=\mathfrak{su}(m,n),\mathfrak{h}=\mathfrak{su}(m,k)+\mathfrak{su}(n-k)+\mathfrak{z}_{L}$, we have $\mathbf{T}\equiv Z_{K}\subset Z_{L}\equiv\mathbf{T}^{2}$. Here, $Z_{K}\subset L$, and, $\tau_{|_{L}}$ irreducible, forces $\tau=\pi_{\Lambda_{1}}^{SU(m)}\boxtimes\pi_{\chi}^{Z_{K}}\boxtimes\pi_{\rho_{SU(n)}}^{SU(n)}$ ; for $\mathfrak{g}\ncong\mathfrak{su}(m,n)$ and $G/K$ a Hermitian symmetric space, we have to consider the next two examples.

For both cases we have $K_{2}=Z_{K}(K_{2})_{ss}$ and $Z_{K}\nsubseteq L$.

1. When $\mathfrak{g}=\mathfrak{so}(2m,2),\mathfrak{h}=\mathfrak{so}(2m,1)$ and $\Psi_{\lambda}=\Psi_{\pm}$, then $\mathfrak{k}_{1}=\mathfrak{so}(2m)$, $\mathfrak{k}_{2}=\mathfrak{z}_{K}$ and obviously $res_{L}(\tau)$ is always an irreducible representation. Here, $\pi_{\Lambda_{2}}^{K_{2}}$ is one dimensional representation.

2. When $\mathfrak{g}=\mathfrak{su}(2,2n),\mathfrak{h}=\mathfrak{sp}(1,n)$, $\Psi_{\lambda}=\Psi_{1}$, then $\mathfrak{k}_{1}=\mathfrak{su}_{2}(\alpha_{max})$, $\mathfrak{k}_{2}=\mathfrak{su}(2n)+\mathfrak{z}_{\mathfrak{k}}$, $\mathfrak{l}\equiv\mathfrak{su}_{2}(\alpha_{max})+\mathfrak{sp}(n)$. Examples of $\tau_{|_{L}}$ irreducible are666This was pointed out by the referee, $\tau=\pi_{b\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\chi}^{Z_{K}}\boxtimes\pi_{\rho_{SU(2n)}+a\widetilde{\Lambda}_{1}}^{SU(2n)}$, $\tau=\pi_{b\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\chi}^{Z_{K}}\boxtimes\pi_{\rho_{SU(2n)}+a\widetilde{\Lambda}_{2n-1}}^{SU(2n)},a\geq 0,b\geq 2$, $\Lambda_{1}$ (resp. $\widetilde{\Lambda}_{1}$) is the highest weight of the first fundamental representation of $\mathfrak{su}_{2}$ (resp. $\mathfrak{su}(2n)$), $\widetilde{\Lambda}_{2n-1}$ is the highest weight of the dual representation to the first fundamental representation for $\mathfrak{su}(2n)$.

For $\mathfrak{g}=\mathfrak{so}(2m,2n),\mathfrak{h}=\mathfrak{so}(2m,2n-1),n>1,$ $\Psi_{\lambda}\cap\Phi_{n}=\{\epsilon_{i}\pm\delta_{j}\},$ $\mathfrak{k}_{1}=\mathfrak{so}(2m)$, and if $\lambda$ is so that $\lambda+\rho_{n}^{\lambda}=ic(\tau)=(\sum c_{i}\epsilon_{i},k(\delta_{1}+\dots+\delta_{n-1}\pm\delta_{n}))+\rho_{K}$, then $res_{L}(\tau)$ is irreducible and $\pi_{\Lambda_{2}}^{K_{2}}=\pi_{k(\delta_{1}+\dots+\delta_{n-1}\pm\delta_{n})+\rho_{K_{2}}}^{K_{2}}$ is not a one dimensional representation for $k>0$. It follows from the classical branching laws that these are the unique $\tau^{\prime}s$ such that $res_{L}(\tau)$ is irreducible.

We do not know the pairs $(\pi_{\lambda},\tau)$ so that $\pi_{\lambda}$ is $H$-admissible and $res_{L}(\tau)$ is irreducible. We believe that for $\mathfrak{g}\ncong\mathfrak{so}(2m,2n)$ or $\mathfrak{g}\ncong\mathfrak{su}(m,n)$ we could conclude that $\tau$ is the tensor product of a irreducible representation of $K_{1}$ times a one dimensional representation of $K_{2}$. That is, $\tau\equiv\pi_{\phi_{1}}^{K_{1}}\boxtimes\pi_{\rho_{K_{2}}}^{K_{2}}\otimes\pi_{\chi}^{Z_{K_{2}}}$.

In 5.3.1 we show that whenever a symmetric pair $(G,H)$ is so that some Discrete Series for $G$ is $H$-admissible, then there exists $H$-admissible Discrete Series for $G$ so that its lowest $K$-type restricted to $L$ is irreducible.

4.1.4. Analysis of $\mathcal{U}(\mathfrak{h}_{0})W$, $\mathcal{L}_{\lambda}$, existence of $D$, case $\tau_{|_{L}}$ is irreducible

As before, our hypothesis is $(G,H)$ is a symmetric space and $\pi_{\lambda}^{G}$ is $H$-admissible. For this paragraph we add the hypothesis $\tau_{|_{L}}=res_{L}(\tau)$ is irreducible. We recall that $\mathcal{U}(\mathfrak{h}_{0})W=L_{\mathcal{U}(\mathfrak{h}_{0})}(H^{2}(G,\tau)[W])$, $\mathcal{L}_{\lambda}=\oplus_{\mu\in Spec_{H}(\pi_{\lambda})}H^{2}(G,\tau)[V_{\mu}^{H}][V_{\mu+\rho_{n}^{\mu}}^{L}]$. We claim:

a) if a $H$-irreducible discrete factor of $V_{\lambda}^{G}$ contains a copy of $\tau_{|L}$, then $\tau_{|L}$ is the lowest $L$-type of such factor.

b) the multiplicity of $res_{L}(\tau)$ in $H^{2}(G,\tau)$ is one.

c) $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is $H_{0}$-equivalent to $H^{2}(H_{0},\tau)$.

d) $\mathcal{L}_{\lambda}$ is $L$-equivalent to $H^{2}(H_{0},\tau)_{L-fin}$.

e) $\mathcal{L}_{\lambda}$ is $L$-equivalent to $\mathcal{U}(\mathfrak{h}_{0})W$. Thus, $D$ exists.

We rely on:

Remark 4.8*.*

1. Two Discrete Series for $H$ are equivalent if and only if their respective lowest $L$-types are equivalent. [31].

2. For any Discrete Series $\pi_{\lambda}$, the highest weight (resp. infinitesimal character) of any $K$-type is equal to the highest weight of the lowest $K$-type (resp. the infinitesimal character of the lowest $K$-type) plus a sum of noncompact roots in $\Psi_{\lambda}$ [27, Lemma 2.22].

From now on $ic(\phi)$ denotes the infinitesimal character (Harish-Chandra parameter) of the representation $\phi$.

Let $V_{\mu}^{H}$ a discrete factor for $res_{H}(\pi_{\lambda})$ so that $\tau_{|_{L}}$ is a $L$-type. Then, Theorem 4.2 implies $V_{\mu+\rho_{n}^{H}}^{L}$ is a $L$-type for $H^{2}(H_{0},\tau)$. Hence, after we apply Remark 4.8, we obtain

$\mu+\rho_{n}^{H}+B_{1}=ic(\tau_{|_{L}})$ with $B_{1}$ a sum of roots in $\Psi_{H,\lambda}\cap\Phi_{n}$.

$\mu+\rho_{n}^{H}=ic(\tau_{|_{L}})+B_{0}$ with $B_{0}$ a sum of roots in $\Psi_{H_{0},\lambda}\cap\Phi_{n}$.

Thus, $B_{0}+B_{1}=0$, whence $B_{0}=B_{1}=0$ and $\mu+\rho_{n}^{H}=ic(\tau_{|_{L}})$, we have verified a).

Due to $H$-admissibility hypothesis, we have $\mathcal{U}(\mathfrak{h})W$ is a finite sum of irreducible underlying modules of Discrete Series for $H$. Now, Corollary 1 to Lemma 3.6, yields that a copy of a $V_{\mu}^{H}$ contained in $\mathcal{U}(\mathfrak{h})W$ contains a copy of $V_{\lambda}^{G}[W]$. Thus, a) implies $\tau_{|_{L}}$ is the lowest $L$-type of such $V_{\mu}^{H}$. Hence, $H^{2}(H,\tau)$ is nonzero. Now, Theorem 4.2 together with the fact that the lowest $L$-type of a Discrete Series has multiplicity one yields that $\dim Hom_{H}(H^{2}(H,\tau),V_{\lambda}^{G})=1$. Also, we obtain $\dim Hom_{H_{0}}(H^{2}(H_{0},\tau),V_{\lambda}^{G})=1$. Thus, whenever $\tau_{|L}$ occurs in $res_{L}(V_{\lambda}^{G})$, we have $\tau_{|_{L}}$ is realized in $V_{\lambda}^{G}[W]$. In other words, the isotypic compoent $V_{\lambda}^{G}[\tau_{|_{L}}]\subset V_{\lambda}^{G}[W]$. Hence, b) holds.

Owing our hypothesis, we may write $\mathcal{U}(\mathfrak{h}_{0})W=N_{1}+...+N_{k}$, with each $N_{j}$ being the underlying Harish-Chandra module of a irreducible square integrable representation for $H_{0}$. Since Lemma 3.6 shows $r_{0}$ is injective in $\mathcal{U}(\mathfrak{h}_{0})W$, we have $r_{0}(\mathrm{Cl}(N_{j}))$ is a Discrete Series in $L^{2}(H_{0}\times_{res_{L}(\tau)}W)$, hence Frobenius reciprocity implies $\tau_{|_{L}}$ is a $L$-type for $N_{j}$. Hence, b) and a) yields $\mathcal{U}(\mathfrak{h}_{0})W$ is $\mathfrak{h}_{0}$-irreducible and c) follows.

By definition, the subspace $\mathcal{L}_{\lambda}$ is the linear span of the subspaces $V_{\lambda}^{G}[V_{\mu}^{H}][V_{\mu+\rho_{n}^{H}}^{L}]$ with $\mu\in Spec_{H}(\pi_{\lambda})$. Since,

[TABLE]

and, both $L$-modules are isotypical, d) follows. Finally, e) follows from c) and d).

Whenever $\pi_{\Lambda_{2}}^{K_{2}}$ is the trivial representation, Theorem 4.2 and Lemma 4.5 justifies:

[TABLE]

the infinitesimal character of $H^{2}(H_{0},\tau)$ is $(\lambda_{1},\rho_{K_{2}\cap L})+q_{\mathfrak{u}}(\rho_{n}^{\lambda})-\rho_{n}^{H_{0}}=(\lambda_{1},\rho_{K_{2}\cap L})+\rho_{n}^{H}$. Thus, $H^{2}(H_{0},\tau)\equiv V_{(\lambda_{1},\rho_{K_{2}\cap L})+\rho_{n}^{H}}^{H_{0}}$.

4.1.5.

Analysis of $\mathcal{U}(\mathfrak{h}_{0})W$, $\mathcal{L}_{\lambda}$, existence of $D$, for general $(\tau,W)$

We recall that by definition, $\mathcal{L}_{\lambda}=\oplus_{\mu\in Spec_{H}(\pi_{\lambda})}H^{2}(G,\tau)[V_{\mu}^{H}][V_{\mu+\rho_{n}^{\mu}}^{L}]$, $\mathcal{U}(\mathfrak{h}_{0})W=L_{\mathcal{U}(\mathfrak{h}_{0})}(H^{2}(G,\tau)[W])$.

Proposition 4.9**.**

*The hypothesis is: $(G,H)$ is a symmetric pair and $\pi_{\lambda}$ a $H$-admissible square integrable representation of lowest $K$-type $(\tau,W)$. We write

xxxxx $res_{L}(\tau)=q_{1}\sigma_{1}+\cdots+q_{r}\sigma_{r},$ with $(\sigma_{j},Z_{j})\in\widehat{L},q_{j}>0$. Then,*

a) if a $H$-irreducible discrete factor for $res_{H}(\pi_{\lambda})$ contains a copy of $\sigma_{j}$, then $\sigma_{j}$ is the lowest $L$-type of such factor.

b) the multiplicity of $\sigma_{j}$ in $res_{L}(H^{2}(G,\tau))$ is equal to $q_{j}$.

c) $r_{0}:\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)\rightarrow\mathbf{H}^{2}(H_{0},\tau)$ is a $H_{0}$-equivalence.

d) $\mathcal{L}_{\lambda}$ is $L$-equivalent to $\mathbf{H}^{2}(H_{0},\tau)_{L-fin}$.

e) $\mathcal{L}_{\lambda}$ is $L$-equivalent to $\mathcal{U}(\mathfrak{h}_{0})W$. Therefore, $D$ exists.

Proof.

Let $V_{\mu}^{H}$ a discrete factor for $res_{H}(\pi_{\lambda})$ so that some irreducible factor of $\tau_{|_{L}}$ is a $L$-type. Then, Theorem 4.2 implies $V_{\mu+\rho_{n}^{H}}^{L}$ is a $L$-type for $\mathbf{H}^{2}(H_{0},\tau)=\oplus_{j}q_{j}H^{2}(H_{0},\sigma_{j})$. Let’s say $V_{\mu+\rho_{n}^{H}}^{L}$ is a subrepresentation of $H^{2}(H_{0},\sigma_{i})$. We recall $ic(\phi)$ denotes the infinitesimal character (Harish-Chandra parameter) of the representation $\phi$. Hence, after we apply Remark 4.8 we obtain

$\mu+\rho_{n}^{H}+B_{1}=ic(\sigma_{j})$ with $B_{1}$ a sum of roots in $\Psi_{H,\lambda}\cap\Phi_{n}$.

$\mu+\rho_{n}^{H}=ic(\sigma_{i})+B_{0}$ with $B_{0}$ a sum of roots in $\Psi_{H_{0},\lambda}\cap\Phi_{n}$.

Thus, $B_{0}+B_{1}=ic(\sigma_{j})-ic(\sigma_{i})$. Now, since $\mathfrak{k}=\mathfrak{k}_{1}+\mathfrak{k}_{2}$, $\mathfrak{k}_{1}\subset\mathfrak{l}$, $\tau=\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\Lambda_{2}}^{K_{2}}$, we may write $\sigma_{s}=\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\phi_{s},$ with $\phi_{s}\in\widehat{L\cap K_{2}}$, hence, $ic(\sigma_{j})-ic(\sigma_{i})=ic(\phi_{j})-ic(\phi_{i})$. Since, each $\phi_{t}$ is a irreducible factor of $res_{L\cap K_{2}}(\pi_{\Lambda_{2}}^{K_{2}})$, we have $ic(\phi_{j})-ic(\phi_{i})$ is equal to the difference of two sum of roots in $\Phi(\mathfrak{k}_{2},\mathfrak{t}\cap\mathfrak{k}_{2})$. The hypothesis forces that the simple roots for $\Psi_{\lambda}\cap\Phi(\mathfrak{k}_{2},\mathfrak{t}\cap\mathfrak{k}_{2})$ are compact simple roots for $\Psi_{\lambda}$ (see [5]) whence $ic(\sigma_{j})-ic(\sigma_{i})$ is a linear combination of compact simple roots for $\Psi_{\lambda}$. On the other hand, $B_{0}+B_{1}$ is a sum of noncompact roots in $\Psi_{\lambda}$. Now $B_{0}+B_{1}$ can not be a linear combination of compact simple roots, unless $B_{0}=B_{1}=0$. Thus, $ic(\sigma_{i})=ic(\sigma_{j})$ and $Z_{j}\equiv V_{\sigma_{j}}^{L}$ is the lowest $L$-type of $V_{\mu}^{H}$, we have verified a).

Due to $H$-admissibility hypothesis, we have $\mathcal{U}(\mathfrak{h})W$ is a finite sum of irreducible underlying Harish-Chandra modules of Discrete Series for $H$. Thus, a copy of certain $V_{\mu}^{H}$ contained in $\mathcal{U}(\mathfrak{h})W$ contains $W[\sigma_{j}]$. Whence, $\sigma_{j}$ is the lowest $L$-type of such $V_{\mu}^{H}$. Hence, $H^{2}(H,\sigma_{j})$ is nonzero and it is equivalent to a subrepresentation of $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$.

We claim, for $i\not=j$, no $\sigma_{j}$ is a $L$-type of $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)[H^{2}(H,\sigma_{i})]$.

Indeed, if $\sigma_{j}$ were a $L$-type in $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)[H^{2}(H,\sigma_{i})]$, then, $\sigma_{j}$ would be a $L$-type of a Discrete Series of lowest $L$-type equal to $\sigma_{i}$, according to a) this forces $i=j$, a contradiction. Now, we compute the multiplicity of $H^{2}(H,\sigma_{j})$ in $H^{2}(G,\tau)$. For this, we apply Theorem 4.2. Thus, $\dim Hom_{H}(V_{\lambda}^{G},H^{2}(H,\sigma_{j}))=\sum_{i}q_{i}\dim Hom_{L}(\sigma_{j},H^{2}(H,\sigma_{i}))=q_{j}$.

In order to realize the isotypic component corresponding to $H^{2}(H,\sigma_{j})$ we write $V_{\lambda}^{G}[W][\sigma_{j}]=R_{1}+\cdots+R_{q_{j}}$ a explicit sum of $L$-irreducibles modules. Then, owing to a), $L_{\mathcal{U}(\mathfrak{h})}(R_{r})$ contains a copy $N_{r}$ of $H^{2}(H,\sigma_{j})$ and $R_{r}$ is the lowest $L$-type of $N_{r}$. Therefore, the multiplicity computation yields $H^{2}(G,\tau)[H^{2}(H,\sigma_{j})]=N_{1}+\cdots+N_{q_{j}}$. Hence, b) holds. A corollary of this computation is:

xxxxxxxxxxxxx $Hom_{H}(H^{2}(H,\sigma_{j}),(\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W))^{\perp})=\{0\}$.

Verification of c). After we recall Lemma 3.6, we have $r_{0}:\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)\rightarrow L^{2}(H_{0}\times_{\tau}W)$ is injective and we apply to the algebra $\mathfrak{h}:=\mathfrak{h}_{0}$, the statement b) together with the computation to show b), we make the choice of the $q_{j}^{\prime}s$ subspaces $Z_{j}$ as a lowest $L$-type subspace of $W[Z_{j}]$. Thus, the image via $r_{0}$ of $\mathcal{U}(\mathfrak{h}_{0})Z_{j}$ is a subspace of $L^{2}(H_{0}\times_{\sigma_{j}}Z_{j})$. Since, Hotta-Parthasarathy [13], Atiyah-Schmid [1], Enright-Wallach [6] have shown $H^{2}(H_{0},\sigma_{j})$ has multiplicity one in $L^{2}(H_{0}\times_{\sigma_{j}}Z_{j})$ we obtain the image of $r_{0}$ is equal to $\mathbf{H}^{2}(H_{0},\tau)$.

The proof of d) and e) are word by word as the one for 4.1.4. ∎

Corollary 4.10**.**

*The multiplicity of $H^{2}(H,\sigma_{j})$ in $res_{H}(H^{2}(G,\tau))$ is equal to

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx $q_{j}=\dim Hom_{L}(\sigma_{j},H^{2}(G,\tau))$.*

Corollary 4.11**.**

For each $\sigma_{j}$, $\mathcal{L}_{\lambda}[Z_{j}]=\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)[Z_{j}]=H^{2}(G,\tau)[W][Z_{j}]="W"[Z_{j}]$. Thus, we may fix $D=I_{"W"[Z_{j}]}:\mathcal{L}_{\lambda}[Z_{j}]\rightarrow\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)[Z_{j}]$.

4.2. Explicit inverse map to $r_{0}^{D}$

We consider three cases: $res_{L}(\tau)$ is irreducible, $res_{L}(\tau)$ is multiplicity free, and general case. Formally, they are quite alike, however, for us it has been illuminating to consider the three cases. As a byproduct, we obtain information on the compositions $r^{\star}r,r_{0}^{\star}r_{0}$; a functional equation that must be satisfied by the kernel of a holographic operator; for some particular discrete factor $H^{2}(H,\sigma)$ of $res_{H}(\pi_{\lambda})$ the reproducing kernel for $H^{2}(G,\tau)$ is a extension of the reproducing kernel for $H^{2}(H,\sigma)$ as well as that the holographic operator from $H^{2}(H,\sigma)$ into $H^{2}(G,\tau)$ is just plain extension of functions.

4.2.1. Case $(\tau,W)$ restricted to $L$ is irreducible

In Tables 1,2,3, we show the list of the triples $(G,H,\pi_{\lambda})$ such that $(G,H)$ is a symmetric pair, and $\pi_{\lambda}$ is $H$-admissible. In 5.3.1 we show that if there exists $(G,H,\pi_{\lambda})$ so that $\pi_{\lambda}$ is $H$-admissible, then there exists a $H$-admissible $\pi_{\lambda^{\prime}}$ so that its lowest $K$-type restricted to $L$ is irreducible and $\lambda^{\prime}$ is dominant with respect to $\Psi_{\lambda}$. We denote by $\eta_{0}$ the Harish-Chandra parameter for $H^{2}(H_{0},\tau)\equiv\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$.

We set $d(\pi)$ for the formal degree of a irreducible square integrable representation $\pi$ and define $c=d(\pi_{\lambda})\dim W/d(\pi_{\eta_{0}}^{H_{0}})$. Next, we show

Proposition 4.12**.**

*We assume the setting as well as the hypothesis in Theorem 3.1, and further $(\tau,W)$ restricted to $L$ is irreducible.

Let $T_{0}\in Hom_{L}(Z,H^{2}(H_{0},\tau))$, then the kernel $K_{T}$ corresponding to $T:=(r_{0}^{D})^{-1}(T_{0})\in Hom_{H}(H^{2}(H,\sigma),H^{2}(G,\tau))$ is*

[TABLE]

Proof.

We systematically apply Theorem 4.9. Under our assumptions, we have: $\mathbf{H}^{2}(H_{0},\tau)$ is a irreducible representation and $\mathbf{H}^{2}(H_{0},\tau)=H^{2}(H_{0},\tau)$;

$\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})(H^{2}(G,\tau)[W]))$ is $H_{0}$-irreducible; We define

$\tilde{r}_{0}:=rest(r_{0}):\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})H^{2}(G,\tau)[W])\rightarrow H^{2}(H_{0},\tau)$ is a isomorphism. To follow, we notice the inverse of $\tilde{r}_{0}$, is up to a constant, equal to $r_{0}^{\star}$ restricted to $H^{2}(H_{0},\tau)$. This is so, because functional analysis yields the equalities $\mathrm{Cl}(Im(r_{0}^{\star}))=ker(r_{0})^{\perp}=\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$, $Ker(r_{0}^{\star})=Im(r_{0})^{\perp}=H^{2}(H_{0},\tau)^{\perp}$. Thus, Schur’s lemma applied to the irreducible modules $H^{2}(H_{0},\tau),\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ implies there exists non zero constants $b,d$ so that $(\tilde{r}_{0}r_{0}^{\star})_{|_{H^{2}(H_{0},\tau)}}=bI_{H^{2}(H_{0},\tau)}$, $r_{0}^{\star}\tilde{r}_{0}=dI_{\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)}$. Whence, the inverse to $\tilde{r}_{0}$ follows. In 4.2.2, we show $b=d=d(\pi_{\lambda})\dim W/d(\pi_{\eta_{0}}^{H_{0}})=c$.

For $x\in G,f\in H^{2}(G,\tau)$, the identity $f(x)=\int_{G}K_{\lambda}(y,x)f(y)dy$ holds. Thus, $r_{0}(f)(p)=f(p)=\int_{G}K_{\lambda}(y,p)f(y)dy,\,\text{for}\,p\in H_{0},f\in H^{2}(G,\tau)$, and, we obtain

$K_{r_{0}}(x,h_{0})=K_{\lambda}(x,h_{0}),\,\,\,K_{r_{0}^{\star}}(h_{0},x)=K_{r_{0}}(x,h_{0})^{\star}=K_{\lambda}(h_{0},x)$.

Hence, for $g\in H^{2}(H_{0},\tau)$ we have,

xxx $\tilde{r}_{0}^{-1}(g)(x)=\frac{1}{c}\int_{H_{0}}K_{r_{0}^{\star}}(h_{0},x)g(h_{0})dh_{0}=\frac{1}{c}\int_{H_{0}}K_{\lambda}(h_{0},x)g(h_{0})dh_{0}$.

Therefore, for $T_{0}\in Hom_{L}(Z,H^{2}(H_{0},\tau))$, the kernel $K_{T}$ of the element $T$ in $Hom_{H}(H^{2}(H,\sigma),H^{2}(G,\tau))$ such that $r_{0}^{D}(T)=T_{0}$, satisfies for $z\in Z$

[TABLE]

More explicitly, after we recall $K_{T}(e,h^{-1}x)=K_{T}(h,x)$,

[TABLE]

∎

Corollary 4.13**.**

For any $T$ in $Hom_{H}(H^{2}(H,\sigma),H^{2}(G,\tau))$ we have

[TABLE]

Corollary 4.14**.**

When $D$ is the identity map, we obtain

[TABLE]

The equality in the conclusion of Proposition 4.12 is equivalent to

xxxxxx $D(K_{T}(e,\cdot))(y)=\int_{H_{0}}\frac{1}{c}K_{\lambda}(h_{0},y)D(K_{T}(e,\cdot))(h_{0})dh_{0},y\in G$.

Hence, we have derived a formula that let us to recover the kernel $K_{T}$ (resp. $D(K_{T}(e,\cdot))(\cdot)$) from $K_{T}(e,\cdot)$ (resp. $D(K_{T}(e,\cdot))(\cdot)$) restricted to $H_{0}$!

Remark 4.15*.*

We notice,

[TABLE]

Since we are assuming $\tau_{|_{L}}$ is irreducible, we have $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is irreducible, hence, Lemma 3.6 let us to obtain that a scalar multiple of $r_{0}^{\star}r_{0}$ is the orthogonal projector onto the irreducible factor $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$.

Whence, the orthogonal projector onto $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is given by $\frac{d(\pi_{\eta_{0}}^{H_{0}})}{d(\pi_{\lambda})\dim W}\,\,r_{0}^{\star}r_{0}$.

Thus, the kernel $K_{\lambda,\eta_{0}}$ of the orthogonal projector onto $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$ is

xxxxxx $K_{\lambda,\eta_{0}}(x,y):=\frac{d(\pi_{\eta_{0}}^{H_{0}})}{d(\pi_{\lambda})\dim W}\int_{H_{0}}K_{\lambda}(p,y)K_{\lambda}(x,p)dp$.

Doing $H:=H_{0}$ we obtain a similar result for the kernel of the orthogonal projector onto $\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W)$.

The equality $(r_{0}r_{0}^{\star})_{|_{H^{2}(H_{0},\tau)}}=cI_{H^{2}(H_{0},\tau)}$ yields the first claim in:

Proposition 4.16**.**

*Assume $res_{L}(\tau)$ is irreducible. Then,

a) for every $g\in H^{2}(H_{0},\tau_{|_{L}})$ (resp. $g\in H^{2}(H,\tau_{|_{L}})$), the function $r_{0}^{\star}(g)$ (resp. $r^{\star}(g)$), is an extension of a scalar multiple of $g$.

b) The kernel $K_{\lambda}^{G}$ is a extension of a scalar multiple of $K_{\tau_{|_{L}}}^{H}$.*

When we restrict holomorphic Discrete Series, this fact naturally happens, see [22], [25, Example 10.1] and references therein.

Proof.

Let $r:H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\tau}W)$ the restriction map. The duality $H,H_{0}$, and Theorem 3.1 applied to $H:=H_{0}$ implies $H^{2}(H,\tau)=r(\mathrm{Cl}(\mathcal{U}(\mathfrak{h})W))$, as well as that there exists, up to a constant, a unique $T\in Hom_{H}(H^{2}(H,\tau),H^{2}(G,\tau))\equiv Hom_{L}(W,H^{2}(H_{0},\tau))\equiv\mathbb{C}$. It follows from the proof of Proposition 4.12, that, up to a constant, $T=r^{\star}$ restricted to $H^{2}(H,\tau)$. After we apply the equality $T(K_{\mu}^{H}(\cdot,e)^{\star}z)(x)=K_{T}(e,x)z$, (see [24]), we obtain,

xxxxxxxxx $r^{\star}(K_{\mu}^{H}(\cdot,e)^{\star}z)(y)=K_{\lambda}(y,e)^{\star}z$.

Also, Schur’s lemma implies $rr^{\star}$ restricted to $H^{2}(H,\tau)$ is a constant times the identity map. Thus, for $h\in H$, we have $rr^{\star}(K_{\mu}^{H}(\cdot,e)^{\star}w)(h)=qK_{\mu}^{H}(h,e)^{\star}w$. For the value of $q$ see 4.2.2. Putting together, we obtain,

xxcccxxxx $K_{\lambda}(h,e)^{\star}z=r(K_{\lambda}(\cdot,e)^{\star}z)(h)=qK_{\mu}^{H}(h,e)^{\star}z$.

Whence, for $h,h_{1}\in H$ we have

$K_{\lambda}(h_{1},h)^{\star}z=K_{\lambda}(h^{-1}h_{1},e)^{\star}z=qK_{\mu}^{H}(h^{-1}h_{1},e)^{\star}z=qK_{\mu}^{H}(h_{1},h)^{\star}z$

as we have claimed. ∎

By the same token, after we set $H:=H_{0}$ we obtain:

For $res_{L}(\tau)$ irreducible, $(\sigma,Z)=(res_{L}(\tau),W)$, and $V_{\eta_{0}}^{H_{0}}=H^{2}(H_{0},\sigma)$, the kernel $K_{\lambda}$ extends a scalar multiple of $K_{\eta_{0}}^{H_{0}}$. Actually, $r_{0}(K_{\lambda}(\cdot,e)^{\star}w)=cK_{\eta_{0}}^{H_{0}}(\cdot,e)^{\star}w$.

Remark 4.17*.*

We would like to point out that the equality

xxxxxxxxxxxxx $r^{\star}(K_{\mu}^{H}(\cdot,e)^{\star}(z))(y)=K_{\lambda}(y,e)^{\star}z$

implies $res_{H}(\pi_{\lambda})$ is $H$-algebraically discretely decomposable. Indeed, we apply a Theorem shown by Kobayashi [16, Lemma 1.5], the Theorem says that when $(V_{\lambda}^{G})_{K-fin}$ contains an irreducible $(\mathfrak{h},L)$ irreducible submodule, then $V_{\lambda}^{G}$ is discretely decomposable. We know $K_{\lambda}(y,e)^{\star}z$ is a $K$-finite vector, the equality implies $K_{\lambda}(y,e)^{\star}z$ is $\mathfrak{z}(\mathcal{U}(\mathfrak{h}))$-finite. Hence, owing to Harish-Chandra [32, Corollary 3.4.7 and Theorem 4.2.1], $H^{2}(G,\tau)_{K-fin}$ contains a nontrivial irreducible $(\mathfrak{h},L)$-module and the fact shown by Kobayashi applies.

4.2.2. Value of $b=d=c$ when $res_{L}(\tau)$ is irreducible

We show $b=d=d(\pi_{\lambda})\dim W/d(\pi_{\eta_{0}}^{H_{0}})=c$. In fact, the constant $b,d$ satisfies $(r_{0}^{\star}r_{0})_{\mathcal{U}(\mathfrak{h}_{0})W}=dI_{\mathcal{U}(\mathfrak{h}_{0})W}$, $(r_{0}r_{0}^{\star})_{|_{H^{2}(H_{0},\tau)}}=bI_{H^{2}(H_{0},\tau)}$. Now, it readily follows $b=d$. To evaluate $r_{0}^{\star}r_{0}$ at $K_{\lambda}(\cdot,e)^{\star}w$, for $h_{1}\in H_{0}$ we compute, for $h_{1}\in H_{0}$,

$bK_{\lambda}(h_{1},e)^{\star}w=r_{0}^{\star}r_{0}(K_{\lambda}(\cdot,e)^{\star}w)(h_{1})$ $=\int_{H_{0}}K_{\lambda}(h_{0},h_{1})K_{\lambda}(h_{0},e)^{\star}dh_{0}w$

nnnnnnnnnnnnnnnnnnnnnnnnnnn $=d(\pi_{\lambda})^{2}\int_{H_{0}}\Phi(h_{1}^{-1}h_{0})\Phi(h_{0})^{\star}dh_{0}w$.

Here, $\Phi$ is the spherical function attached to the lowest $K$-type of $\pi_{\lambda}$. Since, we are assuming $res_{L}(\tau)$ is a irreducible representation, we have $\mathcal{U}(\mathfrak{h}_{0})W$ is a irreducible $(\mathfrak{h}_{0},L)$-module and it is equivalent to the underlying Harish-Chandra module for $H^{2}(H_{0},res_{L}(\tau))$. Thus, the restriction of $\Phi$ to $H_{0}$ is the spherical function attached to the lowest $L$-type of the irreducible square integrable representation $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)\equiv H^{2}(H_{0},res_{L}(\tau))$. We fix a orthonormal basis $\{w_{i}\}$ for $\mathcal{U}(\mathfrak{h}_{0})W[W]$. We recall,

$\Phi(x)w=P_{W}\pi(x)P_{W}w=\sum_{1\leq i\leq\dim W}(\pi(x)w,w_{i})_{L^{2}}w_{i}$,

$\Phi(x^{-1})=\Phi(x)^{\star}$.

For $h_{1}\in H_{0}$, we compute, to justify steps we appeal to the invariance of Haar measure and to the orthogonality relations for matrix coefficients of irreducible square integrable representations and we recall $d(\pi_{\eta_{0}}^{H_{0}})$ denotes the formal degree for $H^{2}(H_{0},res_{L}(\tau))$.

[TABLE]

Thus,

[TABLE]

The functions $K_{\lambda}(\cdot,e)^{\star}w,r_{0}^{\star}r_{0}(K_{\lambda}(\cdot,e)^{\star}w)(\cdot)$ belong to $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$, the injectivity of $r_{0}$ on $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)$, forces, for every $x\in G$

xxxxxxx $r_{0}^{\star}r_{0}(K_{\lambda}(\cdot,e)^{\star}w)(x)=d(\pi_{\lambda})\dim W/d(\pi_{\eta_{0}}^{H_{0}})K_{\lambda}(x,e)^{\star}w$.

Hence, we have computed $b=d=c$.

4.2.3. Analysis of $r_{0}^{D}$ for arbitrary $(\tau,W),(\sigma,Z)$

We recall the decomposition $W=\sum_{\nu_{2}^{\prime}\in Spec_{L\cap K_{2}}(\pi_{\Lambda_{2}}^{K_{2}})}W[\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}^{\prime}}^{L\cap K_{2}}]$.

A consequence of Proposition 4.9 is that $r_{0}^{\star}$ maps $\mathbf{H}^{2}(H_{0},W[\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}}])$ into $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W[\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}}])$. In consequence, $r_{0}r_{0}^{\star}$ restricted to $\mathbf{H}^{2}(H_{0},W[\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}}])$ is a bijective $H_{0}$-endomorphism $C_{j}$. Hence, the inverse map of $r_{0}$ restricted to $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W[\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}}])$ is $r_{0}^{\star}C_{j}^{-1}$. Since, $H^{2}(H_{0},\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}})$ has a unique lowest $L$-type, we conclude $C_{j}$ is determined by an element of $Hom_{L}(\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}},H^{2}(H_{0},\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}})[\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}}])$. Since for $D\in\mathcal{U}(\mathfrak{h}_{0}),w\in W$ we have  $C_{j}(L_{D}w)=L_{D}C_{j}(w)$, we obtain $C_{j}$ is a zero order differential operator on the underlying Harish-Chandra module of $H^{2}(H_{0},\pi_{\Lambda_{1}}^{K_{1}}\boxtimes\pi_{\nu_{2}}^{L\cap K_{2}})$. Summing up, we have that the inverse to $r_{0}:\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)\rightarrow\mathbf{H}^{2}(H_{0},\tau)$ is the function $r_{0}^{\star}(\oplus_{j}C_{j}^{-1})$.

For $T\in Hom_{H}(H^{2}(H,\sigma),H^{2}(G,\tau))$ and $T_{0}\in Hom_{L}(Z,\mathbf{H}^{2}(H_{0},\tau))$ so that $r_{0}^{D}(T)=T_{0}$ we obtain the equalities

[TABLE]

[TABLE]

When $D$ is the identity the formula simplifies as the one in the second Corollary to Proposition 4.12.

4.2.4. Eigenvalues of $r_{0}^{\star}r_{0}$

For general case, we recall $r_{0}^{\star}r_{0}$ intertwines the action of $H_{0}$. Moreover, Proposition 4.9 and its Corollary gives that for each $L$-isotypic component $Z_{1}\subseteq W$ of $res_{L}(\tau)$, we have $\mathcal{U}(\mathfrak{h}_{0})W[Z_{1}]=Z_{1}$. Thus, each isotypic component of $res_{L}((\mathcal{U}(\mathfrak{h}_{0})W)[W])$ is invariant by $r_{0}^{\star}r_{0}$, in consequence, $r_{0}^{\star}r_{0}$ leaves invariant the subspace $"W"=H^{2}(G,\tau)[W]=\{K_{\lambda}(\cdot,e)^{\star}w,w\in W\}$. Since, $Ker(r_{0})=(\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W))^{\perp}$, we have $r_{0}^{\star}r_{0}$ is determined by the values it takes on $"W"$. Now, we assume $res_{L}(\tau)$ is a multiplicity free representation, we write $Z_{1}^{\perp}=Z_{2}\oplus\dots\oplus Z_{q}$, where $Z_{j}$ are $L$-invariant and $L$-irreducible. Thus, Proposition 4.9 implies $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)=\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{1})\oplus\cdots\oplus\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{q})$. This a orthogonal decomposition, each summand is irreducible and no irreducible factor is equivalent to other. For $1\leq i\leq q$, let $\eta_{i}$ denote the Harish-Chandra parameter for $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i})$.

Proposition 4.18**.**

When $res_{L}(\tau)$ is a multiplicity free representation, the linear operator $r_{0}^{\star}r_{0}$ on $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i})$ is equal to $\frac{d(\pi_{\lambda})\dim Z_{i}}{d(\pi_{\eta_{i}}^{H_{0}})}$ times the identity map.

Proof.

For the subspace $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})W)[W]$, we choose a $L^{2}(G)$-orthonormal basis $\{w_{j}\}_{1\leq j\leq\dim W}$ equal to the union of respective $L^{2}(G)$-orthonormal basis for $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i})[Z_{i}]$. Next, we compute and freely make use of the notation in 4.2.2. Owing to our multiplicity free hypothesis, we have that $r_{0}^{\star}r_{0}$ restricted to $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i})$ is equal to a constant $d_{i}$ times the identity map. Hence, on $w\in\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i})[Z_{i}]$ we have $d_{i}w=d(\pi_{\lambda})^{2}\int_{H_{0}}\Phi(h_{0})\Phi(h_{0})^{\star}wdh_{0}$.

Now, $\Phi(h_{0})=(a_{ij})=((\pi_{\lambda}(h_{0})w_{j},w_{i})_{L^{2}(G)})$, Hence, the $pq$-coefficient of the product $\Phi(h_{0})\Phi(h_{0})^{\star}$ is equal to

$\sum_{1\leq j\leq\dim W}(\pi_{\lambda}(h_{0})w_{j},w_{p})_{L^{2}(G)}\overline{(\pi_{\lambda}(h_{0})w_{j},w_{q})}_{L^{2}(G)}$.

Let $I_{i}$ denote the set of indexes $j$ so that $w_{j}\in Z_{i}$. Thus, $\{1,\dots,\dim W\}$ is equal to the disjoint union $\cup_{1\leq i\leq q}I_{i}$. A consequence of Proposition 4.9 is the $L^{2}(G)$-orthogonality of the subspaces $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{j})$, hence, for $t\in I_{a},q\in I_{d}$ and $a\not=d$ we have $(\pi_{\lambda}(h_{0})w_{q},w_{t})_{L^{2}(G)}=0$. Therefore, the previous observation and the disjointness of the sets $I_{r}$, let us obtain that for $i\not=d,p\in I_{i},q\in I_{d}$ each summand in

$\sum_{1\leq j\leq\dim W}\int_{H_{0}}(\pi_{\lambda}(h_{0})w_{j},w_{p})_{L^{2}(G)}\overline{(\pi_{\lambda}(h_{0})w_{j},w_{q})}_{L^{2}(G)}dh_{0}$

is equal to zero.

For $p,q\in I_{i}$, we apply the previous computation and the orthogonality relations to the irreducible representation $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i})$. We obtain

$\sum_{1\leq j\leq\dim W}\int_{H_{0}}(\pi_{\lambda}(h_{0})w_{j},w_{p})_{L^{2}(G)}\overline{(\pi_{\lambda}(h_{0})w_{j},w_{q})}_{L^{2}(G)}dh_{0}$

$=\sum_{j\in I_{i}}\int_{H_{0}}(\pi_{\lambda}(h_{0})w_{j},w_{p})_{L^{2}(G)}\overline{(\pi_{\lambda}(h_{0})w_{j},w_{q})}_{L^{2}(G)}dh_{0}$

$=\sum_{j\in I_{i}}\frac{1}{d(\pi_{\eta_{i}}^{H_{0}})}(w_{j},w_{j})_{L^{2}(G)}(w_{q},w_{p})_{L^{2}(G)}=\frac{\dim Z_{i}}{d(\pi_{\eta_{i}}^{H_{0}})}\delta_{pq}$.

Thus, we have shown Proposition 4.18. ∎

Remark 4.19*.*

Even, when $res_{L}(\tau)$ is not multiplicity free, the conclusion in Proposition 4.18 holds. In fact, let us denote the $L$-isotypic component of $res_{L}(\tau)$ again by $Z_{i}$. Now, the proof goes as the one for Proposition 4.18 till we need to compute

$=\sum_{j\in I_{i}}\int_{H_{0}}(\pi_{\lambda}(h_{0})w_{j},w_{p})_{L^{2}(G)}\overline{(\pi_{\lambda}(h_{0})w_{j},w_{q})}_{L^{2}(G)}dh_{0}$.

For this, we decompose $"Z_{i}"=\sum_{s}Z_{i,s}$ as a $L^{2}(G)$-orthogonal sum of irreducible $L$-modules and we choose the orthonormal basis for $"Z_{i}"$ as a union of orthonormal basis for each $Z_{i,s}$. Then, we have the $L^{2}(G)$-orthogonal decomposition $\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i})=\sum_{s}\mathrm{Cl}(\mathcal{U}(\mathfrak{h}_{0})Z_{i,s})$. Then, the proof follows as in the case $res_{L}(\tau)$ is multiplicity free.

5. Examples

We present three type of examples. The first is: Multiplicity free representations. A simple consequence of the duality theorem is that it readily follows examples of symmetric pair $(G,H)$ and square integrable representation $\pi_{\lambda}^{G}$ so that $res_{H}(\pi_{\lambda})$ is $H$-admissible and the multiplicity of each irreducible factor is one. This is equivalent to determinate when the representation $res_{L}(\mathbf{H}^{2}(H_{0},\tau))$ is multiplicity free. The second is: Explicit examples. Here, we compute the Harish-Chandra parameters of the irreducible factors for some $res_{H}(H^{2}(G,\tau))$. The third is: Existence of representations so that its lowest $K$-types restricted to $L$ is a irreducible representation.

In order to present the examples we need information on certain families of representations.

5.1. Multiplicity free representations

In this paragraph we generalize work of T. Kobayashi and his coworkers in the setting of Hermitian symmetric spaces and holomorphic Discrete Series.

Before we present the examples, we would like to comment.

a) Assume a Discrete Series $\pi_{\lambda}$ has an admissible restriction to a subgroup $H$. Then, any Discrete Series $\pi_{\lambda^{\prime}}$ for $\lambda^{\prime}$ dominant with respect to $\Psi_{\lambda}$ is $H$-admissible [15].

b) If $res_{H}(\pi_{\lambda})$ is $H$-admissible and a multiplicity free representation. Then the restriction to $L$ of the lowest $K$-type for $\pi_{\lambda}$ is multiplicity free. This follows from the duality theorem.

c) In the next paragraphs we will list families $\mathcal{F}$ of Harish-Chandra parameters of Discrete Series for $G$ so that each representation in the family has a multiplicity free restriction to $H$. We find that it may happen that $\mathcal{F}$ is the whole set of Harish-Chandra parameters on a Weyl chamber or $\mathcal{F}$ is a proper subset of a Weyl Chamber. Information on $\mathcal{F}$ for holomorphic reprentations is in [18], [19].

d) Every irreducible $(\mathfrak{g},K)$-module for either $\mathfrak{g}\equiv\mathfrak{su}(n,1)$ or $\mathfrak{g}\equiv\mathfrak{so}(n,1)$, restricted to $K$, is a multiplicity free representation.

5.1.1. Holomorphic representations

For $G$ so that $G/K$ is a Hermitian symmetric space, it has been shown by Harish-Chandra that $G$ admits Discrete Series representations with one dimensional lowest $K$-type. For this paragraph we further assume that the smooth imbedding $H/L\rightarrow G/K$ is holomorphic, equivalently the center of $K$ is contained in $L$, and $\pi_{\lambda}$ is a holomorphic representation. Under this hypothesis, it was shown by Kobayashi [17] that a holomorphic Discrete Series for $G$ has a multiplicity free restriction to the subgroup $H$ whenever it is a scalar holomorphic Discrete Series. Moreover, in [17, Theorem 8.8] computes the Harish-Chandra parameter of each irreducible factor. Also, from the work of Kobayashi and Nakahama we find a description of the restriction to $H$ of a arbitrary holomorphic Discrete Series representations. As a consequence, we find restrictions which are not multiplicity free.

In [19] we find a complete list of the pairs $(\mathfrak{g},\mathfrak{h})$ so that $H/L\rightarrow G/K$ is a holomorphic embedding. From the list in [17], it can be constructed the list bellow.

Also, Theorem 3.1 let us verify that the following pairs $(\mathfrak{g},\mathfrak{h})$ are so that $res_{H}(\pi_{\lambda})$ is multiplicity free for any holomorphic $\pi_{\lambda}$. For this, we list the associated $\mathfrak{h}_{0}$.

$(\mathfrak{su}(m,n),\mathfrak{s}(\mathfrak{u}(m-1,n)+\mathfrak{u}(1)))$, $\mathfrak{h}_{0}=\mathfrak{su}(1,n)+\mathfrak{su}(m-1)+\mathfrak{u}(1)$.

$(\mathfrak{su}(m,n),\mathfrak{s}(\mathfrak{u}(m,n-1)+\mathfrak{u}(1)))$, $\mathfrak{h}_{0}=\mathfrak{su}(n-1)+\mathfrak{su}(m,1)+\mathfrak{u}(1)$.

The list is correct, owing to any Discrete Series for $SU(n,1)$ restricted to $K$ is a multiplicity free representation.

5.1.2. Quaternionic real forms, quaternionic representations

In [9], the authors considered and classified quaternionic real forms, and also, they made a careful study of quaternionic representations. To follow we bring out the essential facts for us. From [9] we read that the list of Lie algebra of quaternionic groups is: $\mathfrak{su}(2,n)$, $\mathfrak{so}(4,n)$, $\mathfrak{sp}(1,n)$, $\mathfrak{e}_{6(2)}$, $\mathfrak{e}_{7(-5)}$, $\mathfrak{e}_{8(-24)}$, $\mathfrak{f}_{4(4)}$, $\mathfrak{g}_{2(2)}$. For each quaternionic real form $G$, there exists a system of positive roots $\Psi\subset\Phi(\mathfrak{g},\mathfrak{t})$ so that the maximal root $\alpha_{max}$ in $\Psi$ is compact, $\alpha_{max}$ is orthogonal to all compact simple roots and $\alpha_{max}$ is not orthogonal to each noncompact simple roots. Hence, $\mathfrak{k}_{1}(\Psi)\equiv\mathfrak{su}_{2}(\alpha_{max})$. The system $\Psi$ is not unique. We appel such a system of positive roots a quaternionic system.

Let us recall that a quaternionic representation is a Discrete Series for a quaternionic real form $G$ so that its Harish-Chandra parameter is dominant with respect to a quaternionic system of positive roots, and so that its lowest $K$-type is equivalent to a irreducible representation for $K_{1}(\Psi)$ times the trivial representation for $K_{2}$. A fact shown in [9] is: Given a quaternionic system of positive roots, for all but finitely many representations $(\tau,W)$ equivalent to the tensor of a nontrivial representation for $K_{1}(\Psi)$ times the trivial representation of $K_{2}$, it holds: $\tau$ is the lowest $K$-type of a quaternionic (unique) irreducible square integrable representation $H^{2}(G,\tau)$. We define a generalized quaternionic representation to be a Discrete Series representation $\pi_{\lambda}$ so that its Harish-Chandra parameter is dominant with respect to a quaternionic system of positive roots.

From Table 1,2 we readily read the pairs $(\mathfrak{g},\mathfrak{h})$ so that $\mathfrak{g}$ is a quaternionic Lie algebra and hence, we have a list of generalized quaternionic representations of $G$ with admissible restriction to $H$.

Let $(G,H)$ denote a symmetric pair so that a quaternionic representation $(\pi_{\lambda},H^{2}(G,\tau))$ is $H$-admissible. Then, from [30] [5] [4] we have: $\mathfrak{k}_{1}(\Psi_{\lambda})\equiv\mathfrak{su}_{2}(\alpha_{max})\subset\mathfrak{l}$ and $\pi_{\lambda}$ is $L$-admissible. In consequence, [16], $\pi_{\lambda}$ is $H_{0}$-admissible. By definition, for a quaternionic representation $\pi_{\lambda}$, we have $\tau_{|_{L}}$ is irreducible, hence, $\mathbf{H}^{2}(H_{0},\tau)$ is irreducible. Moreover, after checking on [30] or Tables 1,2, the list of systems $\Psi_{H_{0},\lambda}$, it follows that $H^{2}(H_{0},\tau)$ is again a quaternionic representation. Finally, in order to present a list of quaternionic representations with multiplicity free restriction to $H$ we recall that it follows from the duality Theorem that $res_{H}(H^{2}(G,\tau))$ is multiplicity free if and only if $res_{L}(H^{2}(H_{0},\tau))$ is a multiplicity free representation, and that on [9, Page 88] it is shown that a quaternionic representation for $H_{0}$ is $L$-multiplicity free if and only if $\mathfrak{h}_{0}=\mathfrak{sp}(n,1),n\geq 1$.

To follow, we list pairs $(\mathfrak{g},\mathfrak{h})$ where multiplicity free restriction holds for all quaternionic representations.

$(\mathfrak{su}(2,2n),\mathfrak{sp}(1,n))$, $\mathfrak{h}_{0}=\mathfrak{sp}(1,n)$, $n\geq 1$.

$(\mathfrak{so}(4,n),\mathfrak{so}(4,n-1))$, $\mathfrak{h}_{0}=\mathfrak{so}(4,1)+\mathfrak{so}(n-1)$ ($n$ even or odd).

$(\mathfrak{sp}(1,n),\mathfrak{sp}(1,k)+\mathfrak{sp}(n-k))$, $\mathfrak{h}_{0}=\mathfrak{sp}(1,n-k)+\mathfrak{sp}(k)$.

$(\mathfrak{f}_{4(4)},\mathfrak{so}(5,4))$, $\mathfrak{h}_{0}=\mathfrak{sp}(1,2)\oplus\mathfrak{su}(2)$.

$(\mathfrak{e}_{6(2)},\mathfrak{f}_{4(4)})$, $\mathfrak{h}_{0}=\mathfrak{sp}(3,1)$.

A special pair is:

$(\mathfrak{su}(2,2),\mathfrak{sp}(1,1)),$ $\mathfrak{h}_{0}=\mathfrak{sp}(1,1)$.

Here, multiplicity free holds for any $\pi_{\lambda}$ so that $\lambda$ is dominant with respect to a system of positive roots that defines a quaternionic structure on $G/K$. For details see [30, Table 2] or Explicit example II.

5.1.3. More examples of multiplicity free restriction

Next, we list pairs $(\mathfrak{g},\mathfrak{h})$ and systems of positive roots $\Psi\subset\Phi(\mathfrak{g},\mathfrak{t})$ so that $\pi_{\lambda^{\prime}}$ is $H$- admissible and multiplicity free for every element $\lambda^{\prime}$ dominant with respect to $\Psi$. We follow either Table 1,2,3 or [19]. For each $(\mathfrak{g},\mathfrak{h})$ we list the corresponding $\mathfrak{h}_{0}$.

$(\mathfrak{su}(m,n),\mathfrak{su}(m,n-1)+\mathfrak{u}(1))$, $\Psi_{a},\tilde{\Psi}_{b}$,

xxxxxxxxxxxxxxxxxxxx $\mathfrak{h}_{0}=\mathfrak{su}(m,1)+\mathfrak{su}(n-1)+\mathfrak{u}(1)$.

$(\mathfrak{so}(2m,2n+1),\mathfrak{so}(2m,2n))$, $\Psi_{\pm}$, $\mathfrak{h}_{0}=\mathfrak{so}(2m,1)+\mathfrak{so}(2n)$.

$(\mathfrak{so}(2m,2),\mathfrak{so}(2m,1))$, $\Psi_{\pm}$, $\mathfrak{h}_{0}=\mathfrak{so}(2m,1)$.

$(\mathfrak{so}(2m,2n),\mathfrak{so}(2m,2n-1)),n>1$, $\Psi_{\pm}$, $\mathfrak{h}_{0}=\mathfrak{so}(2m,1)+\mathfrak{so}(2n-1)$.

5.2. Explicit examples

5.2.1. Quaternionic representations for $Sp(1,b)$

For further use we present a intrinsic description for the $Sp(1)\times Sp(b)$-types of a quaternionic representation for $Sp(1,b)$, a proof of the statements is in [8]. The quaternionic representations for $Sp(1,b)$ are the representations of lowest $Sp(1)\times Sp(b)$-type $S^{n}(\mathbb{C}^{2})\boxtimes\mathbb{C},n\geq 1$. We label the simple roots for the quaternionic system of positive roots $\Psi$ as in [9], $\beta_{1},\dots,\beta_{b+1}$, the long root is $\beta_{b+1}$, $\beta_{1}$ is adjacent to just one simple root and the maximal root $\beta_{max}$ is adjacent to $-\beta_{1}$. Let $\Lambda_{1},\dots,\Lambda_{d+1}$ the associated fundamental weights. Thus, $\Lambda_{1}=\frac{\beta_{max}}{2}$. Let $\tilde{\Lambda}_{1},\dots,\tilde{\Lambda}_{b}$ denote the fundamental weights for $"\Psi\cap\Phi(\mathfrak{sp}(b))"$. The irreducible $L=Sp(1)\times Sp(b)$-factors of

xxx $H^{2}(Sp(1,b),\pi_{n\frac{\beta_{max}}{2}}^{Sp(1)}\boxtimes\pi_{\rho_{Sp(b)}}^{Sp(b)})=H^{2}(Sp(1,b),S^{n-1}(\mathbb{C}^{2})\boxtimes\mathbb{C})$

are

$\{S^{n-1+m}(\mathbb{C}^{2})\boxtimes S^{m}(\mathbb{C}^{2b})\equiv\pi_{(n+m)\frac{\beta_{max}}{2}}^{Sp(1)}\boxtimes\pi_{m\tilde{\Lambda}_{1}+\rho_{Sp(b)}}^{Sp(b)},\,m\geq 0\}$.

The multiplicity of each $L$-type in $H^{2}(Sp(1,b),S^{n-1}(\mathbb{C}^{2})\boxtimes\mathbb{C})$ is one.

5.2.2. Explicit example I

We develop this example in detail. We restrict quaternionic representations for $Sp(1,d)$ to $Sp(1,k)\times Sp(d-k)$. For this, we need to review definitions and facts in [8][19] [30]. The group $G:=Sp(1,d)$ is a subgroup of $GL(\mathbb{C}^{2+2d})$. A maximal compact subgroup of $Sp(1,d)$ is the usual immersion of $Sp(1)\times Sp(d)$. Actually, $Sp(1,d)$ is a quaternionic real form for $Sp(\mathbb{C}^{1+d})$. $Sp(1,d)$ has a compact Cartan subgroup $T$ and there exists a orthogonal basis $\delta,\epsilon_{1},\dots,\epsilon_{d}$ for $i\mathfrak{t}^{\star}$ so that

$\Phi(\mathfrak{sp}(d+1,\mathbb{C}),\mathfrak{t})=\{\pm 2\delta,\pm 2\epsilon_{1},\dots,\pm 2\epsilon_{d},\pm(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq d,\pm(\delta\pm\epsilon_{s}),1\leq s\leq d\}$.

We fix $1\leq k<d$. We consider the usual immersion of $H:=Sp(1,k)\times Sp(d-k)$ into $Sp(1,d)$. Thus,

$\Phi(\mathfrak{h},\mathfrak{t}):=\{\pm 2\delta,\pm 2\epsilon_{1},\dots,\pm 2\epsilon_{d},\pm(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq k,\,\\ \phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxc}\text{or},\,k+1\leq i<j\leq d,\pm(\delta\pm\epsilon_{s}),1\leq s\leq k\}$.

Then, $H_{0}$ is isomorphic to $Sp(1,d-k)\times Sp(k)$. We have

$\Phi(\mathfrak{h}_{0},\mathfrak{t}):=\{\pm 2\delta,\pm 2\epsilon_{1},\dots,\pm 2\epsilon_{d},\pm(\epsilon_{i}\pm\epsilon_{j}),k+1\leq i<j\leq d,\,\\ \phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxc}\text{or},\,1\leq i<j\leq k,\pm(\delta\pm\epsilon_{s}),k+1\leq s\leq d\}$.

From now on, we fix the quaternionic system of positive roots

$\Psi:=\{2\delta,2\epsilon_{1},\dots,2\epsilon_{d},(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq d,(\delta\pm\epsilon_{s}),1\leq s\leq d\}$.

Then, $\alpha_{max}=2\delta$, $\rho_{n}^{\Psi}=d\delta$. The Harish-Chandra parameter $\lambda$ of a quaternionic representation $\pi_{\lambda}$ is dominant with respect to $\Psi$. Hence, $\Psi_{\lambda}=\Psi$. The systems in Theorem 3.10 are $\Psi_{H,\lambda}=\Phi(\mathfrak{h},\mathfrak{t})\cap\Psi$, $\Psi_{H_{0},\lambda}=\Phi(\mathfrak{h}_{0},\mathfrak{t})\cap\Psi$. Also, [5], $\Phi(\mathfrak{k}_{1}:=\mathfrak{k}_{1}(\Psi),\mathfrak{t}_{1}:=\mathfrak{t}\cap\mathfrak{k}_{1})=\{\pm 2\delta\}$, $\Phi(\mathfrak{k}_{2}:=\mathfrak{k}_{2}(\Psi),\mathfrak{t}_{2}:=\mathfrak{t}\cap\mathfrak{k}_{2})=\{\pm 2\epsilon_{1},\dots,\pm 2\epsilon_{d},\pm(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq d\}$. Thus, $K_{1}(\Psi)\equiv SU_{2}(2\delta)\equiv Sp(1)\subset H$, $K_{2}\equiv Sp(d)$. Hence, for a Harish-Chandra parameter $\lambda=(\lambda_{1},\lambda_{2}),\lambda_{j}\in i\mathfrak{t}_{j}^{\star}$ dominant with respect to $\Psi$, the representation $\pi_{\lambda}$ is $H$-admissible.

The lowest $K$-type of a generalized quaternionic representation $\pi_{\lambda}$ is the representation $\tau=\pi_{\lambda+\rho_{n}^{\lambda}}^{K}=\pi_{\lambda_{1}+d\delta}^{K_{1}}\boxtimes\pi_{\lambda_{2}}^{K_{2}}$. Since, $\rho_{K_{2}}=d\epsilon_{1}+(d-1)\epsilon_{2}+\dots+\epsilon_{d}$, for $n\geq d+1$, the functional $\mathfrak{t}^{\star}\ni\lambda_{n}:=n\delta+\rho_{K_{2}}$ is a Harish-Chandra parameter dominant with respect to $\Psi$ and the lowest $K$-type $\tau_{n}$ of $\pi_{\lambda_{n}}$ is $\pi_{(n+d)\delta}^{K_{1}}\boxtimes\pi_{\rho_{K_{2}}}^{K_{2}}$. That is, $\pi_{\lambda_{n}+\rho_{n}^{\lambda}}^{K}$ is equal to a irreducible representation of $K_{1}\equiv Sp(1)=SU_{2}(2\delta)$ times the trivial representation of $K_{2}\equiv Sp(d)$. The family $(\pi_{\lambda_{n}})_{n}$ exhausts, up to equivalence, the set of quaternionic representations for $Sp(1,d)$. Now, $\mathbf{H}^{2}(H_{0},\tau_{n})$ is the irreducible representation of lowest $L$-type equal to the irreducible representation $\pi_{(n+d)\delta}^{K_{1}}$ of $K_{1}$ times the trivial representation of $K_{2}\cap L$. Actually, $\mathbf{H}^{2}(H_{0},\pi_{(n+d)\delta}^{K_{1}}\boxtimes\pi_{\rho_{K_{2}}}^{K_{2}})$ is a realization of the quaternionic representation $H^{2}(Sp(1,n-k),\pi_{(n+d)\delta}^{Sp(1)}\boxtimes\pi_{\rho_{Sp(n-k)}}^{Sp(n-k)})$ for $Sp(1,d-k)$ times the trivial representation of $Sp(k)$. In [8, Proposition 6.3] it is shown that the representation $H^{2}(Sp(1,n-k),\pi_{(n+d)\delta}^{Sp(1)}\boxtimes\pi_{\rho_{Sp(n-k)}}^{Sp(n-k)})$ restricted to $L$ is a multiplicity free representation, and also, they list the highest weights of the totality of $L$-irreducible factors. To follow we explicit such a computation. For this we recall 5.2.1 and notice $b=d-k$; $\Lambda_{1}=\delta$, $\beta_{max}=2\delta$, $\tilde{\Lambda}_{1}=\epsilon_{1}$; as $Sp(1)$-module, $S^{p}(\mathbb{C}^{2})\equiv\pi_{(p+1)\delta}^{SU_{2}(2\delta)}$; for $p\geq 1$, as $Sp(p)$-module $S^{m}(\mathbb{C}^{2p})\equiv\pi_{m\epsilon_{1}+\rho_{Sp(p)}}^{Sp(p)}$. Then,

the irreducible $L=Sp(1)\times Sp(d-k)\times Sp(k)$-factors of

$\mathbf{H}^{2}(H_{0},\pi_{(n+d)\delta}^{K_{1}}\boxtimes\pi_{\rho_{K_{2}}}^{K_{2}})\equiv H^{2}(Sp(1,d-k),\pi_{(n+d)\delta}^{Sp(1)}\boxtimes\pi_{\rho_{Sp(d-k)}}^{Sp(d-k)})\boxtimes\mathbb{C}$

are multiplicity free and it is the set of inequivalent representations

$\{S^{n+d-1+m}(\mathbb{C}^{2})\boxtimes S^{m}(\mathbb{C}^{2(d-k)})\boxtimes\mathbb{C}\\ \phantom{xxxxxxxxxxxxxxxxx}\equiv\pi_{(n+d+m)\delta}^{Sp(1)}\boxtimes\pi_{m\epsilon_{k+1}+\rho_{Sp(d-k)}}^{Sp(d-k)}\boxtimes\pi_{\rho_{Sp(k)}}^{Sp(k)},\,m\geq 0\}$.

Here, $\rho_{Sp(d-k)}=(d-k)\epsilon_{k+1}+(d-k-1)\epsilon_{k+2}+\dots+\epsilon_{d}$ and $\rho_{Sp(k)}=k\epsilon_{1}+(k-1)\epsilon_{2}+\dots+\epsilon_{k}$.

We compute $\Psi_{H,\lambda}=\{2\delta,2\epsilon_{1},\dots,2\epsilon_{d},(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq k\,\text{or}\,k+1\leq i<j\leq d,(\delta\pm\epsilon_{s}),1\leq s\leq k\}$. $\rho_{n}^{\mu}=\rho_{n}^{H}=k\delta$. Now, from Theorem 3.1 we have $Spec_{H}(\pi_{\lambda})+\rho_{n}^{H}=Spec_{L}(\mathbf{H}^{2}(H_{0},\tau))$, whence, we conclude:

The representation $res_{Sp(1,k)\times Sp(d-k)}(\pi_{\lambda_{n}}^{Sp(1,d)})$ is a multiplicity free representation and the totality of Harish-Chandra parameters of the $Sp(1,k)\times Sp(d-k)$-irreducible factors is the set

$\{(n+d+m)\delta+m\epsilon_{k+1}+\rho_{Sp(k)}+\rho_{Sp(d-k)}\}-\rho_{n}^{H}=\\ \phantom{xxx}(n+d+m-k)\delta+m\epsilon_{k+1}+(d-k)\epsilon_{k+1}+\dots+\epsilon_{d}+k\epsilon_{1}+\dots+\epsilon_{k},m\geq 0\}$.

Whence, $res_{Sp(1,k)\times Sp(d-k)}(\pi_{\lambda_{n}}^{Sp(1,d)})$ is equivalent to the Hilbert sum

$\oplus_{m\geq 0}V_{(n+d+m-k)\delta+m\epsilon_{k+1}+\rho_{Sp(k)}+\rho_{Sp(d-k)}}^{Sp(1,k)\times Sp(d-k)}\\ \phantom{xxx}\equiv\oplus_{m\geq 0}H^{2}(Sp(1,k)\times Sp(d-k),\pi_{(n+d+m)\delta+m\epsilon_{k+1}+\rho_{Sp(k)}+\rho_{Sp(d-k)}}^{Sp(1)\times Sp(k)\times Sp(d-k)})$.

A awkward point of our decomposition is that does not provide an explicit description of the $H$-isotypic components for $res_{H}(V_{\lambda}^{G})$.

5.2.3. Explicit example II

We restrict from $Spin(2m,2),m\geq 2$, to $Spin(2m,1)$. We notice the isomorphism between $(Spin(4,2),Spin(4,1))$ and the pair $(SU(2,2),Sp(1,1))$. In this setting $K=Spin(2m)\times Z_{K}$, $L=Spin(2m)$, $Z_{K}\equiv\mathbb{T}$. Obviously, we may conclude that any irreducible representation of $K$ is irreducible when restricted to $L$. In this case $H_{0}\equiv Spin(2m,1)$, and (for $m=2$, $H_{0}\equiv Sp(1,1)$) and $\mathbf{H}^{2}(H_{0},\tau)$ is irreducible. Therefore, the duality theorem together with that any irreducible representation for $Spin(2m,1)$ is $L=Spin(2m)$-multiplicity free [29, page 11], we obtain:

Any $Spin(2m,1)$-admissible representation $\pi_{\lambda}^{Spin(2m,2)}$ is multiplicity free.

For $(Spin(2m,2),Spin(2m,1))$ in [30, Table 2 ], [19] it is verified that any $\pi_{\lambda}$, with $\lambda$ dominant with respect to one of the systems $\Psi_{\pm}$ (see proof of 5) has admissible restriction to $Spin(2m,1)$ and no other $\pi_{\lambda}$ has admissible restriction to $Spin(2m,1)$.

In [30, Table 2 ] [15] [16] it is verified that any square integrable representation $\pi_{\lambda}$ with $\lambda$ dominant with respect to a quaternionic system for $SU(2,2)$, has admissible restriction to $Sp(1,1)$. As in 5.2.2, we may compute the Harish-Chandra parameters for the irreducible components of $res_{Sp(1,1)}(\pi_{\lambda}^{SU(2,2)})$.

5.2.4. Explicit example III

$(\mathfrak{e}_{6(2)},\mathfrak{f}_{4(4)}).$ We fix a compact Cartan subgroup $T\subset K$ so that $U:=T\cap H$ is a compact Cartan subgroup of $L=K\cap H$. Then, there exist a quaternionic and Borel de Siebenthal positive root system $\Psi_{BS}$ for $\Phi(\mathfrak{e}_{6},\mathfrak{t})$ so that, after we write the simple roots as in Bourbaki (see [8][30]), the compact simple roots are $\alpha_{1},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6}$ (They determinate the $A_{5}$-Dynkin sub-diagram) and $\alpha_{2}$ is noncompact. $\alpha_{2}$ is adjacent to $-\alpha_{max}$ and to $\alpha_{4}$. In [30], it is verified $\Psi_{BS}$ is the unique system of positive roots such that $\mathfrak{k}_{1}(\Psi_{BS})=\mathfrak{su}_{2}(\alpha_{max}).$

The automorphism $\sigma$ of $\mathfrak{g}$ acts on the simple roots as follows

[TABLE]

Hence, $\sigma(\Psi_{BS})=\Psi_{BS}.$ Let $h_{2}\in i\mathfrak{t}^{\star}$ be so that $\alpha_{j}(h_{2})=\delta_{j2}$ for $j=1,\dots,6.$ Then, $h_{2}=\frac{2H_{\alpha_{max}}}{(\alpha_{max},\alpha_{max})}$ and $\theta=Ad(exp(\pi ih_{2})).$ A straightforward computation yields: $\mathfrak{k}\equiv\mathfrak{su}_{2}(\alpha_{max})+\mathfrak{su}(6)$, $\mathfrak{l}\equiv\mathfrak{su}_{2}(\alpha_{max})+\mathfrak{sp}(3)$; the fix point subalgebra for $\theta\sigma$ is isomorphic to $\mathfrak{sp}(1,3)$. Thus, the pair $(\mathfrak{e}_{6(2)},\mathfrak{sp}(1,3))$ is the associated pair to $(\mathfrak{e}_{6(2)},\mathfrak{f}_{4(4)}).$ Let $q_{\mathfrak{u}}$ denote the restriction map from $\mathfrak{t}^{\star}$ into $\mathfrak{u}^{\star}$. Then, for $\lambda$ dominant with respect to $\Psi_{BS}$, the simple roots for $\Psi_{H,\lambda}=\Psi_{\mathfrak{f}_{4(4)},\lambda}$, respectively $\Psi_{\mathfrak{sp}(1,3),\lambda},$ are:

$\alpha_{2},\,\,\alpha_{4},\,\,q_{\mathfrak{u}}(\alpha_{3})=q_{\mathfrak{u}}(\alpha_{5}),\,\,q_{\mathfrak{u}}(\alpha_{1})=q_{\mathfrak{u}}(\alpha_{6}).$

$\beta_{1}=q_{\mathfrak{u}}(\alpha_{2}+\alpha_{4}+\alpha_{5})=q_{\mathfrak{u}}(\alpha_{2}+\alpha_{4}+\alpha_{3}),\,\,\beta_{2}=q_{\mathfrak{u}}(\alpha_{1})=q_{\mathfrak{u}}(\alpha_{6})$,

$\,\,\beta_{3}=q_{\mathfrak{u}}(\alpha_{3})=q_{\mathfrak{u}}(\alpha_{5}),\,\,\beta_{4}=\alpha_{4}.$

The fundamental weight $\tilde{\Lambda}_{1}$ associated to $\beta_{1}$ is equal to $\frac{1}{2}\beta_{max}$. Hence, $\tilde{\Lambda}_{1}=\beta_{1}+\beta_{2}+\beta_{3}+\frac{1}{2}\beta_{4}=\alpha_{2}+\frac{3}{2}\alpha_{4}+\alpha_{3}+\alpha_{5}+\frac{1}{2}(\alpha_{1}+\alpha_{6})$.

Thus, from the Duality Theorem, for the quaternionic representation

xxxxxxxxxxxxxxxxxxxx$H^{2}(E_{6(2)},\pi_{n\frac{\alpha_{max}}{2}+\rho_{SU(6)}}^{SU_{2}(\alpha_{max})\times SU(6)})$

the set of Harish-Chandra parameters of the irreducible $F_{4(4)}$-factors is equal to:

$-\rho_{n}^{H}$* plus the set of infinitesimal characters of the $L\equiv SU_{2}(\alpha_{max})\times Sp(3)$-irreducible factors for*

xxxxxxxxx $res_{SU_{2}(\alpha_{max})\times Sp(3)}(H^{2}(Sp(1,3),\pi_{n\frac{\alpha_{max}}{2}+\rho_{Sp(3)}}^{SU_{2}(\alpha_{max})\times Sp(3)}))$.

Here, $-\rho_{n}^{H}=-d_{H}\frac{\alpha_{max}}{2}$, $d_{H}=d_{\mathfrak{f}_{4(4)}}=7$ (see [9]).

Therefore, from the computation in 5.2.1, we obtain:

[TABLE]

Here, $\rho_{Sp(3)}=3\beta_{2}+5\beta_{3}+3\beta_{4}=\frac{3}{2}(\alpha_{1}+\alpha_{6})+\frac{5}{2}(\alpha_{3}+\alpha_{5})+3\alpha_{4}$.

5.2.5. Comments on admissible restriction of quaternionic representations

As usual $(G,H)$ is a symmetric pair, $H$ is not a compact group and $(\pi_{\lambda},H^{2}(G,\tau))$ is a $H$-admissible, non-holomorphic, square integrable representation. We further assume $G/K$ as well as $H/L$ holds a quaternionic structure and the inclusion $H/L\hookrightarrow G/K$ respects the respective quaternionic structures. Then, from Tables 1,2,3 it follows:

a) $\lambda$ is dominant with respect to a quaternionic system of positive roots. That is, $\pi_{\lambda}$ is a generalized quaternionic representation.

b) $H_{0}/L$ has a quaternionic structure.

c) Each system $\Psi_{H,\lambda}$, $\Psi_{H_{0},\lambda}$ is a quaternionic system.

d) The representation $\mathbf{H}^{2}(H_{0},\tau)$ is a sum of generalized quaternionic representations.

e) When $\pi_{\lambda}$ is quaternionic, then the representation $\mathbf{H}^{2}(H_{0},\tau)$ is equal to $H^{2}(H_{0},res_{L}(\tau))$, hence, it is quaternionic. Moreover, in [8] is computed the highest weight and the respective multiplicity of each of its $L$-irreducible factors.

f) Thus, the duality Theorem 3.1 together with a)—e) let us compute the Harish-Chandra parameters of the irreducible $H$-factors for a quaternionic representation $\pi_{\lambda}$. Actually, the computation of the Harish-Chandra parameters is quite similar to the computation in Explicit example I, Explicit example III.

In the following paragraph we consider particular quaternionic symmetric pairs. One pair is $(\mathfrak{f}_{4(4)},\mathfrak{so}(5,4))$. Here, $\mathfrak{h}_{0}\equiv\mathfrak{sp}(1,2)+\mathfrak{su}(2)$. Thus, for any Harish-Chandra parameter $\lambda$ dominant with respect to the quaternionic system of positive roots, we have $\pi_{\lambda}$ restricted to $SO(5,4)$ is an admissible representation and the Duality theorem allows us compute either multiplicities or Harish-Chandra parameters of the restriction. Moreover, since quaternionic Discrete Series for $Sp(1,2)\times SU(2)$ are multiplicity free, [8], we have that quaternionic Discrete Series for $\mathfrak{f}_{4(4)}$, restricted to $SO(5,4)$ are multiplicity free. It seems that it can be deduced from the branching rules for the pair $(Sp(3),Sp(1)\times Sp(2))$ that a generalized quaternionic representation, $res_{SO(5,4)}(\pi_{\lambda})$ is multiplicity free if and only $\pi_{\lambda}$ is quaternionic.

For the pair $(\mathfrak{f}_{4(4)},\mathfrak{so}(5,4))$, if we attempt to deduce our decomposition result from the work of [9], we have to consider the group of Lie algebra $\mathfrak{g}^{d}\equiv\mathfrak{f}_{4(-20)}$, its maximal compactly embedded subalgebra is isomorphic to $\mathfrak{so}(9)$, a simple Lie algebra, hence no Discrete Series for $G^{d}$ has an admissible restriction to $H_{0}$ (see [18] [5]). Thus, it is not clear to us how to deduce our Duality result from the Duality Theorem in [8].

For the pairs $(\mathfrak{e}_{6(2)},\mathfrak{so}(6,4)+\mathfrak{so}(2)),$ $(\mathfrak{e}_{7(-5)},\mathfrak{e}_{6(2)}+\mathfrak{so}(2))$, for each $G$, generalized quaternionic representations do exist and they are $H$-admissible. For these pairs, the respective $\mathfrak{h}_{0}$ are: $\mathfrak{su}(2,4)+\mathfrak{su}(2),\mathfrak{su}(6,2)$. In these two cases, the Maple soft developed by Silva-Vergne[2], allows to compute the $L$-Harish-Chandra parameters and respective multiplicity for each Discrete Series for $H_{0}\equiv SU(p,q)\times SU(r)$, hence, the duality formula yields the Harish-Chandra Parameters for $res_{H}(\pi_{\lambda})$ and their multiplicity.

5.2.6. Explicit example IV

The pair $(SO(2m,n),SO(2m,n-1))$. This pair is considered in [9]. We recall their result and we sketch how to derive the result from our duality Theorem. We only consider the case $\mathfrak{g}=\mathfrak{so}(2m,2n+1)$. Here, $\mathfrak{k}=\mathfrak{so}(2m)+\mathfrak{so}(2n+1)$, $\mathfrak{h}=\mathfrak{so}(2m,2n)$, $\mathfrak{h}_{0}=\mathfrak{so}(2m,1)+\mathfrak{so}(2n)$, $\mathfrak{l}=\mathfrak{so}(2m)+\mathfrak{so}(2n)$. We fix a Cartan subalgebra $\mathfrak{t}\subset\mathfrak{l}\subset\mathfrak{k}$. Then, there exists a orthogonal basis $\epsilon_{1},\dots,\epsilon_{m},\delta_{1},\dots,\delta_{n}$ for $i\mathfrak{t}^{\star}$ so that

$\Delta=\{(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq m,(\delta_{r}\pm\delta_{s}),1\leq r<s\leq n\}\cup\{\delta_{j}\}_{1\leq j\leq n}.$

[TABLE]

The systems of positive roots $\Psi_{\lambda}$ so that $\pi_{\lambda}^{G}$ is an admissible representation of $H$ are the systems $\Psi_{\pm}$ associated to the lexicographic orders $\epsilon_{1}>\dots>\epsilon_{m}>\delta_{1}>\dots>\delta_{n}$, $\epsilon_{1}>\dots>\epsilon_{m-1}>-\epsilon_{m}>\delta_{1}>\dots>\delta_{n-1}>-\delta_{n}.$ Here, for $m\geq 3$, $\mathfrak{k}_{1}(\Psi_{\pm})=\mathfrak{so}(2m).$ For $m=2,\mathfrak{k}_{1}(\Psi_{\pm})=\mathfrak{su}_{2}(\epsilon_{1}\pm\epsilon_{2}).$ Then,

$\Psi_{H,+}=\{(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq m,(\delta_{r}\pm\delta_{s}),1\leq r<s\leq n\}\cup\{(\epsilon_{r}\pm\delta_{s}),r=1,\dots,m,s=1,\dots,n\}$,

$\Psi_{H_{0},+}=\{(\epsilon_{i}\pm\epsilon_{j}),1\leq i<j\leq m,(\delta_{r}\pm\delta_{s}),1\leq r<s\leq n\}\cup\{\epsilon_{j},j=1,\dots,m\}.$

$\mathfrak{g}^{d}=\mathfrak{so}(2m+2n,1)$. Thus, from either our duality Theorem or from [9], we infer that whenever $res_{H}(\pi_{\lambda})$ is $H$-admissible, then, $res_{H}(\pi_{\lambda})$ is a multiplicity free representation. Whence, we are left to compute the Harish-Chandra parameters for $res_{SO(2m,2n)}(H^{2}(SO(2m,2n+1),\pi_{\Lambda_{1}}^{SO(2m)}\boxtimes\pi_{\Lambda_{2}}^{SO(2n+1)}))$. For this, according to the duality Theorem, we have to compute the infinitesimal characters of each irreducible factor of the underlying $L$-module in

[TABLE]

. The branching rules for $res_{SO(2m)}(H^{2}(SO(2m,1),\pi_{\Lambda_{1}}^{SO(2m)}))$ are found in [29] and other references, the branching rule for $res_{SO(2n)}(\pi_{\Lambda_{2}}^{SO(2n+1)})$ can be found in [29]. From both computations, we deduce: [9, Proposition 3], for $\lambda=\sum_{1\leq i\leq m}\lambda_{i}\epsilon_{i}+\sum_{1\leq j\leq n}\lambda_{m+j}\delta_{j}$, then $V_{\mu}^{H}$ is a $H$-subrepresentation of $H^{2}(G,\tau)\equiv V_{\lambda}^{SO(2m,2n+1)}$ ($\mu=\sum_{1\leq i\leq m}\mu_{i}\epsilon_{i}+\sum_{1\leq j\leq n}\mu_{j+m}\delta_{j}$) if and only if

xxxx$\mu_{1}>\lambda_{1}>\dots>\mu_{m}>\lambda_{m},\lambda_{m+1}>\mu_{m+1}>\dots\lambda_{m+n}>|\mu_{m+n}|.$

5.3. Existence of Discrete Series whose lowest $K$-type restricted to $K_{1}(\Psi)$ is irreducible

Let $G$ a semisimple Lie group that admits square integrable representations. This hypothesis allows to fix a compact Cartan subgroup $T\subset K$ of $G$. In [5] it is defined for each system of positive roots $\Psi\subset\Phi(\mathfrak{g},\mathfrak{t})$ a normal subgroup $K_{1}(\Psi)\subset K$ so that for a symmetric pair $(G,H)$, with $H$ a $\theta$-invariant subgroup, it holds: for any Harish-Chandra parameter dominant with respect to $\Psi$, the representation $res_{H}(\pi_{\lambda})$ is $H$-admissible if and only if $K_{1}(\Psi)$ is a subgroup of $H$. For a holomorphic system $\Psi$, $K_{1}(\Psi)$ is equal to the center of $K$; for a quaternionic system of positive roots $K_{1}(\Psi)\equiv SU_{2}(\alpha_{max})$. Either for the holomorphic family or for a quaternionic real forms we find that among the $H$-admissible Discrete Series for $G$, there are many examples of the following nature: the lowest $K$-type of $\pi_{\lambda}$ is equal to a irreducible representation of $K_{1}(\Psi)$ tensor with the trivial representation for $K_{2}$, [9]. To follow, under the general setting at the beginning of this paragraph, we verify.

5.3.1.

For each system of positive roots $\Psi\subset\Phi(\mathfrak{g},\mathfrak{t})$, there exist Discrete Series with Harish-Chandra parameter dominant with respect to $\Psi$ and so that its lowest $K$-type is equal to a irreducible representation of $K_{1}(\Psi)$ tensor with the trivial representation for $K_{2}(\Psi)$.

We may assume $K_{1}(\Psi)$ is a proper subgroup of $K$. Then, when $K_{1}(\Psi)=Z_{K}$, Harish-Chandra showed there exists such a representation. For $G$ a quaternionic real form, $\Psi$ a quaternionic system of positive roots, $K_{1}(\Psi)=SU_{2}(\alpha_{max})$, then, in [9] we find a proof of the statement. From the tables in [5][30], we are left to consider the triples $(G,K,K_{1}(\Psi))$ so that their respective Lie algebras are in the triples

$(\mathfrak{su}(m,n),\mathfrak{su}(m)+\mathfrak{su}(n)+\mathfrak{u}(1),\mathfrak{su}(m)),m>2$,

$(\mathfrak{sp}(m,n),\mathfrak{sp}(m)+\mathfrak{sp}(n),\mathfrak{sp}(m))$.

$(\mathfrak{so}(2m,n),\mathfrak{so}(2m)+\mathfrak{so}(n),\mathfrak{so}(2m))$.

We analyze the second triple of the list. To follow, $G$ is so that its Lie algebra is $\mathfrak{sp}(m,n)$, $n\geq 2,m>1$. We want to produce Discrete Series representations so that the lowest $K$-type restricted to $K_{1}(\Psi)$ is still irreducible. Here, $\mathfrak{k}=\mathfrak{sp}(m)+\mathfrak{sp}(n)$. We fix maximal torus $T\subset K$ and describe the root system as in [30]. For the system of positive roots $\Psi:=\{\epsilon_{i}\pm\epsilon_{j},i<j,\delta_{r}\pm\delta_{s},r<s,\epsilon_{a}\pm\delta_{b},2\epsilon_{a},2\delta_{b},1\leq a,i,j\leq m,1\leq b,r,s\leq n\}$, we have $K_{1}(\Psi)=K_{1}\equiv Sp(m),K_{2}(\Psi)=K_{2}\equiv Sp(n)$. Obviously, there exists a system of positive roots $\tilde{\Psi}$ so that $K_{1}(\tilde{\Psi})\equiv Sp(n),K_{2}(\tilde{\Psi})\equiv Sp(m)$. For any other system of positive roots in $\Phi(\mathfrak{g},\mathfrak{t})$ we have that the associated subgroup $K_{1}$ is equal to $K$. It readily follows that $\lambda:=\sum_{1\leq j\leq m}a_{j}\epsilon_{j}+\rho_{K_{2}}$ is a $\Psi$-dominant Harish-Chandra parameter when the coefficients $a_{j}$ are all integers so that $a_{1}>\dots>a_{m}>>0$. Since $\rho_{n}^{\lambda}$ belongs to $span_{\mathbb{C}}\{\epsilon_{1},\dots,\epsilon_{m}\}$, it follows that the lowest $K$ type of $\pi_{\lambda}$ is equivalent to a irreducible representation for $Sp(m)$ times the trivial representation for $Sp(n)$. With the same proof it is verified that the statement holds for the third triple. For the first triple, we further assume $G=SU(p,q)$. Thus, $K$ is the product of two simply connected subgroups times a one dimensional torus $Z_{K}$, we notice $\rho_{n}^{\Psi_{a}}=\rho_{\mathfrak{g}}^{\lambda}-\rho_{K}$, hence, $\rho_{n}^{\Psi_{a}}$ lifts to a character of $K$. Thus, as in the case $\mathfrak{sp}(m,n)$, we obtain $\pi_{\lambda}$ with $\lambda$ dominant with respect to $\Psi_{a}$ so that its lowest $K=SU(p)SU(q)Z_{K}$-type is the tensor product of a irreducible representation for $SU(p)Z_{K}$ times the trivial representation for $SU(q)$. Since $\rho_{n}^{\Psi_{a}}$ lifts to a character of $K$, after some computation the claim follows.

6. Symmetry breaking operators and normal derivatives

For this subsection $(G,H)$ is a symmetric pair and $\pi_{\lambda}$ is a square integrable representation. Our aim is to generalize a result in [22, Theorem 5.1]. In [20] are considered symmetry breaking operators expressed by means of normal derivatives, they obtain results for holomorphic embedding of a rank one symmetric pairs. As before, $H_{0}=G^{\sigma\theta}$ is the associated subgroup. We recall $\mathfrak{h}\cap\mathfrak{p}$ is orthogonal to $\mathfrak{h}_{0}\cap\mathfrak{p}$ and that $\mathfrak{h}\cap\mathfrak{p}\equiv T_{eL}(H/L)$, $\mathfrak{h}_{0}\cap\mathfrak{p}\equiv T_{eL}(H_{0}/L)$. Hence, for $X\in\mathfrak{h}_{0}\cap\mathfrak{p}$, more generally for $X\in\mathcal{U}(\mathfrak{h}_{0})$, we say $R_{X}$ is a normal derivative to $H/L$ differential operator. For short, normal derivative. Other ingredient necessary for the next Proposition are the subspaces $\mathcal{L}_{\lambda}$ and $\mathcal{U}(\mathfrak{h}_{0})W$. The latter subspace is contained in the subspace of $K$-finite vectors, whereas, according to a well known conjecture, whenever $res_{H}(\pi_{\lambda})$ is not discretely decomposable, the former subspace is disjoint to the subspace of $G$-smooth vectors. When, $res_{H}(\pi_{\lambda})$ is $H$-admissible $\mathcal{L}_{\lambda}$ is contained in the subspace of $K$-finite vectors. However, it might not be equal to $\mathcal{U}(\mathfrak{h}_{0})W$ as we have pointed out. The next Proposition and its converse, dealt with consequences of the equality $\mathcal{L}_{\lambda}=\mathcal{U}(\mathfrak{h}_{0})W$.

Proposition 6.1**.**

We assume $(G,H)$ is a symmetric pair. We also assume there exists a irreducible representation $(\sigma,Z)$ of $L$ so that $H^{2}(H,\sigma)$ is a irreducible factor of $H^{2}(G,\tau)$ and $H^{2}(G,\tau)[H^{2}(H,\sigma)][Z]=\mathcal{L}_{\lambda}[Z]=\mathcal{U}(\mathfrak{h}_{0})W[Z]=L_{\mathcal{U}(\mathfrak{h}_{0})}(H^{2}(G,\tau)[W])[Z]$. Then, $res_{H}(\pi_{\lambda})$ is $H$-admissible. Moreover, any symmetry breaking operator from $H^{2}(G,\tau)$ into $H^{2}(H,\sigma)$ is represented by a normal derivative differential operator.

We show a converse to Proposition 6.1 in 6.0.1.

Proof.

We begin by recalling that $H^{2}(G,\tau)[W]=\{K_{\lambda}(\cdot,e)^{\star}w,w\in W\}$ is a subspace of $H^{2}(G,\tau)_{K-fin}$, hence $L_{\mathcal{U}(\mathfrak{h}_{0})}(H^{2}(G,\tau)[W])[Z]$ is a subspace of $H^{2}(G,\tau)_{K-fin}$. Owing to our hypothesis we then have $\mathcal{L}_{\lambda}[Z]$ is a subspace of $H^{2}(G,\tau)_{K-fin}$. Next, we quote a result of Harish-Chandra: a $\mathcal{U}(\mathfrak{h})-$finitely generated, $\mathfrak{z}(\mathcal{U}(\mathfrak{h}))-$finite, module has a finite composition series. Thus, $H^{2}(G,\tau)_{K-fin}$ contains an irreducible $(\mathfrak{h},L)$-submodule. For a proof (cf. [32, Corollary 3.4.7 and Theorem 4.2.1]). Now, in [15, Lemma 1.5] we find a proof of: if a $(\mathfrak{g},K)-$module contains an irreducible $(\mathfrak{h},L)-$submodule, then the $(\mathfrak{g},K)-$module is $\mathfrak{h}-$algebraically decomposable. Thus, $res_{H}(\pi_{\lambda})$ is algebraically discretely decomposable. In [16, Theorem 4.2], it is shown that under the hypothesis $(G,H)$ is a symmetric pair, for Discrete Series, $\mathfrak{h}$-algebraically discrete decomposable is equivalent to $H$-admissibility, hence $res_{H}(\pi_{\lambda})$ is $H$-admissible. Let $S:H^{2}(G,\tau)\rightarrow H^{2}(H,\sigma)=V_{\mu}^{H}$ a continuous intertwining linear map. Then, we have shown in 3.1, for $z\in Z$, $K_{S}(\cdot,e)^{\star}z\in H^{2}(G,\tau)[V_{\mu}^{H}][Z]$. We fix a orthonormal basis $\{z_{p}\},\,p=1,\dots,\dim Z$ for $Z$. The hypothesis

ccccccccc $H^{2}(G,\tau)[V_{\mu}^{H}][Z]=L_{\mathcal{U}(\mathfrak{h}_{0})}(H^{2}(G,\tau)[W])[Z]$

implies for each $p$, there exists $D_{p}\in\mathcal{U}(\mathfrak{h}_{0})$ and $w_{p}\in W$ so that $K_{S}(\cdot,e)^{\star}z_{p}=L_{D_{p}}K_{\lambda}(\cdot,e)^{\star}w_{p}$. Next, we fix $f_{1}\in H^{2}(G,\tau)^{\infty},h\in H$ and set $f:=L_{h^{-1}}(f_{1})$, then $f(e)=f_{1}(h)$ and we recall $D^{\star}$ (resp. $\check{D}$) is the formal adjoint of $D\in\mathcal{U}(\mathfrak{g})$, (resp. is the image under the anti homomorphism of $\mathcal{U}(\mathfrak{g})$ that extends minus the identity of $\mathfrak{g}$). We have,

[TABLE]

Thus, for each $z\in Z$ and $f_{1}$ smooth vector we obtain

$(S(f_{1})(h),z)_{Z}=\sum_{p}(S(f_{1})(h),(z,z_{p})_{Z}z_{p})_{Z}=\sum_{p}(R_{\check{D}_{p}^{\star}}f_{1}(h),w_{p})_{W}(z_{p},z)_{Z}$.

As in [25, Proof of Lemma 2] we conclude that for any $f\in H^{2}(G,\tau)$

[TABLE]

Since $D_{p}\in\mathcal{U}(\mathfrak{h}_{0})$ such a expression of $S(f)$ is an representation in terms of normal derivatives. ∎

6.0.1. Converse to Proposition 6.1

We want to show: If every element in $Hom_{H}(H^{2}(G,\tau),H^{2}(H,\sigma))$ has a expression as differential operator by means of "normal derivatives", then, the equality $\mathcal{L}_{\lambda}[Z]=H^{2}(G,\tau)[H^{2}(H,\sigma)][Z]=\mathcal{U}(\mathfrak{h}_{0})W[Z]$ holds.

In fact, the hypothesis $S(f)(h)=\sum_{1\leq p\leq\dim Z}(R_{\check{D}_{p}^{\star}}f(h),w_{p})_{W}\,\,z_{p}$, $D_{p}\in\mathcal{U}(\mathfrak{h}_{0})$, yields $K_{S}(\cdot,e)^{\star}z=L_{D_{z}}K_{\lambda}(\cdot,e)^{\star}w_{z}$, $D_{z}\in\mathcal{U}(\mathfrak{h}_{0})$, $w_{z}\in W$. The fact that $(\sigma,Z)$ has multiplicity one in $H^{2}(H,\sigma)$ gives

xx $\dim Hom_{H}(H^{2}(G,\tau),H^{2}(H,\sigma))=\dim Hom_{L}(Z,H^{2}(G,\tau)[H^{2}(H,\sigma)][Z])$.

Hence, the functions

xxxxxx $\{K_{S}(\cdot,e)^{\star}z,z\in Z,S\in Hom_{H}(H^{2}(G,\tau),H^{2}(H,\sigma))\}$

span $H^{2}(G,\tau)[H^{2}(H,\sigma)][Z]$. Therefore, $H^{2}(G,\tau)[H^{2}(H,\sigma)][Z]$ is contained in $\mathcal{U}(\mathfrak{h}_{0})W[Z]=L_{\mathcal{U}(\mathfrak{h}_{0})}H^{2}(G,\tau)[W][Z]$. Owing to Theorem 3.1, both spaces have the same dimension, whence, the equality holds.

The pairs so that Proposition 6.1777In work in progress we have shown the Proposition holds for $(\mathfrak{sp}(m,1),\mathfrak{sp}(m-1,1)+\mathfrak{sp}(1))$ and a quaternionic representation. holds for scalar holomorphic Discrete Series are $(\mathfrak{su}(m,n),\mathfrak{su}(m,l)+\mathfrak{su}(n-l)+\mathfrak{u}(1)),$ $(\mathfrak{so}(2m,2),\mathfrak{u}(m,1)),$ $(\mathfrak{so}^{\star}(2n),\mathfrak{u}(1,n-1)),$ $(\mathfrak{so}^{\star}(2n),\mathfrak{so}(2)+\mathfrak{so}^{\star}(2n-2)),$ $(\mathfrak{e}_{6(-14)},\mathfrak{so}(2,8)+\mathfrak{so}(2)).$ See [30, (4.6)].

6.0.2. Comments on the interplay among the subspaces, $\mathcal{L}_{\lambda}$, $\mathcal{U}(\mathfrak{h}_{0})W$, $H^{2}(G,\tau)_{K-fin}$ and symmetry breaking operators

It readily follows that the subspace $\mathcal{L}_{\lambda}[Z]=V_{\lambda}^{G}[H^{2}(H,\sigma)][Z]$ is equal to the closure of the linear span of

$\mathcal{K}_{Sy}(G,H):=\{K_{S^{\star}}(e,\cdot)z=K_{S}(\cdot,e)^{\star}z,z\in Z,S\in Hom_{H}(V_{\lambda}^{G},V_{\mu}^{H})\}$.

(1)  $H^{2}(G,\tau)_{K-fin}\cap\mathcal{L}_{\lambda}[Z]$ is equal to the linear span of the elements in $\mathcal{K}_{Sy}(G,H)$ so that the corresponding symmetry breaking operator is represented by a differential operator. See [25, Lemma 4.2].

(2)   $\mathcal{U}(\mathfrak{h}_{0})W\cap\mathcal{L}_{\lambda}[Z]$ is equal to the linear span corresponding to the elements $K_{S}$ in $\mathcal{K}_{Sy}(G,H)$ so that $S$ is represented by normal derivative differential, operator. This is shown in Proposition 6.1 and its converse.

(3) The set of symmetry breaking operators represented by a differential operator is not the null space if and only if $res_{H}(\pi_{\lambda})$ is $H$-discretely decomposable. See [25, Theorem 4.3] and the proof of Proposition 6.1.

(4) We believe that from Nakahama’s thesis, it is possible to construct examples of $V_{\lambda}^{G}[H^{2}(H,\sigma)][Z]\cap\mathcal{U}(\mathfrak{h}_{0})W[Z]\not=\{0\}$, so that the equality $V_{\lambda}^{G}[H^{2}(H,\sigma)][Z]=\mathcal{U}(\mathfrak{h}_{0})W[Z]$ does not hold! That is, there are symmetry breaking operators represented by plain differential operators and some of the operators are not represented by normal derivative operators.

6.0.3. A functional equation for symmetry breaking operators

Notation is as in Theorem 3.1. We assume $(G,H)$ is a symmetric pair and $res_{H}(\pi_{\lambda})$ is admissible. The objects involved in the equation are: $H_{0}=G^{\sigma\theta}$, $Z=V_{\mu+\rho_{n}^{H}}^{L}$ the lowest $L$-type for $V_{\mu}^{H}$, $\mathcal{L}_{\lambda}=\sum_{\mu}H^{2}(G,\tau)[V_{\mu}^{H}][V_{\mu+\rho_{n}^{H}}^{L}]$, $\mathcal{U}(\mathfrak{h}_{0})W=L_{\mathcal{U}(\mathfrak{h}_{0})}H^{2}(G,\tau)[W]$, $L$-isomorphism $D:\mathcal{L}_{\lambda}[Z]\rightarrow\mathcal{U}(\mathfrak{h}_{0})W[Z]$, a $H$-equivariant continuous linear map $S:H^{2}(G,\tau)\rightarrow H^{2}(H,\sigma)$, the kernel $K_{S}:G\times H\rightarrow Hom_{\mathbb{C}}(W,Z)$ corresponding to $S$, 3.1 implies $K_{S}(\cdot,e)^{\star}z\in\mathcal{L}_{\lambda}[Z]$, finally, we recall $K_{\lambda}:G\times G\rightarrow Hom_{\mathbb{C}}(W,W)$ the kernel associated to the orthogonal projector onto $H^{2}(G,\tau)$. Then,

Proposition 6.2**.**

For $z\in Z$, $y\in G$ we have

[TABLE]

Here, $c=d(\pi_{\lambda})\dim W/d(\pi_{\eta_{0}}^{H_{0}}).$ When, $D$ is the identity map, the functional equation turns into

[TABLE]

The functional equation follows from Proposition 4.12 applied to $T:=S^{\star}$. The second equation follows after we compute the adjoint of the first equation.

We note, that as in the case of holographic operators, a symmetry breaking operator can be recovered from its restriction to $H_{0}$.

We also note that [22] has shown a different functional equation for $K_{S}$ for scalar holomorphic Discrete Series and holomorphic embedding $H/L\rightarrow G/K$.

7. Tables

For an arbitrary symmetric pair $(G,H),$ whenever $\pi_{\lambda}^{G}$ is an admissible representation of $H,$ we define,

[TABLE]

In the next tables we present the 5-tuple satisfying: $(G,H)$ is a symmetric pair, $H_{0}$ is the associated group to $H,$ $\Psi_{\lambda}$ is a system of positive roots such that $\pi_{\lambda}^{G}$ is an admissible representation of $H,$ and $K_{1}=Z_{1}(\Psi_{\lambda})K_{1}(\Psi_{\lambda}).$ Actually, instead of writing Lie groups we write their respective Lie algebras. Each table is in part a reproduction of tables in [18] [30]. The tables can also be computed by means of the techniques presented in [5]. Note that each table is "symmetric" when we replace $H$ by $H_{0}.$ As usual, $\alpha_{max}$ denotes the highest root in $\Psi_{\lambda}.$ Unexplained notation is as in [30].

[TABLE]

Table 1. Case $U=T,\Psi_{\lambda}$ non holomorphic.

[TABLE]

Table 2, Case $U\not=T,\Psi_{\lambda}$ non holomorphic.

[TABLE]

Table 3, $\pi_{\lambda}^{G}$ holomorphic Discrete Series.

The last two lines show the unique holomorphic pairs so that $U\not=T.$

8. Partial list of symbols and definitions

• $(\tau,W),$ $(\sigma,Z)$, $L^{2}(G\times_{\tau}W),L^{2}(H\times_{\sigma}Z)$ (cf. Section 2).

-$\,H^{2}(G,\tau)=V_{\lambda}=V_{\lambda}^{G}$, $H^{2}(H,\sigma)=V_{\mu}^{H},\pi_{\mu}^{H},\pi_{\nu}^{K}.$ (cf. Section 2).

-$\pi_{\lambda}=\pi_{\lambda}^{G}$, $d_{\lambda}=d(\pi_{\lambda})$ formal degree of $\pi_{\lambda},$ $P_{\lambda},P_{\mu},K_{\lambda},K_{\mu},$ (cf. Section 2).

-$P_{X}$ orthogonal projector onto subspace $X$.

-$\Phi(x)=P_{W}\pi(x)P_{W}$ spherical function attached to the lowest $K$-type $W$ of $\pi_{\lambda}$.

-$K_{\lambda}(y,x)=d(\pi_{\lambda})\Phi(x^{-1}y)$.

-$M_{K-fin}(resp.M^{\infty})$ $K-$finite vectors in $M$ (resp. smooth vectors in $M$).

-$dg,dh$ Haar measures on $G$, $H$.

-A unitary representation is square integrable, equivalently a Discrete Series representation, (resp. integrable) if some nonzero matrix coefficient is square integrable (resp. integrable) with respect to Haar measure on the group in question.

-$\Theta_{\pi_{\mu}^{H}}(...)$ Harish-Chandra character of the representation $\pi_{\mu}^{H}.$

-For a module $M$ and a simple submodule $N$, $M[N]$ denotes the isotypic component of $N$ in $M$. That is, $M[N]$ is the sum of all irreducible submodules isomorphic to $N.$ If topology is involved, we define $M[N]$ to be the closure of $M[N].$

-$M_{H-disc}$ is the closure of the linear subspace spanned by the totality of $H-$irreducible submodules. $M_{disc}:=M_{G-disc}$

-A representation $M$ is $H-$discretely decomposable if $M_{H-disc}=M.$

-A representation is $H-$admissible if it is $H-$discretely decomposable and each isotypic component is equal to a finite sum of $H-$irreducible representations.

-$\mathcal{U}(\mathfrak{g})$ (resp. $\mathfrak{z}(\mathcal{U}(\mathfrak{g}))=\mathfrak{z}_{\mathfrak{g}}$) universal enveloping algebra of the Lie algebra $\mathfrak{g}$(resp. center of universal enveloping algebra).

-$\mathrm{Cl}(X)=$closure of the set $X$.

-$I_{X}$ identity function on set $X$.

-$\mathbb{T}$ one dimensional torus.

-$Z_{S}$ identity connected component of the center of the group $S$.

$S^{(r)}(V)$ the $r^{th}$-symmetric power of the vector space $V$.