Branching problems in reproducing kernel spaces

TL;DR

This paper investigates how discrete series representations of semisimple Lie groups decompose when restricted to subgroups, linking discrete decomposability to differential operator representations and reproducing kernel conditions.

Contribution

It establishes new conditions for discrete decomposability of discrete series representations upon restriction, connecting classical and recent results with differential operator frameworks.

Findings

Proves discrete decomposability under Harish-Chandra's cusp form condition

Links decomposability to differential operator representations

Provides new branching laws for discrete series representations

Abstract

For a semisimple Lie group $G$ satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for discrete series when restricted to a subgroup $H$ of the same type by combining classical results with recent work of T. Kobayashi; in particular, we prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernel. We show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators.

Equations

$$\begin{gathered} L^2(G{\times }_{\tau }W):=\{f\in L^2(G)\otimes W: \\ f(gk)=\tau (k{)}^{-1}f(g),g\in G,k\in K\}. \end{gathered}$$

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

$$\begin{gathered} H^2(G,\tau )=\{f\in L^2(G)\otimes W:f(gk)=\tau (k{)}^{-1}f(g) \\ g\in G,k\in K,R_{{\Omega }_G}f=[(\lambda ,\lambda )-(\rho ,\rho )]f\}. \end{gathered}$$

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

$$(P_{\lambda }(f)(x),w{)}_W={\int }_G(f(y),K_{\lambda }(y,x{)}^{\star }w{)}_{W}dy,f\in L_2(G{\times }_{\tau }W).$$

$$\Vert f(x){\Vert }_W$$

$$\le ({\int }_G\Vert K_{\lambda }(y,x){\Vert }^{2}dy{)}^{\frac{1}{2}}({\int }_G\Vert f(y){\Vert }^{2}dy{)}^{\frac{1}{2}}$$

$$\le ({\int }_G\Vert K_{\lambda }(y,e){\Vert }^{2}dy{)}^{\frac{1}{2}}({\int }_G\Vert f(y){\Vert }^{2}dy{)}^{\frac{1}{2}}$$

$$=\Vert K_{\lambda }(\cdot ,e){\Vert }_{L^2(G)}\Vert f{\Vert }_2.$$

$$K_T:H\times G\to Ho{m}_{\mathbb{C}}(E,W)$$

$$(Tg(x),w{)}_W={\int }_H(g(h),K_T(h,x{)}^{\star }w{)}_{Z}dh.$$

$$Tg(x)={\int }_H{K}_T(h,x)g(h)dh,x\in G.$$

$$(T^{\star }f,g{)}_{L^2(H{\times }_{\nu }E)}={\int }_G{\int }_H(f(x),K_T(h,x)g(h){)}_{W}dhdx.$$

$$S(f)(h)=\sum _{a,b,c}{d}_{a,b,c}([R_{D_b}f](h),w_c{)}_W{z}_a\forall h\in H.$$

$$\begin{gathered} \sum _{a,b,c}{d}_{a,b,c}{z}_a\otimes D_b\otimes w_c \\ \mapsto (C^{\infty }(G{\times }_{\tau }W)\ni f\mapsto (h\mapsto \sum _{a,b,c}{d}_{a,b,c}([R_{D_b}f](h),w_c{)}_W{z}_a)). \end{gathered}$$

$$r_n:H^2(G,\tau )\to L^2(H{\times }_{{\tau }_n}Ho{m}_{\mathbb{C}}((\mathfrak{p}/{\mathfrak{p}}^{\prime }{)}^{(n)},W))$$

$$K_{r_n}:G\times H\to Ho{m}_{\mathbb{C}}(W,Ho{m}_{\mathbb{C}}((\mathfrak{p}/{\mathfrak{p}}^{\prime }{)}^{(n)},W))$$

$$K_{r_n}(y,h)(w,D)=R_{\lambda (D)}^{(2)}(K_{\lambda }(y,h)w).$$

$${\int }_G(f(y),K_S(y,e{)}^{\star }{z}_j{)}_{W}dy$$

$$=\sum d_{j,b,i}(L_{{\check{D}}_b}f(e),w_i{)}_W$$

$$=\sum _{b,i}{d}_{j,b,i}{\int }_G(L_{{\check{D}}_b}f(y),K_{\lambda }(y,e{)}^{\star }{w}_i{)}_{W}dy$$

$$=\sum _{b,i}{d}_{j,b,i}{\int }_G(f(y),L_{{\check{D}}_b^{\star }}^{(1)}{K}_{\lambda }(y,e{)}^{\star }{w}_i{)}_{W}dy.$$

$$\begin{gathered} S(f)(h^{-1})=S(L_{h}f)(e)=\sum _j(S(L_{h}f)(e),z_j)z_j \\ =\sum _j{\int }_G(L_{h}f(y),K_S(y,e{)}^{\star }{z}_j{)}_W{z}_j=\sum _j{\int }_G(L_{h}f(y),L_{D_j}{K}_{\lambda }(y,e{)}^{\star }w{)}_W{z}_j \\ =\sum _j{\int }_G(L_{D_j^{\star }}(L_{h}f)(y),K_{\lambda }(y,e{)}^{\star }w{)}_{W}dy{z}_j=\sum _j(L_{D_j^{\star }}(L_{h}f)(e),w{)}_W{z}_j \\ =\sum _j((R_{\check{D_j^{\star }}})(L_{h}f)(e),w{)}_W{z}_j \\ =\sum _j((R_{\check{D_j^{\star }}})(f)(h^{-1}),w{)}_W{z}_j. \end{gathered}$$

$$S(f)(h)=$$

$$=\sum _j{\int }_G(\tilde{f}(y),K_S(y,e{)}^{\star }{z}_j{)}_{W}dy{z}_j$$

$$=\sum _j{\int }_G(\tilde{f}(y),[L_{C_{j,i}}^{(1)}{K}_{\lambda }](y,e{)}^{\star }{w}_i{)}_{W}dy{z}_j$$

$$=\sum _j{\int }_G(L_{C_{j,i}^{\star }}\tilde{f}(y),K_{\lambda }(y,e{)}^{\star }{w}_i{)}_{W}dy{z}_j$$

$$=\sum _j([L_{C_{j,i}^{\star }}\tilde{f}](e),w_i{)}_W{z}_j=\sum _j([R_{\check{C_{j,i}^{\star }}}\tilde{f}](e),w_i{)}_W{z}_j$$

$$=\sum _j([R_{\check{C_{j,i}^{\star }}}(f)](h),w_i{)}_W{z}_j.$$

$$\begin{gathered} S(f)(h)=\underset{r}{\lim }S(f_r)(h)=\underset{r}{\lim }\sum _j([R_{D_{j,i}}{f}_r](h),w_i{)}_W{z}_j \\ =\sum _j([R_{D_{j,i}}\underset{r}{\lim }{f}_r](h),w_i{)}_W{z}_j=\sum _j([R_{D_{j,i}}f](h),w_i{)}_W{z}_j. \end{gathered}$$

$$\begin{gathered} {\int }_G(f(y),L_{D^{\star }}^{(1)}{K}_S(y,e{)}^{\star }z{)}_{W}dy={\int }_G(L_{D}f(y),K_S(y,e{)}^{\star }z{)}_{W}dy \\ =(S(L_{D}f)(e),z{)}_Z={\chi }_{\mu }(D)(S(f)(e),z{)}_Z \\ ={\chi }_{\mu }(D){\int }_G(f(y),K_S(y,e{)}^{\star }z{)}_{W}dy. \end{gathered}$$

$$\begin{gathered} {\int }_G(f(y),\frac{v_0(y)}{\Vert v_0\Vert }{)}_{W}dy=(f,T(g_0){)}_{H^2(G,\tau )}=(T^{\star }f,g_0{)}_{H^2(H,\sigma )} \\ =(S(f),\iota (z_0){)}_{H^2(H,\sigma )}=({\iota }^{\star }(Sf),z_0{)}_Z=(S(f)(e),z_0{)}_Z. \end{gathered}$$

$$K_S(y,h{)}^{\star }{z}_j=\sum _{a,b,i}{d}_{j,b,i}{g}_{a,b}(h)(L_{R_a}^{(1)}{K}_{\lambda })(y,h{)}^{\star }{w}_i.$$

$$K_S(y,h{)}^{\star }{z}_j$$

$$=\sum _{a,b,i}{d}_{j,b,i}(L_{Ad(h^{-1})\check{D_b^{\star }}}^{(1)}{K}_{\lambda })(y,h{)}^{\star }{w}_i$$

Full text

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

1. 1 Introduction

2. 2 Preliminaries

3. 3 Structure of intertwining maps

4. 3.1 Symmetry breaking operators

5. 3.2 Operators from $L^{2}(H\times_{\nu}E)$ into $\,H^{2}(G,\tau)$

6. 3.3 Generalized Shintani functions

7. 4 Intertwining operators via differential operators

8. 4.0.1 Differential Operators.

9. 4.1 Symmetry breaking via differential operators

10. 4.2 Equivariant maps into $K-$types

11. 4.3 Extension of an intertwining map to maximal globalization

12. 4.4 Comments on the relation among Hom’s

13. 5 Two results on $L^{2}(G\times_{\tau}W)[V_{\lambda_{1}}^{G}]$

14. 5.1 Analysis for the elements of $Hom_{G}(H^{2}(G,\tau_{1}),L^{2}(G\times_{\tau}W))$

15. 5.2 Kernel for the projector onto $L^{2}(G\times_{\tau}W)[V_{\lambda_{1}}^{G}]$

16. 6 Projector onto isotypic component of $res_{H}(\pi_{\lambda})$

17. 6.1 Projector onto isotypic components via differential operators

18. 6.2 Kernel for the projector onto $\,H^{2}(G,\tau)[V_{\mu}^{H}]$

19. 7 Criteria for discretely decomposable restriction

20. 8 Reproducing kernels and existence of discrete factors

21. 8.1 Examples of discrete factors in $res_{H}(\pi_{\lambda})$

22. 9 Proof of Proposition 3.7 and Theorem 3.5

23. 10 Other model to realize discrete series representations

24. 10.1 An example of discrete decomposition

25. 11 Appendix

26. 11.1 Kernel linear maps

27. 11.2 Adjoint to a kernel linear map

28. 11.3 Example

29. 11.4 Reproducing kernel subspace

30. 11.5 $L^{2}$-kernel of an elliptic operator

31. 11.6 Kernel for $P_{\lambda}$

32. 12 Notation

1. Introduction

Restricting a unitary irreducible representation $\pi$ of a Lie group $G$ to a closed subgroup $H$ leads to the branching law problem, namely finding the explicit decomposition of $\pi$ into irreducible representations of $H$. This objective generalizes the theory of spectral decomposition of a selfadjoint operator and, in the same way, both a discrete and a continuous spectrum may occur.

In this paper, we shall consider the case of $G,$ a semisimple Lie group, with $\pi$, a representation in the discrete series, that is, $\pi$ occurs as a left-invariant closed irreducible subspace of $L^{2}(G)$; these form the celebrated discrete series of Harish-Chandra. $H$ will be a subgroup of the same type, and we shall assume that both $G$ and $H$ admit discrete series. An aim is to find criteria ensuring that the branching law for $\pi$ gives irreducible $H$-invariant subspaces, perhaps even that the restriction of $\pi$ is a direct sum of such irreducible $H$-factors, in which case we call this discretely decomposable, and, if this happens with finite multiplicities, we call it $H-$admissible. We shall combine the results of Harish-Chandra on the distribution character and also the Plancherel formula for $G$ with recent results of T. Kobayashi on the admissibility of representations to obtain some new results on branching laws in our setting. Some of our proofs rely on work of N. Wallach published in [Wa2]. We shall also apply the theory of reproducing kernels, using some specific models of the discrete series. This approach involves the spherical functions studied by Harish-Chandra. Note that a special case of our situation could be with $H$ as the diagonal subgroup in $G\times G$, so that the branching problem corresponds to decomposing a tensor product; this is already a complicated problem for discrete series, we hope our approach leads to a deeper understanding (as an easy, yet not completely trivial consequence, we see that $\pi\otimes\pi^{*}$ is not discretely decomposable).

There are (at least) two basic facts from Harish-Chandra that we use (1) convolution by the distribution character of $\pi$ gives the orthogonal projection onto the isotypic component of $\pi$, and (2) the trace of the spherical function for the lowest $K$-type $W$ is the convolution (over $K$) of the distribution character and the character of $W$. We also draw on the groundwork on $H-$admissibility of T. Kobayashi [K1], [K2], [K3]. Our interest is only within the category of unitary representations and spectral decomposition in Hilbert spaces; thus, we do not consider the smooth category and the corresponding symmetry-breaking operators of T. Kobayashi. For this paper, a symmetry breaking operator is a continuous $H-$map from a unitary representation of $G$ into one of $H$. Concerning either the structure of symmetry breaking operators or the structure of the orthogonal projector onto an isotypic component, we would like to mention the seminal work of Kobayashi-Pevzner [KobPev] [KobPev2] [KobPev3], Kobayashi-Speh [KobSp1] [KobSp2], Nakahama [N], Peng-Zhang [PZ] and references therein. The branching problem for the tensor product of two holomorphic discrete series of $G=SL_{2}({\mathbb{R}})$ has been studied by Molch$\check{\textrm{a}}$nov [M2], J. Repka and other authors. An account of techniques and results for this example is in Kobayashi-Pevzner [KobPev3]. In particular, it is shown that the expression of the symmetry breaking operators employing Rankin-Cohen brackets is helpful to make explicit the corresponding Plancherel formula. In [KobSp1] [KobSp2], Kobayashi-Speh analyze branching problems from $O(n,1)$ to $O(n-1,1)$ for general unitary irreducible representation $\pi$. In order to study the representations in the discrete spectrum they find an interesting relation with the symmetry-breaking differential operators found by A. Juhl [AJ]. We may say that all of these examples, in addition to the results on wavefront in [HHO] [HHO2], have been a source of inspiration for the questions this paper considers. We want to point out that this paper is a sequel to [OV] and that C.R.A.S., [OV2], has published a partial announcement of this paper.

We recall that applications of the knowledge of symmetry breaking operators to Harmonic Analysis have been part of the research of several authors, among them we mention [KobPev3], [DaOZ], the work of many authors on decomposing as direct integral of irreducible representations the left regular representation of $G$ on $L^{2}(G/H)$, the papers of Molch$\check{\textrm{a}}$nov [M1], [M2].

It remains an interesting question to understand the geometric nature of the differential operators found to exist in connection with symmetry breaking for the family of representations considered here, i.e., they live in spaces of sections over Riemannian symmetric spaces. In addition to finding explicitly these operators, it might be possible to extend their existence to more general Riemannian manifolds and submanifolds - this should then include curvature of such more general manifolds.

We now mention some of the main results in this paper.

This paper splits into twelve sections. The results in Sections 2.0, 5.0, and subsections 3.2, 6.2 are quite technical. In subsection 3.1, we show that under certain hypotheses, symmetry breaking operators agree on dense subspaces with integral operators. In all the sections, we use the fact that a discrete series representation admits a model on a space of solutions to elliptic differential operators. For example, we carry out an analysis of symmetry breaking operators as differential operators (subsection 4.1, 4.2, 4.3). In particular, we show that $H-$admissibility implies that all the elements of some families of symmetry breaking operators are represented by differential operators. We also show a converse statement. It is quite a different situation when we analyze the orthogonal projector onto an isotypic component (Examples 6.2, 6.3); under the hypothesis of admissibility, we may show projectors are represented via truncated Taylor series (section 6) or some version of infinite order differential operators. Concerning necessary and sufficient conditions for a representation to be $H-$discretely decomposable, we obtain criteria saying that some particular symmetry breaking operators are restrictions of differential operators (Theorem 4.3) or in terms of reproducing kernels and cusp forms of Harish-Chandra (Theorem 7.1). We observe that in the family of discrete series representations, the hypothesis $H$-admissible implies $H-$smooth vectors in a isotypic component are $G-$smooth vectors. In section 8, we introduce two functions that are suitable for checking the existence of discrete factors (Theorem 8.4). In two appendices, we collect results on integral operators, reproducing kernel subspaces and kernel of elliptic invariant differential operators, as well as notation. The main results in this paper are: Theorems 3.5, 4.3, Proposition 4.4, Theorem 4.11, Propositions 5.2, 6.6, 6.7, Theorem 7.1, Corollary 7.4.

2. Preliminaries

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 hypotheses, Harish-Chandra showed that the set of equivalence classes of irreducible square integrable representations is parameterized by a lattice in $i\mathfrak{t}_{\mathbb{R}}^{\star}.$ 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 followed by the subindex $\mathbb{R}.$ The complexification of the Lie algebra of a Lie group is 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 $i{\mathfrak{t}}_{\mathbb{R}}^{\star}.$ 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 $\lambda\in i{\mathfrak{t}}_{\mathbb{R}}^{\star}$ 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 $G$ (cf. [Wa1, 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, $\Delta\subset\Psi_{\lambda}.$

Henceforth, $\Psi$ is a system of positive roots for $\Phi(\mathfrak{g},{\mathfrak{t}})$ containing $\Delta,$ and $(\pi_{\lambda}^{G},V_{\lambda}^{G})$ a square integrable representation for $G$ of Harish-Chandra parameter $\lambda$ dominant with respect to $\Psi.$ $(\tau,W):=(\pi_{\lambda+\rho_{n}}^{K},V_{\lambda+\rho_{n}}^{K})$ denotes 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 which states that $(\tau,W)$ determines $(\pi_{\lambda},V_{\lambda}^{G})$ up to unitary equivalence [Vo]. 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 $U(\mathfrak{g})$ for the Lie algebra $\mathfrak{g}$. As usual, $\Omega_{G}$ denotes the Casimir operator for $\mathfrak{g}.$ Following Hotta, Enright-Wallach [OV], we realize $V_{\lambda}:=V_{\lambda}^{G}$ as the space

[TABLE]

Henceforth, for $f\in H^{2}(G,\tau)$, we write $\pi_{\lambda}(x)(f):=L_{x}(f).$ We 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. Appendix 11.4). Therefore, the orthogonal projector $P_{\lambda}$ onto $\,H^{2}(G,\tau)$ is an integral map (integral operator) represented by the smooth Carleman matrix kernel or reproducing kernel (cf. Appendix 11.1, Appendix 11.4, Appendix 11.6).

[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 analyze the restriction to $H.$ In particular, we study the irreducible $H-$subrepresentations for $\pi_{\lambda}$. A known result is that any irreducible $H-$subrepresentation of $V_{\lambda}$ is a square integrable representation for $H$, for a proof (cf. [GW]). 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 $L=H\cap K.$ Next, we consider a square integrable representation $H^{2}(H,\sigma)\subset L^{2}(H\times_{\sigma}Z)$ of lowest $L-$type $(\sigma,Z).$ An aim of this paper is to understand the nature of the intertwining operators between the unitary $H-$representations $H^{2}(H,\sigma)$ and $\,H^{2}(G,\tau),$ the adjoint of such intertwining operators and consequences of their structure.

Remark 2.1*.*

To give a glance on the nature of the elements in $\,H^{2}(G,\tau)$, we show: Every $f\in\,H^{2}(G,\tau)$ is a bounded function.

In fact, let $K_{\lambda}(y,x)$ be as in 2.1. Thus, $f(x)=\int_{G}K_{\lambda}(y,x)f(y)dy.$ Then, Schwarz inequality, $K_{\lambda}(\cdot,x)$ is square integrable, the equality $K_{\lambda}(y,x)=K_{\lambda}(x^{-1}y,e)$ and the invariance of Haar measure, justify the inequalities

[TABLE]

3. Structure of intertwining maps

For this section, $G,K,(\tau,W),(\pi_{\lambda},\,H^{2}(G,\tau))=(\pi_{\lambda}^{G},V_{\lambda}^{G}),H,L$ are as in Section 2. Besides, we fix a finite dimensional representation $(\nu,E)$ for $L$, and a continuous intertwining linear $H-$map $T:L^{2}(H\times_{\nu}E)\rightarrow\,H^{2}(G,\tau)$.

Fact 3.1*.*

We show $T$ is a Carleman kernel map. (cf. Appendix 11.1).

In fact, for each $x\in G,w\in W$, the linear functional $L^{2}(H\times_{\nu}E)\ni g\mapsto(Tg(x),w)_{W}$ is continuous. Riesz representation Theorem shows there exists a function

[TABLE]

so that the map $h\mapsto K_{T}(h,x)^{\star}(w)$ belongs to $L^{2}(H\times_{\nu}E)$ and for $g\in L^{2}(H\times_{\nu}E),x\in G,w\in W$ we have the absolutely convergent integral and the equality

[TABLE]

That is, $T$ is the integral map

[TABLE]

3.1. Symmetry breaking operators

Examples shows that the adjoint $T^{\star}$ of integral linear map $T$ need not be an integral map (cf. Appendix 10.3), whence, we would like to know when $T^{\star}$ is an integral linear map. Formally, we may write $T^{\star}f(h)=\int_{G}K_{T}(h,x)^{\star}f(x)dx,$ where the convergence of the integral is in the weak sense. That is, for each $g\in L^{2}(H\times_{\nu}E),f\in\,H^{2}(G,\tau),$ we have the convergence of the iterated integral

[TABLE]

Thus, the adjoint $T^{\star}$ of any continuous $T:L^{2}(H\times_{\nu}E)\rightarrow\,H^{2}(G,\tau)$ is a weak integral map.

In order to study branching problems, T. Kobayashi has introduced the concept of symmetry breaking operator. In our setting, a symmetry breaking operator is a continuous $H-$map $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$. For a symmetry breaking operator $S$, the above considerations applied to $T:=S^{\star}$ let us conclude: under our hypothesis, a symmetry breaking operator is always a weak integral map.

In [Fo, page 45], we find an exposition on Bargmann transform $B:L^{2}(\mathbb{R}^{n})\rightarrow\mathcal{H}_{2}(\mathbb{C}^{n})$. $B$ is an example of integral map, such that $B^{\star}$ may not be an integral map. However, Bargmann has shown $B^{\star}$ restricted to certain dense subspace is an integral map. In Appendix 11.3, we provide an example of a continuous integral map into a reproducing kernel space such is that its adjoint is not an integral map on the whole space. However, it is equal to an integral map on a certain subspace. Theorem 3.5 shows that a quite common feature for the kind of integral maps $T$ under our consideration, that its adjoint $T^{\star}$ is an integral map on certain dense subspace of $\,H^{2}(G,\tau).$ In order to state our results we need a few definitions.

Definition 3.2**.**

A unitary representation $(\pi,V)$ of $G$ is discretely decomposable over $H$, if there exists an orthogonal family of closed, $H-$invariant, $H-$irreducible subspaces of $V$ so that the closure of its algebraic sum is equal to $V.$

Definition 3.3**.**

A unitary representation $(\pi,V)$ of $G$ is $H-$admissible if the representation is discretely decomposable over $H$ and the multiplicity of each irreducible $H$-factor is finite.

In [KO], we find a complete list of triples $(G,H,\pi)$ such that $(G,H)$ is a symmetric pair and $\pi$ is an $H-$admissible representation. For example, for the pair $(SO(2n,1),SO(2k)\times SO(2n-2k,1))$ there is no $\pi_{\lambda}$ with an admissible restriction to $H.$ Whereas, for the pair $(SU(m,n),S(U(m,k)\times U(n-k)))$ there are exactly $m$ Weyl chambers $C_{1},\dots,C_{m}$ in $i\mathfrak{t}_{\mathbb{R}}^{\star}$, so that $\pi_{\lambda}$ is $H-$admissible if and only $\lambda$ belongs to $C_{1}\cup\dots\cup C_{m}.$ For both cases $0<k<n$.

Definition 3.4**.**

A representation $(\pi,V)$ of $G$ is integrable, if some nontrivial matrix coefficient is an integrable function with respect to Haar measure.

It is a Theorem of Trombi-Varadarajan, Hecht-Schmid, [HS]: $(\pi_{\lambda},V_{\lambda})$ is an integrable representation if and only if $|(\lambda,\beta)|>\frac{1}{2}\sum_{\alpha\in\Phi(\mathfrak{g},{\mathfrak{t}}):(\alpha,\beta)>0}(\alpha,\beta)$ for every noncompact root $\beta.$

The next result gives more information on symmetry breaking operators. We recall (cf. section 2) that the action $\pi_{\lambda}$ of $G$ on $H^{2}(G,\tau)$ by left translation provides an explicit realization of $(\pi_{\lambda},V_{\lambda})$.

Theorem 3.5**.**

Let $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$ be a continuous intertwining linear $H-$map. Then,

a) If the restriction to $H$ of $(\pi_{\lambda},\,H^{2}(G,\tau))$ is discretely decomposable, then $S$ is an integral map.

b) If $(\pi_{\lambda},\,H^{2}(G,\tau))$ is an integrable representation, then $S$ restricted to the subspace of smooth vectors is an integral linear map.

We defer the proof of Theorem 3.5 to section 9.

Example 3.6*.*

An application for Theorem 3.5 is: for a continuous $H-$map, we write the polar decomposition for $T=VP:L^{2}(H\times_{\sigma}Z)\rightarrow\,H^{2}(G,\tau)$ where $P=\sqrt{T^{\star}T}$ and $V:L^{2}(H\times_{\sigma}Z)\rightarrow\,H^{2}(G,\tau)$ is a partial isometry. $V$ is usually called a generalized Bargmann transform. Then, $V$ is an integral map, and, whenever $\pi_{\lambda}^{G}$ is an integrable discrete series, the linear map $V^{\star}$, as well as $T^{\star}$, restricted to the subspace of smooth vectors in $\,H^{2}(G,\tau)$, is equal to an integral map. A particular case of this is $T=r^{\star}$, where $r$ is the restriction map $r:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{res_{L}(\tau)}W)$ as in Example 4.1. Here, $r^{\star}=V\sqrt{rr^{\star}}$, $r=V^{\star}\sqrt{r^{\star}r}$, and $r,r^{\star},rr^{\star},r^{\star}r,\sqrt{r^{\star}r},\sqrt{rr^{\star}}$ are integral maps (cf. [He]). Theorem 3.5 shows that when $\pi_{\lambda}$ is integrable, $V^{\star}$ is a kernel map on the subspace of smooth vectors. For $G=SU(1,1),H=A$, in Example 9.1, we verify $V^{\star}$ restricted to the subspace of $K-$finite vectors is an integral map, despite $\pi_{\lambda}$ is not an integrable representation.

3.2. Operators from $L^{2}(H\times_{\nu}E)$ into $\,H^{2}(G,\tau)$

In this somewhat technical subsection we assemble properties of the Carleman matrix kernel that represent a $H-$map into $\,H^{2}(G,\tau)$ and its adjoint.

Proposition 3.7**.**

*Let $T:L^{2}(H\times_{\nu}E)\rightarrow\,H^{2}(G,\tau)$ be a continuous intertwining linear $H-$map. Then, the function $K_{T}$ satisfies:

a) $K_{T}(h,x)^{\star}w=T^{\star}(y\mapsto K_{\lambda}(y,x)^{\star}w)(h).$

b) The function $h\mapsto K_{T}(h,e)^{\star}w$ is an $L-$finite vector in $L^{2}(H\times_{\nu}E).$

c) $K_{T}$ is a smooth map. Further, $K_{T}(\cdot,x)^{\star}w$ is a smooth vector.

d) There exists a constant $C$ and finitely many functions $\phi_{a,b}:G\rightarrow{\mathbb{C}}$ so that for every $x\in G$, $\|K_{T}(e,x)^{\star}\|_{Hom_{\mathbb{C}}(W,Z)}\leq C\|T^{\star}\|\sum_{a,b}|\phi_{a,b}(x)|.$

e) The function $G\ni x\mapsto\|K_{T}(\cdot,x)^{\star}\|_{L^{2}(H\times_{\tau^{\star}\otimes\nu}Hom_{\mathbb{C}}(W,E))}$ is bounded.

f) $K_{T}(hh_{1}s,hxk)=\tau(k^{-1})K_{T}(h_{1},x)\nu(s),\,x\in G,s\in L,h,h_{1}\in H,k\in K.$

g) If $T^{\star}$ is a kernel map, with kernel $K_{T^{\star}}:G\times H\rightarrow Hom(W,E)$ and $K_{T^{\star}}(\cdot,h)^{\star}z\in\,H^{2}(G,\tau).$ Then, $L_{D}^{(2)}K_{T}(h,\cdot)=\chi_{\lambda}(D)K_{T}(h,\cdot)$ for every $D$ in the center of $U(\mathfrak{g}).$ Here, $\chi_{\lambda}$ is the infinitesimal character of $\pi_{\lambda}.$*

Note. The functions $\phi_{a,b}$ are defined as follows. We fix linear basis $\{X_{b}\}_{1\leq b\leq N}$ (resp. $\{Y_{a}\}_{1\leq a\leq M}$) for the space of elements in $U(\mathfrak{h})$ of degree less or equal than $\dim\mathfrak{h}$ (resp. for the space of elements in $U(\mathfrak{g})$ of degree less or equal than $\dim\mathfrak{h}$). Then $\phi_{a,b}$ are defined by $Ad(x^{-1})(X_{b})=\sum_{1\leq a\leq M}\phi_{a,b}(x)Y_{a},b=1,\cdots,N.$

We postpone the verification of Proposition 3.7 to section 9.

Remark 3.8*.*

We assume $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$ is a continuous symmetry breaking operator represented by a Carleman kernel $K_{S}$. Thus, owing to Proposition 3.7 $S^{\star}$ is a Carleman map. Hence, $K_{S}(x,h)=K_{S^{\star}}(h,x)^{\star}$ and hence we may conclude:

a) $K_{S}$ is a smooth map.

b) $K_{S}(hxk,hh_{1}s)=\nu(s^{-1})K_{S}(s,h_{1})\tau(k),h,h_{1}\in H,x\in G,s\in L.$

c) For $z\in E$, the function $G\ni x\mapsto K_{S}(x,e)^{\star}z\in W$ belongs to $\,H^{2}(G,\tau)$ and it is a $L-$finite function.

Remark 3.9*.*

Any continuous $H-$map from $\,H^{2}(G,\tau)$ into $H^{2}(H,\sigma)$ or vice versa, always, it is represented by a smooth Carleman kernel that enjoys the properties in Remark 3.8 or Proposition 3.7. We also show, in Theorem 4.3, that symmetry breaking operators can be represented by differential operators under the hypothesis of $res_{H}(\pi_{\lambda})$ being $H-$admissible.

With respect to compute the dimension of the space of symmetry breaking operators, we refer to [DV], [DGV], [GW], [N] and references therein. Quite often, the dimension of the space of symmetry breaking operators is large, (cf. [DGV]) and references therein.

3.3. Generalized Shintani functions

Kobayashi in [K5] has began deep analysis of functions in relation to $H-$intertwining linear operators $R:V_{G}\rightarrow V_{H}$ between two smooth irreducible and class one representations $(\pi_{G},V_{G})$ $(\pi_{H},V_{H})$ for $G,$ $H$ respectively. For this, he fixes $v_{G}\in V_{G},v_{H}\in V_{H}$ nonzero vectors fixed by $K,L$ respectively, and he defines the Shintani function $S(g)=(R(\pi_{G}(g)v_{G}),v_{H})_{V_{H}},g\in G.$ The function $S$ satisfies: right invariant under $L$; left invariant under $K$; eigenfunction for $R_{D},D\in\mathfrak{z}(U(\mathfrak{g}))$; eigenfunction for $L_{D},D\in\mathfrak{z}(U(\mathfrak{h})).$ Here, $\mathfrak{z}(U(\mathfrak{g}))$ denotes the center of the universal enveloping algebra for $\mathfrak{g}.$ Let $S:\,H^{2}(G,\tau)\rightarrow H^{2}(H,\sigma)$ a continuous intertwining linear map. Since $H^{2}(H,\sigma)$ is a reproducing kernel subspace, $S$ is an integral map represented by a kernel $K_{S}:G\times H\rightarrow Hom_{\mathbb{C}}(W,Z)$ that satisfies the statements in Remark 3.8. We now verify that properties of the function $y\mapsto K_{S}(y,e)$ suggest a natural generalization for the concept of Shintani function. In fact, we set $\Phi(y):=K_{S}(y,e),y\in G.$ Thus, $\Phi$ maps $G$ into $Hom_{\mathbb{C}}(W,Z)$. Among the properties of the function $\Phi$ are:

1. (1)

$\Phi$ is a smooth function. 2. (2)

$\Phi(syk)=\sigma(s)\Phi(y)\tau(k),s\in L,k\in K,y\in G.$ 3. (3)

$R_{D}\Phi=\chi_{\lambda}(D)\Phi$ for $D\in\mathfrak{z}(U(\mathfrak{g})).$ 4. (4)

$L_{D}\Phi=\tilde{\chi}_{\mu}(D)\Phi$ for $D\in\mathfrak{z}(U(\mathfrak{h})).$ 5. (5)

$\Phi$ as well as its restriction to $H$ are square integrable functions.

The previous considerations let us define a generalized Shintani function. For this, we fix representations $(\tau,W)$ of $K$, $(\sigma,Z)$ of $L$ and infinitesimal characters $\chi_{\lambda},\chi_{\mu}$ for $\mathfrak{g},\mathfrak{h}$ respectively. A generalized Shintani function is a function $\Phi:G\rightarrow Hom_{\mathbb{C}}(W,Z)$ that satisfies the five conditions enumerated in the previous paragraph. When we dealt with discrete series representations, the trivial representation of a maximal compact subgroup of the ambient group, never occurs as a $K-$type. However, it is clear that the first four stated properties are a generalization of the concept of Shintani’s function. The obvious result is: the space of generalized Shintani functions attached to $(\tau,W),(\sigma,Z),\chi_{\lambda},\chi_{\mu}$ is isomorphic to the space of continuous $H-$maps from $V_{\lambda}^{G}$ to $V_{\mu}^{H}.$

4. Intertwining operators via differential operators

For this section, let $G,K,H,L,W,\pi_{\lambda},\,H^{2}(G,\tau)$ be as in Section 2. Besides, we fix a finite dimensional representation $(\nu,E)$ for $L$. In [N], [KobPev],

[K6], the authors have constructed $H-$intertwining maps between holomorphic discrete series by means of differential operators. Some of these authors also considered the case of intertwining maps between two principal series representations [KobSp1], [KobSp2]. Motivated by the fact that discrete series can be modeled as function spaces, an aim of this section is to analyze to what extent symmetry breaking linear maps agree with restriction of linear differential operators. In [K6], is presented a general conjecture on the subject, we present a partial solution to the conjecture for the particular case of discrete series representations.

4.0.1. Differential Operators.

For the purpose of this paper a differential operator is a linear map $S:C^{\infty}(G\times_{\tau}W)\rightarrow C^{\infty}(H\times_{\nu}E)$ so that there exists finitely many elements $D_{b}\in U(\mathfrak{g}),$ a basis $\{w_{c}\}\,\text{for}\,W,$ finitely   many   elements $\{z_{a}\}\,\text{in}\,E,$ $d_{a,b,c}\in{\mathbb{C}},$ and for any $f\in C^{\infty}(G\times_{\tau}W)$ we have

[TABLE]

Sometimes we will allow the constants $d_{a,b,c}$ to be smooth functions on $H.$

The definition is motivated by the following result of Kobayashi, for a proof (cf. [KobPev, Theorem 2.9]). They present an isomorphism between the space $Hom_{L}(E^{\vee},Ind_{\mathfrak{k}}^{\mathfrak{g}}(W^{\vee}))$ and the set of $H-$invariant differential operators from $C^{\infty}(G\times_{\tau}W)$ into $C^{\infty}(H\times_{\nu}E)$. As usual, $M^{\vee}$ denotes the contragredient representation. The isomorphism is:

[TABLE]

Here, $d_{a,b,c}\in{\mathbb{C}},\{z_{a}\}\,\text{is basis for}\,E,D_{b}\in U(\mathfrak{g}),\{w_{c}\}\,\text{is basis for}\,W,h\in H.$

It is obvious that a differential operator according to our definition, with $d_{a,b,c}\in{\mathbb{C}}\,\,\forall a,b,c$ is represented by an element of $Hom_{L}(E^{\vee},Ind_{\mathfrak{k}}^{\mathfrak{g}}(W^{\vee}))$.

Example 4.1*.*

Examples of differential operators are the normal derivatives considered in [OV]. For this, we write the Cartan decomposition as $\mathfrak{g}={\mathfrak{k}}+\mathfrak{p}$ and $\mathfrak{h}=\mathfrak{l}+\mathfrak{p}^{\prime}$. We have $\mathfrak{p}^{\prime}=\mathfrak{p}\cap\mathfrak{h}.$ Let $(\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)}$ denote the $n-$th symmetric power of the orthogonal to $\mathfrak{p}^{\prime}$ with respect to the Killing form in $\mathfrak{p}.$ Let $\tau_{n}$ denote the natural representation of $L$ in $Hom_{\mathbb{C}}((\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)},W)$. Let $\lambda:S(\mathfrak{g})\rightarrow U(\mathfrak{g})$ denote the symmetrization map. Then, for each $D\in(\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)},$ $f\in\,H^{2}(G,\tau),h\in H$ we compute the normal derivative of $f$ in the direction $D$ at the point $h,$ $r_{n}(f)(D)(h):=R_{\lambda(D)}f(h).$ In [OV], it is shown that $r_{n}(f)\in L^{2}(H\times_{\tau_{n}}Hom_{\mathbb{C}}((\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)},W))$ and the resulting map

[TABLE]

is $H-$equivariant and continuous for $L^{2}-$topologies. As before, $K_{\lambda}$ is the matrix kernel of $P_{\lambda}$. The map $r_{n}$ is represented by the matrix kernel

[TABLE]

given by

[TABLE]

Here, the upper index $(2)$ means we compute the derivative of $K_{\lambda}$ on the second variable.

4.1. Symmetry breaking via differential operators

In this subsection we show the equivalence between $res_{H}(\pi_{\lambda})$ being a $H-$admissible representation and that every symmetry breaking operator from $\,H^{2}(G,\tau)$ into an arbitrary representation $H^{2}(H,\sigma)$ is the restriction of a differential operator. For this, (cf. Lemma 4.2), we characterize those symmetry braking operators that are restriction of differential operators. Next, we formulate the main result in Theorem 4.3. We recall (cf. Theorem 3.5 and 3.8 c)) we have shown that if $res_{H}(\pi_{\lambda})$ is a $H-$discretely decomposable representation, then any symmetry breaking operator $S$ is a Carleman integral map satisfying: for every $z\in E$, $y\mapsto K_{S}(y,e)^{\star}z$ is a $L-$finite vector for $\pi_{\lambda}.$

Lemma 4.2**.**

$G,H,\,H^{2}(G,\tau),L$* are as in Section 2. $(\nu,E)$ is a finite dimensional representation for $L$. Let $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$ be a not necessarily continuous intertwining $H-$map such that $S$ is the restriction of a differential operator. Then,

a) $S$ is a Carleman map. That is, there exists $K_{S}:G\times H\rightarrow Hom_{\mathbb{C}}(W,E)$ so that $y\mapsto K_{S}(y,h)^{\star}z\in\,H^{2}(G,\tau)\,\text{for}\,h\in H,z\in E$ and for $\,f\in\,H^{2}(G,\tau)$

$(S(f)(h),z)_{Z}=\int_{G}(f(y),K_{S}(y,h)^{\star}z)_{W}dy\,z\in E.$

b) For every $z\in E$, $y\mapsto K_{S}(y,e)^{\star}z$ is a $K-$finite vector for $\pi_{\lambda}.$

c) $K_{S}$ is a smooth function.

d) $S$ is continuous in $L^{2}-$topologies.

Conversely, if $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$ is a Carleman $H-$map so that for each $z\in E$, $y\mapsto K_{S}(y,e)^{\star}z$ is a $K-$finite vector for $\pi_{\lambda}.$ Then, $S$ is the restriction of a differential operator and $S$ is continuous in $L^{2}-$topologies.*

We defer the proof of Lemma 4.2 till we formulate the main results for this subsection.

Theorem 4.3**.**

We recall $G$ is a semisimple matrix Lie group, $(\pi_{\lambda},\,H^{2}(G,\tau))$ is an irreducible square integrable representation for $G$, $H$ is a closed, equal rank, reductive subgroup of $G$ and $L$ is a maximal compact subgroup for $H$. We have: $res_{H}(\pi_{\lambda})$ is an $H-$admissible representation if and only if for every irreducible representation $(\sigma,Z)$ of $L$, it holds that any continuous $H-$map from $\,H^{2}(G,\tau)$ into $H^{2}(H,\sigma)$ is the restriction of a differential operator.

Theorem 4.3 readily follows from Lemma 4.2 and the next Proposition.

In [N, Theorem 3.10], Nakahama has shown the direct implication in Theorem 4.3 under the hypotheses $(G,H)$ is a holomorphic symmetric pair and $\pi_{\lambda}$ is a scalar holomorphic discrete series.

The proof of Theorem 4.3 is based in the following fact.

Proposition 4.4**.**

*Let $G,K,\,H^{2}(G,\tau),\pi_{\lambda},H,L,(\nu,E)$ be as is in Lemma 4.2. Let $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$ denote an everywhere defined intertwining linear $H-$map. The following statements holds:

If $res_{H}(\pi_{\lambda})$ is an $H-$admissible representation and $S$ is continuous. Then $S$ is the restriction of a linear differential operator.

For a converse statement, we have,

i) If we assume for some $\nu$, that, some nonzero linear intertwining $H-$map $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)_{disc}$ is the restriction of a linear differential operator. Then, $res_{H}(\pi_{\lambda})$ is discretely decomposable.

ii) If we assume for some irreducible representation $(\sigma,Z)$ for $L$, that, every nonzero linear intertwining $H-$map $S:\,H^{2}(G,\tau)\rightarrow H^{2}(H,\sigma)$ is the restriction of a linear differential operator. Then, the multiplicity of $(\pi_{\mu}^{H},H^{2}(H,\sigma))$ in $res_{H}(\pi_{\lambda})$ is finite.*

In Proposition 4.4, the hypothesis on image of $S$ is quite essential. Examples and counterexamples are provided by $r,r_{1},r_{2},...$.

An example for a map $S$ where Proposition 4.4 may be applied, is the normal map $r_{n}$ defined in Example 4.1. In particular, we obtain: $res_{H}(\pi_{\lambda})$ is discretely decomposable if and only if there exists an $n$ so that the image of $r_{n}$ is contained in $L^{2}(H\times_{\tau_{n}}Hom_{\mathbb{C}}((\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)},W))_{disc}.$

Proof of Lemma 4.2.

We fix $\{z_{j},j=1,\cdots,\dim E\},\{w_{i},i=1,\dots,\dim W\}$ orthonormal basis for $E,W$ respectively. Our hypothesis gives for every $f\in\,H^{2}(G,\tau),h\in H$ the following equality holds

xxxxxxxxxxxxxxxxx $S(f)(h)=\sum_{j,b,i}d_{j,b,i}\,([R_{D_{b}}f](h),w_{i})_{W}\,z_{j}.$

In [At], we find a proof that in the $L^{2}-$kernel of an elliptic operator, $L^{2}-$convergence implies uniform convergence on compact sets of the sequence as well as any of its derivatives. Since the Casimir operator acts on $G/K$ as an elliptic operator, the result on PDE just quoted applies to $\,H^{2}(G,\tau).$ Hence, the equality $(S(f)(h),z_{j})_{Z}=\sum_{b,i}d_{j,b,i}(R_{D_{b}}f(h),w_{i})_{W}$ yields the map $f\mapsto(S(f)(h),z_{j})_{Z}$ is a continuous linear functional on $\,H^{2}(G,\tau).$ Thus, owing to Riesz representation Theorem, there exists a function $K_{S}:G\times H\rightarrow Hom_{\mathbb{C}}(W,E)$ so that $a)$ holds. The hypothesis $S$ is an intertwining map, yields the equality $K_{S}(h_{1}yk,h_{1}hs)=\nu(s^{-1})K_{S}(y,h)\tau(k),h_{1},h\in H,y\in G,s\in L,k\in K.$ The smoothness for $K_{S}$ follows from that, for each $z\in E$, $K_{S}(\cdot,\cdot)^{\star}z$ is equal to the map $(y,h)\mapsto h^{-1}y$ followed by the map $x\mapsto K_{S}(x,e)^{\star}z$ and that $v\mapsto K_{S}(v,e)^{\star}z$ is an element of $\,H^{2}(G,\tau).$ Next, we justify the four equalities in the computation below. The first equality is due to the expression of $S$; the second is due to the identity $L_{\check{D}}(f)(e)=R_{D}(f)(e)$; the third is due to (2.1); finally, we recall for an arbitrary $D\in U(\mathfrak{g})$ any smooth vector $f\in V_{\lambda}^{\infty}$ is in the domain for $L_{D}$, in particular, $y\mapsto K_{\lambda}(y,e)^{\star}w_{i}$ is in the domain for $L_{D}$. These four considerations justifies the following equalities for any smooth vector $f\in\,H^{2}(G,\tau)$,

[TABLE]

We observe, the first and last member of the above equalities defines continuous linear functionals on $\,H^{2}(G,\tau)$ and the two functionals agree on a dense subspace, namely the subspace of smooth vectors, whence

[TABLE]

Since, the right hand side of the above equality is a $K-$finite vector for $\pi_{\lambda},$ we have shown $b)$. To show the continuity of $S$, we notice that $S$ is defined by the Carleman kernel $K_{S}$ (for each $h\in H$, $K_{S}(\cdot,h)^{\star}z_{j}\in\,H^{2}(G,\tau)$) and by hypothesis the domain of the integral operator defined by $K_{S}$ on $L^{2}(G\times_{\tau}W)$ contains $\,H^{2}(G,\tau).$ Since a Carleman kernel determines a closed map on its maximal domain, and $\,H^{2}(G,\tau)$ is a closed subspace, we have $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$ is a closed linear map with domain $\,H^{2}(G,\tau)$, the closed graph Theorem leads us to the continuity of $S.$

To show the converse statement, we make explicit the hypotheses on $S$,

$K_{S}(hx,hh_{1})=K_{S}(x,h_{1}),h,h_{1}\in H,x\in G$; for each $z\in E$, $y\mapsto K_{S}(y,e)^{\star}z$ is a $K-$finite vector in $\,H^{2}(G,\tau)$; domain of $S$ equal to $\,H^{2}(G,\tau)$. We show $S$ is continuous and $S$ the restriction of a differential operator. In fact, since $K_{S}(x,h)=K_{S}(h^{-1}x,e)$ and $K_{S}(\cdot,e)^{\star}z$ belongs to $\,H^{2}(G,\tau)$ we obtain $K_{S}(\cdot,h)$ is square integrable and hence $S$ is a Carleman map. As in the direct implication, we obtain $S$ is continuous. To verify $S$ is the restriction of a differential operator, we fix a nonzero vector $w\in W.$ Since $\,H^{2}(G,\tau)$ is an irreducible representation for $G$, a result of Harish-Chandra shows that the underlying $(\mathfrak{g},K)-$module for $\,H^{2}(G,\tau)$ is $U(\mathfrak{g})-$irreducible. It readily follows that the function $K_{\lambda}(\cdot,e)^{\star}w$ is nonzero (otherwise $K_{\lambda}$ would be the null function) and a $K-$finite vector, therefore, for each $z_{j}$, there exists $D_{j}\in U(\mathfrak{g})$ so that $K_{S}(\cdot,e)^{\star}z_{j}=L_{D_{j}}K_{\lambda}(\cdot,e)^{\star}w.$ For $f$ smooth vector in $\,H^{2}(G,\tau),$ $h\in H$, we have

[TABLE]

Thus, after we fix a linear basis $\{R_{i}\}$ for $U(\mathfrak{g}),$ and we write $\check{D_{j}^{\star}}=\sum_{i}d_{i,j}R_{i}$, for any smooth vector $f\in\,H^{2}(G,\tau),h\in H$ we obtain

xxxxxxxxxxxxxxxx $S(f)(h)=\sum_{j,i}d_{i,j}([R_{R_{i}}(f)](h),w)z_{j}$.

We claim the equality holds for every $f\in\,H^{2}(G,\tau).$ In fact, each $f$ in $\,H^{2}(G,\tau)$ is limit of a sequence $f_{n}$ of smooth vectors. The result of PDE alluded to in the direct proof let us conclude that sequence of functions $\sum_{j,i}d_{i,j}([R_{R_{i}}(f_{n})](h),w)z_{j}$ converges uniformly on compact sets to the right hand side evaluated at $f$. On the other hand, we already know $S$ is continuous, thus, if necessary going to a subsequence, the Riesz-Fischer Theorem yields the left hand side converges pointwise (a.e.) to $S(f)$. Therefore, we have shown $S$ is equal to the restriction of a differential operator. Thus, we have concluded a proof of Lemma 4.2. ∎

Proof of Proposition 4.4..

For the vector spaces $E,W$, we fix orthonormal basis $\{z_{j}\},\{w_{i}\}$ respectively. Since $res_{H}(\pi_{\lambda})$ is discretely decomposable, Theorem 3.5 shows $S$ is an integral map and Remark 3.8 yields $K_{S}(\cdot,e)^{\star}z_{j}$ is a $L-$finite vector. The hypothesis $res_{H}(\pi_{\lambda})$ is an admissible representation allows us to apply [K3, Prop. 1.6]. In this way we obtain that the subspace of $L-$finite vectors in $\,H^{2}(G,\tau)$ is equal to the subspace of $K-$finite vectors. Whence, $x\mapsto K_{S}(x,e)^{\star}z_{j}$ is a $K-$finite vector. By hypothesis, $\,H^{2}(G,\tau)_{K-fin}$ is an irreducible representation under the action of $U(\mathfrak{g})$ and the function $y\mapsto K_{\lambda}(y,e)^{\star}w_{i}$ is nonzero and $K-$finite, hence, for each $i,j$ there exists $C_{j,i}\in U(\mathfrak{g})$ so that $[L_{C_{j,i}}^{(1)}K_{\lambda}](y,e)^{\star}w_{i}=K_{S}(y,e)^{\star}z_{j},\,\text{ for \,all}\,y\in G.$ Therefore, since $x\mapsto K_{S}(x,e)^{\star}z_{j}$ is a smooth vector for $G$, for $f\in V_{\lambda}^{\infty}$ we justify as in the proof of Lemma 4.2 the fifth and seventh equality in the following computation, the sixth equality is due to (2.1), for $h\in H$, we set $\tilde{f}(x)=L_{h^{-1}}(f)(x),$ the hypothesis of $S$ is a $H-$map justifies the first equality in

[TABLE]

After we set $D_{j,i}:=\check{C_{j,i}^{\star}}$ and we recall definition 4.1 we conclude that $S$ restricted to the subspace of smooth vectors agrees with the restriction of a differential operator. In order to show the equality for generic $f\in\,H^{2}(G,\tau)$ we argue as follows: there exists a sequence $f_{r}$ of elements in $V_{\lambda}^{\infty}$ which converges in $L^{2}-$norm to $f$. Owing to the Casimir operator is elliptic on $G/K$, the sequence $f_{r}$, as well as any derivatives of the sequence, converge uniformly on compact subsets. Moreover, owing to Harish-Chandra’s Plancherel Theorem, we have that $L^{2}(H\times_{\nu}E)_{disc}$ is equal to a finite sum of eigenspaces for the Casimir operator for $\mathfrak{h}.$ We know the Casimir operator acts as an elliptic differential operator on $L^{2}(H\times_{\nu}E)$, whence, $L^{2}(H\times_{\nu}E)_{disc}$ is a reproducing kernel subspace, in consequence, point evaluation is a continuous linear functional on $L^{2}(H\times_{\nu}E)_{disc}$ [At]. Finally, the hypothesis on $res_{H}(\pi_{\lambda})$ gives the image of $S$ is contained in $L^{2}(H\times_{\nu}E)_{disc}$. Therefore, we have justified the steps in

[TABLE]

Whence, we have shown the first statement in Proposition 4.4.

To follow we assume for some $\nu$ and some nonzero intertwining $H-$map $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)_{disc}$ is the restriction of a nonzero linear differential operator, we show $res_{H}(\pi_{\lambda})$ is discretely decomposable.

The hypothesis allows us to apply Lemma 4.2. Thus, $y\mapsto K_{S}(y,e)^{\star}z_{j}$ is a $K-$finite vector in $\,H^{2}(G,\tau).$ We claim, $K_{S}(\cdot,e)^{\star}z_{j}$ is $\mathfrak{z}(U(\mathfrak{h}))-$finite. In fact, in a previous paragraph we argued that Harish-Chandra’s Plancherel Theorem shows $L^{2}(H\times_{\nu}E)_{disc}$ is equal to a finite sum of eigenspaces of the Casimir operator for $\mathfrak{h}$. Thus, for $f\in V_{\lambda}^{\infty},D\in\mathfrak{z}(U(\mathfrak{h}))$ so that $S(f)$ belongs to in an irreducible component of an eigenspace for the Casimir operator, we have the equalities

[TABLE]

The third equality holds because by hypothesis $S(f)$ is an eigenfunction for $\mathfrak{z}(U(\mathfrak{h})).$ Since, each function $K_{S}(\cdot,e)^{\star}z,\,\,L_{D^{\star}}^{(1)}K_{S}(y,e)^{\star}z$ belongs to $H^{2}(G,\tau)$, the first and last member of the above equalities determine continuous linear functionals on $\,H^{2}(G,\tau)$, the two linear functionals agree in the dense subspace of smooth vectors. Whence, $y\mapsto K_{S}(y,e)^{\star}z_{j}$ is an eigenfunction for $\mathfrak{z}(U(\mathfrak{h})).$ The general case readily follows. The hypothesis $S$ is nonzero and Remark 3.8 $b)$, gives us $z\in E$ so that $U(\mathfrak{h})K_{S}(\cdot,e)^{\star}z$ is a $\mathfrak{z}(U(\mathfrak{h}))-$finite and a nonzero $U(\mathfrak{h})-$submodule of $\,H^{2}(G,\tau)_{K-fin}.$ We quote a result of Harish-Chandra: a $U(\mathfrak{h})-$finitely generated, $\mathfrak{z}(U(\mathfrak{h}))-$finite, $(\mathfrak{h},L)-$module has a finite composition series. For a proof (cf. [Wa1, Corollary 3.4.7 and Theorem 4.2.1]). Thus, $\,H^{2}(G,\tau)_{K-fin}$ contains an irreducible $(\mathfrak{h},L)$-submodule. Therefore, the subspace $U(\mathfrak{h})K_{S}(\cdot,e)^{\star}z$ contains an irreducible $U(\mathfrak{h})-$submodule. Next, in [K3, 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. The hypothesis $V_{\lambda}$ is unitary yields discrete decomposable [K4, Theorem 4.2.6]. Whence, we have shown $i)$.

We now assume for some $(\sigma,Z)$ and every intertwining linear $H-$map

$S:\,H^{2}(G,\tau)\rightarrow H^{2}(H,\sigma)$ is the restriction of a linear differential operator. We show, the multiplicity of $(\pi_{\mu}^{H},H^{2}(H,\sigma))$ in $res_{H}(\pi_{\lambda})$ is finite.

The hypothesis and $i)$ yields $res_{H}(\pi_{\lambda})$ is discrete decomposable. Let’s assume the multiplicity of $H^{2}(H,\sigma)$ in $res_{H}(\pi_{\lambda})$ is infinite. Thus, there exists $T_{1},T_{2},\dots$ so that $T_{j}:H^{2}(H,\sigma)\rightarrow\,H^{2}(G,\tau)[V_{\mu}^{H}]$ are isometric immersion intertwining linear $H-$maps so that for $r\not=s$ the image of $T_{r}$ is orthogonal to the image of $T_{s}$ and the algebraic sum of the subspaces $T_{r}(H^{2}(H,\sigma)),\,\,r=1,2,\cdots$ is dense in $\,H^{2}(G,\tau)[V_{\mu}^{H}]$. Let $\iota:Z\rightarrow H^{2}(H,\sigma)[Z]$ be the equivariant immersion adjoint to evaluation at the identity. We fix a norm one vector $g_{0}:=\iota(z_{0})\in H^{2}(H,\sigma)[Z]$. There are two possibilities: For some $r$ the function $T_{r}(g_{0})$ is not a $K-$finite vector or else for every $r$ the function $T_{r}(g_{0})$ is a $K-$finite vector. To follow, we analyze the second case, for this, we define $v_{n}:=T_{n}(g_{0})$ and we choose a sequence of nonzero positive real numbers $(a_{n})_{n}$ so that $v_{0}:=\sum_{n}a_{n}v_{n}$ is not the zero vector. Due to the orthogonality for the image of the $T_{r}$ and the choice of the sequence, $v_{0}$ is not a $K-$finite vector. Since the stabilizer of $v_{0}$ in $H$ is equal to the stabilizer of $g_{0}$ on $H$, the correspondence $T:H^{2}(H,\sigma)\rightarrow\,H^{2}(G,\tau)$ defined by $T(h.g_{0})=\frac{1}{\|v_{0}\|}h.v_{0}$ extends to an isometric immersion. We claim $S=T^{\star}$ is not the restriction of a linear differential operator. For this, we show $S(\frac{v_{0}}{\|v_{0}\|})=g_{0}$ and $K_{S}(\cdot,e)^{\star}z_{0}=\frac{v_{0}}{\|v_{0}\|}$. On one hand, we have the following system of equations

xxxxxxx $(S(f)(e),z_{0})_{Z}=\int_{G}(f(y),K_{S}(y,e)^{\star}z_{0})_{W}dy\,\,\,\forall f\in\,H^{2}(G,\tau)$

determine the function $K_{S}(\cdot,e)^{\star}z_{0}$. On the other hand, for arbitrary $f\in H^{2}(G,\tau)$ we have

[TABLE]

Thus, we have shown the equality $K_{S}(\cdot,e)^{\star}z_{0}=\frac{v_{0}}{\|v_{0}\|}.$ Therefore, if $S$ were a differential operator, the fact $v_{0}$ is not a $K-$finite vector, would contradict Lemma 4.2 $b)$. In the first case, a similar argument yields $S:=T_{r}^{\star}$ is not the restriction of a linear differential operator. This concludes the proof of $ii)$ in Proposition 4.4. ∎

Corollary 4.5**.**

Let $\{R_{a}\}$ be a linear basis for $U(\mathfrak{g})$. Then, there exists smooth functions $g_{a,b}$ on $H$ and complex numbers $d_{j,b,i}$ so that

[TABLE]

In fact, we recall the equality $(\natural)$, let $g_{a,b}$ be smooth functions so that $Ad(h^{-1})\check{D_{b}^{\star}}=\sum_{a}g_{a,b}(h)R_{a}.$ We have:

[TABLE]

Example 4.6*.*

For a real form $H/L$ for the Hermitian symmetric space $G/K$ and a holomorphic discrete series $\,H^{2}(G,\tau)$ for $G$, any nonzero intertwining linear $H-$map $S:\,H^{2}(G,\tau)\rightarrow H^{2}(H,\sigma)$ never is the restriction of a differential operator. The statement holds because under our hypothesis $res_{H}(\pi_{\lambda})$ is not discretely decomposable [Ho].

4.2. Equivariant maps into $K-$types

We now show a result related to Lemma 4.2, and by means of a very similar proof. Let $(\vartheta,B)$ be an irreducible representation of $K$. We fix a realization for $(\vartheta,B)$ as a subspace of the space of smooth sections for a bundle $K\times_{\theta}C\rightarrow K/K_{1}$. Here, $K_{1}$ is a closed subgroup of $K$ and $(\theta,C)$ is a finite dimensional representation for $K_{1}.$

Fact 4.7*.*

Assume $(\vartheta,B)$ is a $K-$type for $\,H^{2}(G,\tau)$. Let $S:\,H^{2}(G,\tau)\rightarrow B$ be a continuous intertwining linear map. Then, $S$ is the restriction of a differential operator.

Indeed, we fix $c\in C,k\in K$, then, the map $\,H^{2}(G,\tau)\ni f\mapsto(S(f)(k),c)_{C}$ is a continuous linear functional in $\,H^{2}(G,\tau)$. Thus, there exists $K_{S}:G\times K\rightarrow Hom_{\mathbb{C}}(W,C)$ so that $y\mapsto K_{S}(y,c)^{\star}c$ belongs to $\,H^{2}(G,\tau)$ and $(S(f)(k),c)_{C}=\int_{G}(f(y),K_{S}(y,k)^{\star}c)_{W}dy.$ Actually, $y\mapsto K_{S}(y,c)^{\star}c$ belongs to $\,H^{2}(G,\tau)[B]\subset(V_{\lambda})_{K-fin}.$ The $U(\mathfrak{g})-$irreducibility for $(V_{\lambda})_{K-fin}$ implies there exists $D\in U(\mathfrak{g})$ so that $L_{D}^{(1)}K_{\lambda}(\cdot,e)w=K_{S}(\cdot,e)^{\star}c$ holds. Recalling $K_{S}(y,k)=K_{S}(k^{-1}y,e)$ and proceeding as in the proof of Lemma 4.2 and its Corollary Fact 4.7 follows.

Fact 4.8*.*

Let $(\vartheta,B)$ be as before. We assume $res_{H}(\pi_{\lambda},V_{\lambda})$ is admissible. Then, any intertwining operator from $\,H^{2}(G,\tau)$ into $(\vartheta,B)$ is equal to a restriction of a differential operator.

In fact, the $H-$admissibility forces $V_{\lambda}[B]\subset(V_{\lambda})_{K-fin}$. Now, the proof goes word by word as the one for Fact 4.7.

Remark 4.9*.*

In the setting of Fact 4.7 or Fact 4.8, we further assume $B$ is realized as the kernel of differential operators. Then, any intertwining map $S:\,H^{2}(G,\tau)\rightarrow B$ extends to an intertwining map from the maximal globalization provided by the kernel of the Schmid operator into $B$. The extension is a differential operator. The proof of this remark is as the proof for Theorem 4.11.

Remark 4.10*.*

Any intertwining linear map $S:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\nu}E)$ restricted to the subspace $\,H^{2}(G,\tau)_{H-disc}$ is at the same time the restriction of a Carleman kernel map and of a differential operator.

4.3. Extension of an intertwining map to maximal globalization

A conjecture of Toshiyuki Kobayashi [K6] predicts that under certain hypothesis, each continuous intertwining linear operator between two maximal globalizations of Zuckerman modules, realized via Dolbeault cohomology, is given by restriction of a holomorphic differential operator. In this subsection, we show an analogous statement for the maximal globalization provided by Schmid operators.

The symbols $G,K,(\tau,W),\,H^{2}(G,\tau),\pi_{\lambda},H,L,(\sigma,Z),H^{2}(H,\sigma),\pi_{\mu}^{H}$ are as in Section 2. Let

[TABLE]

be the Schmid operator [Sch] [Wo]. Similarly, we have a Schmid operator $D_{H}:C^{\infty}(H\times_{\sigma}Z)\rightarrow C^{\infty}(H\times_{\sigma_{1}}Z_{1}).$ Since $D_{G}$ is an elliptic operator, $Ker(D_{G})$ is a closed subspace of the space of smooth sections. Thus, $Ker(D_{G})$ becomes a smooth Frechet representation $\ell$ for $G.$ Among the properties of the kernel of the operator $D_{G}$ are: $\,H^{2}(G,\tau)$ is a linear subspace of $Ker(D_{G});$ the inclusion map $\,H^{2}(G,\tau)$ into $Ker(D_{G})$ is continuous; the subspace of $K-$finite vectors in $Ker(D_{G})$ is equal to the subspace of $K-$finite vectors for $\,H^{2}(G,\tau)$; $Ker(D_{G})$ is a maximal globalization for the underlying Harish-Chandra module for $(\pi_{\lambda},\,H^{2}(G,\tau))$. A similar statement holds for $D_{H}$. Now, we are ready to state the corresponding result.

Theorem 4.11**.**

*We assume $res_{H}(\pi_{\lambda})$ is an $H-$admissible representation. Then, the following two statements hold:

a) Any continuous, $H-$intertwining linear map $S:Ker(D_{G})\rightarrow Ker(D_{H})$ is the restriction of a differential operator.

b) Any continuous $H-$intertwining linear map $S:\,H^{2}(G,\tau)\rightarrow H^{2}(H,\sigma)$ extends to a continuous intertwining operator from $Ker(D_{G})$ to $Ker(D_{H}).$*

In [N, Theorem 3.6], Nakahama has shown a similar result under the hypothesis of both $G/K,H/L$ are Hermitian symmetric spaces, the inclusion $H/L$ into $G/K$ is holomorphic, and both representations are holomorphic discrete series

Proof of Theorem 4.11.

We show $a)$. Let $S$ be as in the hypothesis. Since each inclusion $H^{2}(G,\sigma)\subset Ker(D_{G})$, $H^{2}(H,\sigma)\subset Ker(D_{H})$ is continuous, we have for $h\in H,z\in Z$ that the linear functional $\,H^{2}(G,\tau)\ni f\mapsto(S(f)(h),z)_{Z}$ is continuous; whence, Riez representation Theorem implies there exists an element $y\mapsto K_{S}(y,h)^{\star}z$ of $\,H^{2}(G,\tau)$ so that $(S(f)(h),z)_{Z}=\int_{G}(f(y),K_{S}(y,h)^{\star}z)_{W}dy.$ It readily follows that $y\mapsto K_{S}(y,e)^{\star}z$ is an $L-$finite vector. Since, for discrete series, the hypothesis of $H-$admissibility implies $L-$admissibility, [DV], we apply [K3, Proposition 1.6], hence, $y\mapsto K_{S}(y,e)^{\star}z$ is a $K-$finite vector. The $U(\mathfrak{g})-$irreducibility of the subspace of $K-$finite vectors yields $K_{S}(\cdot,e)^{\star}z=L_{D_{z,w}}^{(1)}K_{\lambda}(\cdot,e)^{\star}w$. Next, for a $K-$finite vector $f$ after a computation similar to the one in the proof of Theorem 4.3, we arrive at the equality $(S(f)(h),z)_{Z}=([R_{\check{D_{z,w}^{\star}}}f](h),w)_{W}$. The continuity of $S$ together with $Ker(D_{G})$ is a maximal globalization, let us conclude: $S$ is the restriction of a differential operator. Thus, we have shown $a)$. We now verify $b)$. The hypothesis of $H-$admissibility, let us apply Theorem 4.3. Therefore, $S$ is the restriction of a differential operator. More precisely, $S(f)(h)=\sum_{a,b,i}d_{a,b,i}([R_{D_{b}}f](h),w_{i})_{W}z_{a}$ and $D_{H}(S(f))\equiv 0$ for any $K-$finite vector $f$ in $\,H^{2}(G,\tau)$. We extend $S$ to $Ker(D_{G})$ via the previous equality. Obviously the extension is continuous in smooth topology. We claim: the image of the extension is contained in $Ker(D_{H}).$ Indeed, owing to the subspace of $K-$finite vectors in $\,H^{2}(G,\tau)$ is dense in $Ker(D_{G})$ for smooth topology, we obtain $D_{H}(Sf)\equiv 0$ for every $f\in Ker(D_{G})$. Whence, we have shown Theorem 4.11.∎

Theorem 4.11 is a step in the proof of the conjecture of Kobayashi. Actually, the statement in Theorem 4.11 is a solution to the conjecture of Kobayashi if we choose as maximal globalizations the one constructed via Schmid operator. In order to formulate the conjecture, we need to recall notation as well as results from Schmid thesis [Sch]. The Harish-Chandra parameter $\lambda$ gives rise to $G-$invariant complex structure on $G/T$, as well as a $K-$invariant complex structure on $K/T$ and holomorphic line bundles $\mathcal{L}_{\lambda}\rightarrow G/T$, $\mathcal{L}_{\lambda}\rightarrow K/T$ so that the representation of $K$ in $H^{s}(K/T,\mathcal{O}(\mathcal{L}_{\lambda}))$ is equivalent to $(\tau,W)$ and the representation $\ell^{\star}$ of $G$ on $H^{s}(G/T,\mathcal{O}(\mathcal{L}_{\lambda}))$ is infinitesimally equivalent to $(\pi_{\lambda},V_{\lambda}^{G}).$ Here, $s=\frac{1}{2}dimK/T.$ Owing to the construction of the respective complex structures the inclusion map $i_{K}:K/T\rightarrow G/T$ is holomorphic. After we endow the space of smooth forms on $G/T$ with the smooth topology, the work of Schmid, Wolf and Hon-Wei Wong shows that the image of $\bar{\partial}$ is closed. Thus, $H^{s}(G/T,\mathcal{O}(\mathcal{L}_{\lambda}))$ affords a Frechet representation for $G.$ Next, we describe an equivalence $F_{K,T}$ between the representations $(\ell,Ker(D_{G}))$ and $(\ell^{\star},H^{s}(G/T,\mathcal{O}(\mathcal{L}_{\lambda}))).$ For this, we model $(\tau,W)$ on $(\ell^{\star},H^{s}(K/T,\mathcal{O}(\mathcal{L}_{\lambda})))$. Then, for a smooth $(0,s)-$form $\varphi$ on $G/T$ with values on $\mathcal{L}_{\lambda}$, Schmid associates the function $G\ni g\mapsto F_{K,T}(\varphi)(g):=i_{K}^{\star}(\ell^{\star}_{g}\varphi)\in H^{s}(K/T,\mathcal{O}(\mathcal{L}_{\lambda})).$ Schmid shows that when $\varphi$ is closed we have that $F_{K,T}(\varphi)$ belongs to $Ker(D_{G})$ and the resulting map from $H^{s}(G/T,\mathcal{O}(\mathcal{L}_{\lambda}))$ into $Ker(D_{G})$ is bijective. We do not describe the inverse of the map $F_{K,T}$. Similarly, attached to the representation $V_{\mu}^{H}$, we have a holomorphic line bundle $\mathcal{L}_{\mu}$ over $H/U$, and a map $F_{L,U}:H^{s^{\prime}}(H/U,\mathcal{O}(\mathcal{L}_{\mu}))\rightarrow Ker(D_{H})$. The conjecture of Kobayashi is: assume $res_{H}(\pi_{\lambda})$ is an $H-$admissible representation. Then, for every discrete factor $H^{2}(H,\sigma)$ of $res_{H}(\,H^{2}(G,\tau))$, we have that any $H-$intertwining continuous map from $H^{s}(G/T,\mathcal{O}(\mathcal{L}_{\lambda}))$ into $H^{s^{\prime}}(H/U,\mathcal{O}(\mathcal{L}_{\mu}))$ is a holomorphic differential operator. We are able to show that each intertwining operator is represented by a differential operator. In fact, let $S:H^{s}(G/T,\mathcal{O}(\mathcal{L}_{\lambda}))\rightarrow H^{s^{\prime}}(H/U,\mathcal{O}(\mathcal{L}_{\mu}))$ be a continuous intertwining operator. Then, $F_{L,U}SF_{K,T}^{-1}$ is a continuous intertwining linear map from $\,H^{2}(G,\tau)$ into $H^{2}(H,\sigma)$, Theorem 4.11 yields that this composition is a differential operator in our sense.

4.4. Comments on the relation among Hom’s

As usual, $Hom_{H}(...,...)$ denotes the space of continuous intertwining operators. We have the natural inclusions

[TABLE]

as well as, similar inclusions for $Hom^{\prime}s(H^{2}(H,\sigma),\,H^{2}(G,\tau)).$ We would like to point out that when $res_{H}(\pi_{\lambda})$ is $H-$admissible the above inclusions are equalities. In fact, in [K3, Lemma 1.3, Prop. 1.6], it is shown that the above inclusions are equalities under the hypothesis of $res_{H}(\pi_{\lambda})$ is $L-$admissible. Actually, in [DV], it is shown for discrete series representations $H-$admissible is equivalent to be $L-$admissible. As a consequence, of both facts, we obtain the equalities

[TABLE]

Here, the sum is algebraic. Thus, this work of Kobayashi together with Theorem 4.3 shows that once we know a representation $V_{\lambda}$ is $H-$admissible, the associated branching law problem is algebraic.

In different papers T. Kobayashi, his co-authors and other authors have done a deep study of the space of continuous intertwining linear operators between two principal series representations; their results yields estimates for dimension of such spaces as well as precise computation of such spaces. Next, we present some comments of our work for discrete series representations. For this paper,

[TABLE]

is the space of not necessarily continuous linear intertwining maps that are restriction of differential operators. We have shown in Lemma 4.2 the inclusion "automatic continuity Theorem"

[TABLE]

We have shown that a representation is $H-$admissible if and only if for every $H^{2}(H,\sigma)$ the following equality holds,

[TABLE]

Besides, we have shown that if for some $H^{2}(H,\sigma)$ we have

[TABLE]

then $res_{H}(\pi_{\lambda})$ is discretely decomposable. Therefore, $H-$discretely decomposable and multiplicity of $H^{2}(H,\sigma)$ in $res_{H}(\pi_{\lambda})$ is infinite, forces some symmetry breaking map from $\,H^{2}(G,\tau)$ into $H^{2}(H,\sigma))$ is not the restriction of a differential operator.

Also, for a closed subgroup $L_{1}$ of $K$, under the hypothesis that $\pi_{\lambda}$ is $L_{1}-$admissible, after we realize each irreducible representation $(\vartheta,B)$ for $L_{1}$ in some space of smooth sections, we have shown the equality

xxxxxxxxxx $Diff_{L_{1}}(\,H^{2}(G,\tau),B)=Hom_{L_{1}}(\,H^{2}(G,\tau),B).$

Moreover, each continuous $L_{1}$-map from $\,H^{2}(G,\tau)$ into $B$ extends to a $L_{1}-$differential operator from $Ker(D_{G})$ into $B.$

5. Two results on $L^{2}(G\times_{\tau}W)[V_{\lambda_{1}}^{G}]$

Besides, $(\tau,W),\,H^{2}(G,\tau)$, as in section 2, we consider another square integrable irreducible representation $V_{\lambda_{1}}^{G}$ for $G$ of Harish-Chandra parameter $\lambda_{1}$ and lowest $K-$type $(\tau_{1},W_{1}).$ In this section, we show that any intertwining $G-$map from $H^{2}(G,\tau_{1})$ into $L^{2}(G\times_{\tau}W)$ is an integral operator. In the second part of this section, we compute a kernel for the orthogonal projector onto the isotypic component in $L^{2}(G\times_{\tau}W)$ determined by $V_{\lambda_{1}}^{G}$.

5.1. Analysis for the elements of $Hom_{G}(H^{2}(G,\tau_{1}),L^{2}(G\times_{\tau}W))$

In this subsection, we develop a similar study to the one developed by [K5] on Shintani’s functions. For a Harish-Chandra parameter $\lambda_{1}$ we study continuous intertwining linear map $T:H^{2}(G,\tau_{1})=V_{\lambda_{1}}^{G}\rightarrow L^{2}(G\times_{\tau}W)$. To begin with, we show that the natural inclusion maps

[TABLE]

are bijective. In fact, let $T_{0}$ be a linear map in $Hom_{(\mathfrak{g},K)}(...,...),$ then, due that $ImT_{0}$ is contained in a Hilbert space, we have $(closure(ImT_{0}))_{K-fin}=(ImT_{0})_{K-fin}$. Thus, $closure(ImT_{0})$ is an irreducible unitary representation. In [Wa1, Lemma 3.4.11], we find a proof that $T_{0}$ extends to a continuous intertwining map $T$ from $H^{2}(G,\tau_{1})$ onto the closure of the image of $T_{0}.$ Hence, the claim follows. Next, we show, for $T\in Hom_{G}(V_{\lambda_{1}}^{G},L^{2}(G\times_{\tau}W))$ that

Proposition 5.1**.**

$T$* is represented by a $G-$invariant smooth kernel $k_{T}$. We set $k(x):=K_{T}(x,1).$ The function $k$ satisfies: $k:G\rightarrow Hom_{\mathbb{C}}(W_{1},W)$ is smooth; $k(k_{1}xk_{2})=\tau(k_{1})k(x)\tau_{1}(k_{2}),k_{1},k_{2}\in K,x\in G$; $k$ is a solution to the equation $L_{\Omega_{G}}k=[(\lambda_{1},\lambda_{1})-(\rho,\rho)]k$; and $k$ is square integrable. Conversely, given a function $k$ which satisfies the four previous properties, then $Tf(x)=\int_{G}k(x^{-1}y)f(y)dy$ defines a continuous intertwining continuous linear map from $H^{2}(G,\tau_{1})$ into $L^{2}(G\times_{\tau}W).$*

Proof.

For the direct affirmation we notice that since $T$ is continuous, Schur’s Lemma yields that $T^{\star}T$ is a constant times the identity, hence, we may and will assume $T$ is an isometry into its image. Thus, $Im(T)$ is a closed irreducible left invariant subspace of $L^{2}(G\times_{\tau}W).$ Therefore, $Im(T)$ is included in an eigenspace of the Casimir operator, whence $Im(T)$ is contained in the kernel of an elliptic $G-$invariant operator. Therefore, $T$ is given by a kernel $K_{T}:G\times G\rightarrow Hom_{\mathbb{C}}(W_{1},W)$ so that the map $y\mapsto K_{T}(y,x)^{\star}w$ belongs to $H^{2}(G,\tau_{1})$ for each $w\in W,x\in G.$ Since $T$ is an intertwining map we have the equality $K_{T}(y,x)=K_{T}(x^{-1}y,e).$ Thus, $K_{T}$ is a smooth function. From $K_{T}(ky,xk_{2})=\tau(k_{2}^{-1})K_{T}(y,x)\tau_{1}(k)$ we obtain $k(k_{1}xk_{2})=\tau(k_{1})k(x)\tau_{1}(k_{2}).$ Since $y\mapsto K_{T}(y,x)^{\star}w$ belongs to $H^{2}(G,\tau_{1})$ we obtain that $L_{\Omega_{G}}k=[(\lambda_{1},\lambda_{1})-(\rho,\rho)]k$ and that $k$ is square integrable. Conversely, given $k$ that satisfies the four properties listed. Then, in [HC2, Corollary to Lemma 65], it is shown $k$ is tempered in the sense of Harish-Chandra, the hypothesis $G$ is linear let us apply [OV, Proposition 6] to $k$ and deduce $k\in L^{2-\epsilon}(G,Hom_{\mathbb{C}}(Z,W))$ for $\epsilon$ small. Thus, the Kunze-Stein phenomena, [C], let us conclude that the proposed formula for $T$ defines a continuous linear map. ∎

Actually, it can be shown that the knowledge of one explicit immersion of the Harish-Chandra module for $H^{2}(G,\tau_{1})$ into a non-unitary principal series representation yields that the kernel of each element in $Hom_{G}(H^{2}(G,\tau_{1}),\\ L^{2}(G\times_{\tau}W))$ can be written as a Eisenstein integral.

One way to compute the dimension of the space $Hom_{G}(H^{2}(G,\tau_{1}),L^{2}(G\times_{\tau}W))$ is via Frobenius reciprocity and Blattner’s formula [DHV]. Thus, there is an explicit formula of $dim\,Hom_{G}(H^{2}(G,\tau_{1}),L^{2}(G\times_{\tau}W))$ based on the Harish-Chandra parameter for $H^{2}(G,\tau_{1})$, the highest weight for $W$ and a partition function associated to the noncompact roots with positive inner product with respect to the Harish-Chandra parameter for $H^{2}(G,\tau_{1}).$

5.2. Kernel for the projector onto $L^{2}(G\times_{\tau}W)[V_{\lambda_{1}}^{G}]$

In this section we generalize a Theorem of [WW] where they compute the kernel for orthogonal projector onto the isotypic component $L^{2}(G\times_{\tau}W)[V_{\lambda}^{G}]=\,H^{2}(G,\tau).$ We also extend work of Shimeno [Shi] for the case of line bundles over $G/K$. We fix a representative $(\pi_{\lambda_{1}},V_{\lambda_{1}}^{G})$ of the representation for $G$ of Harish-Chandra parameter $\lambda_{1}$ and we assume $(\tau,W)$ is a $K-$type for $\pi_{\lambda_{1}}.$ We fix an orthonormal basis $\{f_{j}\}_{j=1\dots N}$ for $V_{\lambda_{1}}^{G}[W]$ and recall the spherical trace function $\phi_{1}(z)=d_{\lambda_{1}}\sum_{j}(\pi_{\lambda_{1}}^{G}(z)f_{j},f_{j})_{V_{\lambda_{1}}^{G}}.$ Then,

Proposition 5.2**.**

The orthogonal projector $P$ onto $L^{2}(G\times_{\tau}W)[V_{\lambda_{1}}^{G}]$ is kernel map given the "external kernel"

[TABLE]

and by the matrix valued Carleman kernel

[TABLE]

Proof.

Let $Q$ denote the orthogonal projector onto $V_{\lambda_{1}}^{G}[W]$. Then, each nonzero intertwining $K-$map $b:V_{\lambda_{1}}^{G}[W]\rightarrow W$ yields a map $f_{b}:V_{\lambda_{1}}^{G}\rightarrow L^{2}(G\times_{\tau}W)$ defined by $V_{\lambda_{1}}^{G}\ni v\mapsto(G\ni x\mapsto b(Q(\pi_{\lambda_{1}}(x^{-1}v)):=f_{b}(v,x)).$ It is obvious that $f_{b}$ gives rise to an equivariant embedding of $V_{\lambda_{1}}^{G}$ into $L^{2}(G\times_{\tau}W)[V_{\mu}^{G}]$. After we fix a linear basis for $Hom_{K}(V_{\mu}^{G}[W],W),$ owing to Frobenius reciprocity, the subspace $L^{2}(G\times_{\tau}W)[V_{\lambda_{1}}^{G}]$ is equal to the linear span of the image of the functions $f_{b}$ when $b$ runs over the chosen linear basis for $Hom_{K}(V_{\lambda_{1}}^{G}[W],W)$. Next, the spherical trace function $\phi_{1}$ is $K-$central, hence, it follows that for each $f$ in $L^{2}(G\times_{\tau}W)$, the right hand side of $({\dagger})$ belongs to $\Gamma(G\times_{\tau}W)$. The hypothesis $G$ is a linear group let us conclude that $\phi_{1}\in L^{2-\epsilon}(G)$ for some positive $\epsilon.$ Thus, the Kunze-Stein phenomena [C] yields the right hand side determines a continuous linear operator on $L^{2}(G\times_{\tau}W)$. Next, as in Appendix 11.6, we verify $\int_{G}\phi_{1}(x^{-1}y)f_{b}(v,y)dy=f_{b}(v,x).$ Harish-Chandra Plancherel’s Theorem yields the integral $({\dagger})$ evaluated at a wave package $f$ orthogonal to $L^{2}(G\times_{\tau}W)[V_{\mu}^{G}]\cap(L^{2}(G\times_{\tau}W))_{disc}$ is equal to zero. Thus, the first statement holds. The second statement readily follows. ∎

6. Projector onto isotypic component of $res_{H}(\pi_{\lambda})$

6.1. Projector onto isotypic components via differential operators

In [N], for a scalar holomorphic discrete series, it is shown that the orthogonal projector onto an isotypic component is equal to an infinite order differential operator when restricted to the subspace of smooth vectors. This subsection aims is to analyze the general value of his results. For this, we analyze ways of expressing the orthogonal projector onto an isotypic component using differential operators. To begin with, we study an example.

Example 6.1*.*

Let $G=SU(1,1)$. We fix as $K=$diagonal matrices $(e^{i\varphi},e^{-i\varphi}).$ The characters of $K$ are $e^{in\varphi},n\in{\mathbb{Z}}.$ The discrete series $(\pi_{\lambda},V_{\lambda})$ is presented as a subspace of the space of holomorphic functions $f(z)=a_{0}+a_{1}z+\cdots+a_{n}z^{n}+\cdots$ on the unit disk $D=\{z\in{\mathbb{C}}:|z|<1\}$. The action of $G$ is by homographic transformations. The action of $K$ is such that a typical isotypic subspace is $V_{\lambda}[V_{n}^{K}]={\mathbb{C}}z^{n}.$ It readily follows that $P_{V_{\lambda}[V_{n}^{K}]}(f)(z)=c_{\lambda,n}z^{n}\frac{\partial^{n}f}{\partial z^{n}}(0)$ and $K_{V_{\lambda}[V_{n}^{K}]}(z,w)=d_{\lambda,n}\bar{z}^{n}w^{n}.$ Here, the reproducing kernel is $K_{\lambda}(z,w)=\frac{k_{\lambda}}{(1-\bar{z}w)^{\lambda}}$ and $K_{V_{\lambda}[V_{n}^{K}]}(z,w)=d_{\lambda,n}\bar{z}^{n}w^{n}\frac{\partial^{n}}{\partial\bar{z}^{n}}\frac{\partial^{n}}{\partial{w}^{n}}(K_{\lambda})(0,0)$. We point out that since $K_{\lambda}$ is a rational function and $K_{V_{\lambda}[V_{n}^{K}]}$ is a polynomial function, hence, no derivative of $K_{\lambda}$ is equal to $K_{V_{\lambda}[V_{n}^{K}]}.$ This example expresses $P_{V_{\lambda}[V_{n}^{K}]}(f)(\cdot)=F_{1}(\cdot)D(f)(e)$, $K_{V_{\lambda}[V_{n}^{K}]}(\cdot)=F_{2}(\cdot)\tilde{D}(K_{\lambda})(e,e)$, where $D,\tilde{D}$ are elements of $U(\mathfrak{sl}(2,{\mathbb{R}}))$, and, $F_{1},F_{2}$ are smooth functions. In Fact 6.4, we show this is a common feature for the orthogonal projector onto a $K-$isotypic component as well as for its kernel.

Example 6.2*.*

This example is a sequel to the previous one. We would like to point out that if a differential operator $D:=f_{0}+f_{1}\partial+\cdots+f_{N}\partial^{N}+\dots$ $(\partial=d/dz)$ is equal to orthogonal projector onto ${\mathbb{C}}z^{M}$, then, $D$ ought to be of infinite order. In fact, after a direct computation we obtain

[TABLE]

Example 6.3*.*

Let $G,H,\,H^{2}(G,\tau)$ be as in Section 2. We assume $\pi_{\lambda}$ is an $H-$admissible representation. We fix an isotypic component $M$ for the action of $H.$ To follow, we compute an infinite order differential operator $D$ so that its restriction to the subspace of $K-$finite vectors agrees with the restriction of the orthogonal projector $P_{M}$ onto the isotypic component $M$. For this, we label the isotypic components as $M_{1},M_{2}=M,M_{3},M_{4},\cdots$. We fix a nonzero element $\Omega_{1}$ in the center of $U(\mathfrak{h})$. Thus, $R:=L_{\Omega_{1}}$ acts by a constant $c_{j}$ on the subspace of $H-$smooth vectors in $M_{j}$. We further assume $c_{k}\notin\{c_{1},\dots,c_{k-1}\}$ for $k\geq 2$. We claim: there exists a sequence of numbers $d_{0},d_{1},\dots$ such that

[TABLE]

In fact, the hypothesis $\pi_{\lambda}$ is $H-$admissible together with a result of T. Kobayashi implies that the subspace of $L-$finite vectors in $\,H^{2}(G,\tau)$ is equal to the subspace of $K-$finite vectors. Next, the hypothesis on $c_{1}$ lets us find $d_{0}$ so that $d_{0}(R-c_{1})$ is equal to the identity in the subspace of $K-$finite vectors in $M_{2}$. Then, we find $d_{1}$ so that $d_{0}(R-c_{1})+d_{1}(R-c_{1})(R-c_{2})$ is equal to zero on the subspace of $K-$finite vectors in $M_{3}$, and so on.

The proof of the next fact shows a technique, which, let us compute differential operators with smooth coefficients to describe the orthogonal projector onto a finite dimensional $K-$invariant subspace $M$ of $\,H^{2}(G,\tau)$.

Fact 6.4*.*

Let $(\vartheta,V_{\vartheta}^{K})$ denote an irreducible representation for $K$. Let $P_{V_{\lambda}[V_{\vartheta}^{K}]}$ denotes the orthogonal projector onto $V_{\lambda}[V_{\vartheta}^{K}]$, let $K_{V_{\lambda}[V_{\vartheta}^{K}]}:G\times G\rightarrow Hom_{\mathbb{C}}(W,W)$ denotes the kernel that represents $P_{V_{\lambda}[V_{\vartheta}^{K}]}$. We fix an orthonormal basis $\{w_{i}\}$ for $W.$ Then, there exists a family $\{D_{i,\alpha}\}$ $1\leq i\leq dimW,1\leq\alpha\leq\dim V_{\lambda}[V_{\vartheta}^{K}]$ of elements in $U(\mathfrak{g})$ and complex valued smooth functions $F_{i,\alpha}$ on $G$ so that

[TABLE]

Further,

[TABLE]

Indeed, the identity

[TABLE]

shows the function $y\mapsto K_{V_{\lambda}[V_{\vartheta}^{K}]}(y,x)^{\star}w_{i}$ is orthogonal to any function $f$ orthogonal to $V_{\lambda}[V_{\vartheta}^{K}].$ Thus, $y\mapsto K_{V_{\lambda}[V_{\vartheta}^{K}]}(y,x)^{\star}w_{i}$ belongs to $V_{\lambda}[V_{\vartheta}^{K}]$. The function

[TABLE]

is smooth, because $V_{\lambda}[V_{\vartheta}^{K}]$ is a finite dimensional vector space and the equality $(d)$ shows that the evaluation of coordinate functions in $H_{i}(x)$ yields a smooth function on $G$. Next, we set $k_{w}(\cdot):=K_{\lambda}(\cdot,e)^{\star}w$. Then, $k_{w}$ belongs to $V_{\lambda}[W]$, and it is non zero for each nonzero $w$. We fix $i:1\leq i\leq\dim W.$ The $U(\mathfrak{g})-$irreducibility for $(V_{\lambda})_{K-fin}$ implies there exists a finite dimensional vector subspace $N_{\vartheta,i}$ of $U(\mathfrak{g})$ so that the linear map $R_{i}:N_{\vartheta,i}\rightarrow V_{\lambda}[V_{\vartheta}^{K}]$ defined by $R_{i}(D):=L_{D}(k_{w_{i}})$ is bijective. Therefore, the composition $R_{i}^{-1}H_{i}$ is a smooth function on $G$. To follow, we fix a linear basis $D_{i,\alpha},1\leq\alpha\leq\dim V_{\lambda}[V_{\vartheta}^{K}]$ for $N_{\vartheta,i}$. Hence, $R_{i}^{-1}H_{i}(x)=\sum_{\alpha}F_{i,\alpha}(x)D_{i,\alpha}$ where $F_{i,\alpha}$ are complex valued smooth functions on $G.$ Thus,

[TABLE]

[TABLE]

For $f\in V_{\lambda}^{\infty}$, the following equalities hold.

[TABLE]

Finally, the equality of the first and last member extends to any $f\in H^{2}(G,\tau)$ owing to that, for a fixed $x\in G$, both members are continuous linear functionals on $H^{2}(G,\tau)$ and they agree on smooth vectors. This concludes the proof of Fact 6.4.

Fact 6.5*.*

A completely similar result to Fact 6.4 holds after we replace $V_{\lambda}[V_{\vartheta}^{K}]$ by a finite dimensional subspace $M$ of $(V_{\lambda})_{K-fin}.$ Of course, the elements $F_{i,\alpha},D_{i,\alpha}$ depend on $M$ and they are highly non unique.

On the basis of Fact 6.4, Fact 6.5, we say that a linear map $T$ from a subspace of $\,H^{2}(G,\tau)$ into $\,H^{2}(G,\tau)$ is expressed by means of differential operators if for each orthonormal basis $\{w_{i}\}$ for $W$, it is possible to find finitely many smooth complex valued functions $f_{i,\alpha}$ on $G$ and a finite family $\{D_{i,\alpha}\}$ of elements in $U(\mathfrak{g})$ so that for every $f$ in the domain of $T$ and $x\in G$ we have $T(f)(x)=\sum_{i,\alpha}f_{i,a}(x)(L_{D_{i,\alpha}}(f)(e),w_{i})_{W}w_{i}.$

The next result shows a relation between discretely decomposable and orthogonal projectors represented by means of differential operators.

Proposition 6.6**.**

We assume $res_{H}(\pi_{\lambda})$ is an $H-$admissible representation. Then, for any finite dimensional subspace $M$ of $(V_{\lambda})_{L-fin}$ the orthogonal projector onto $M$ is expressed by means of differential operators. Conversely, if for some Harish-Chandra parameter $\mu$ for $H$, the orthogonal projector onto some nontrivial finite dimensional subspace of $(V_{\lambda}[V_{\mu}^{H}])_{L-fin}$ is expressed by means of differential operators, then, $res_{H}(\pi_{\lambda})$ is discretely decomposable. Furthermore, if for each nontrivial finite dimensional $L-$invariant subspace of $V_{\lambda}[V_{\mu}^{H}]$, its orthogonal projector is expressed by means of differential operators, then, the multiplicity of $V_{\mu}^{H}$ in $res_{H}(\pi_{\lambda})$ is finite.

Proof.

For the direct implication, the hypothesis $H-$admissible let us apply [K3, Proposition 1.6]. Thus, $(V_{\lambda})_{L-fin}=(V_{\lambda})_{K-fin}.$ Whence, Fact 6.5 yields there exists $\{D_{i,\alpha}\}_{,}1\leq i\leq\dim W,1\leq\alpha\leq\dim M$ of elements in $U(\mathfrak{g})$ and complex valued smooth functions $F_{i,\alpha}$ on $G$, so that for every $f\in V_{\lambda}$ we have

[TABLE]

Whence, $P_{M}$ is computed by means of differential operators. For the converse statement, the hypothesis on $M$ gives an expression for $P_{M}$ as above and for $K_{P_{M}}$ ($w,v\in W$) we have

[TABLE]

Hence, for a fixed $x\in G$, the function $y\mapsto K_{P_{M}}(y,x)^{\star}w$ is $K-$finite, because the above expression gives that the function $y\mapsto K_{P_{M}}(y,x)^{\star}w$ is equal to a finite sum of $K-$finite vectors of type $y\mapsto L_{D}K_{\lambda}(y,e)^{\star}w.$ Now, any $H-$smooth element in $V_{\lambda}[V_{\mu}^{H}]_{L-fin}$ is $\mathfrak{z}(U(\mathfrak{h}))-$finite, and, by hypothesis, $M\subset V_{\lambda}[V_{\mu}^{H}]_{L-fin}$. Thus, $K_{P_{M}}(\cdot,x)^{\star}w$ is $K-$finite and $\mathfrak{z}(U(\mathfrak{h}))-$finite. Whence, as in previous Theorems, we conclude $res_{H}(\pi_{\lambda})$ is algebraically decomposable. Thus, $res_{H}(\pi_{\lambda})$ is discretely decomposable. To follow, we show the last statement in Proposition 6.6. The proof goes parallel to the proof of Theorem 4.3 ii). That is, we assume $V_{\lambda}[V_{\mu}^{H}]$ is not $H-$admissible and we built up a finite dimensional $L-$invariant subspace $M$ so that $P_{M}$ is not expressed by means of differential operators. Let $T_{j}:V_{\mu}^{H}\rightarrow V_{\lambda}[V_{\mu}^{H}],$ $j=1,\dots$ be isometric immersions $H-$maps so that the subspaces $T_{j}(V_{\mu}^{H})$ are pairwise orthogonal. We fix $v_{0}\in V_{\mu}^{H}[V_{\sigma}^{L}]$ of norm one. There are two possibilities, either every $T_{j}(v_{0})$ is a $K-$finite vector, or at least one $T_{j}(v_{0})$ is not a $K-$finite vector. In the second case we set $w:=T_{j}(v_{0})$, and $M=\text{linear \,span}\,\pi_{\lambda}(L)w$. In the first case, we choose a sequence of positive real numbers $(a_{n})_{n}$ so that $w:=\sum_{n}a_{n}v_{n}$ is nonzero and we set $M=\text{linear \,span}\,\pi_{\lambda}(L)w$. Given that $\pi_{\lambda}$ is $K-$admissible, we have $w$ is not a $K-$finite vector. As in the proof of Theorem 4.3, $M$ is a finite dimensional subspace of $V_{\lambda}[V_{\mu}^{H}]$ and $P_{M}$ is not the restriction of a differential operator because $K_{P_{M}}(\cdot,e)^{\star}v_{0}=w$ is not a $K-$finite vector. This concludes the proof of Proposition 6.6 ∎

Next, after we assume $res_{H}(\pi_{\lambda})$ is an $H-$admissible representation, we show the orthogonal projector onto an isotypic component for $H$ can be expressed by means of an infinite degree differential operator on the subspace of smooth vectors for $G$. The result generalizes [N, Theorem 3.10] for $\pi_{\lambda}$ a scalar holomorphic discrete series.

The set up for the next Theorem $G,K,(\tau,W),\,H^{2}(G,\tau)=V_{\lambda},\pi_{\lambda},H,L,V_{\mu}^{H}\\ =H^{2}(H,\sigma),\pi_{\mu}^{H}$ are as in Section 2.

Theorem 6.7**.**

We assume $res_{H}(\pi_{\lambda})$ is an admissible representation for $H.$ We denote by $\sigma_{1},\sigma_{2},\cdots$ the $L-$types for $V_{\mu}^{H}$. Let $P_{\lambda,\mu}$ denote the orthogonal projector on the isotypic component $V_{\lambda}[V_{\mu}^{H}]$. Then, for $j\in\mathbb{N}$, $1\leq i\leq\dim W$, $1\leq\alpha\leq\dim V_{\lambda}[V_{\mu}^{H}][V_{\sigma_{j}}^{L}]$, there exists a family $\{D_{j,i,\alpha}\},$ of elements in $U(\mathfrak{g})$ and complex valued smooth functions $F_{j,i,\alpha}$ on $G$, such that for each smooth vector $f\in V_{\lambda}$ we have

[TABLE]

The convergence is in smooth topology. Furthermore, for a $K-$finite function $f$, the sum on the right is finite.

Proof.

In [DV], we find a proof that the hypothesis $H-$admissible implies $V_{\lambda}$ is also $L-$admissible. Thus, our hypothesis leads that the isotypic subspaces $V_{\lambda}[V_{\mu}^{H}][V_{\sigma_{j}}^{L}]$ are nonzero, finite dimensional and we have the Hilbert sum $V_{\lambda}[V_{\mu}^{H}]=\oplus_{j}V_{\lambda}[V_{\mu}^{H}][V_{\sigma_{j}}^{L}]$. Furthermore, the hypothesis of being $H-$admissible forces all the $L-$finite vectors are $K-$finite vectors [K3], hence, each subspace $V_{\lambda}[V_{\mu}^{H}][V_{\sigma_{j}}^{L}]$ is contained in $(V_{\lambda})_{K-fin}.$ We now apply Proposition 6.6 to obtain finitely many elements $D_{j,i,\alpha},i=1,\cdots,\dim W,1\leq\alpha\leq\dim V_{\lambda}[V_{\mu}^{H}][V_{\sigma_{j}}^{L}]$ in $U(\mathfrak{g})$ so that

xxxxxx $P_{V_{\lambda}[V_{\mu}^{H}][V_{\sigma_{j}}^{L}]}(f)(x)=\sum_{i,\alpha}F_{j,i,\alpha}(x)(L_{D_{j,i,\alpha}}(f)(e),w_{i})_{W}\,w_{i}.$

Next, the series $\sum_{j}P_{V_{\lambda}[V_{\mu}^{H}][V_{\sigma_{j}}^{L}]}$ converges pointwise to $P_{\lambda,\mu}$. Further, in [HC2], we find a proof that for a smooth vector $f$, the convergence is absolute in the smooth topology. Whence, we have obtained the first statement in Theorem 6.7. The second statement follows because $L-$admissible implies each isotypic component for an irreducible representation of $K$ is contained in a finite sum of isotypic components for $L.$ ∎

6.2. Kernel for the projector onto $\,H^{2}(G,\tau)[V_{\mu}^{H}]$

Let $G,H,(\tau,W)$, $\\ \,H^{2}(G,\tau),K_{\lambda}$ be as in Section 2. Let $(\pi_{\mu}^{H},V_{\mu}^{H})$ denote an irreducible square integrable representation for $H.$ Let $\,H^{2}(G,\tau)[V_{\mu}^{H}]$ denote isotypic component for $V_{\mu}^{H}$ in $\,H^{2}(G,\tau)$ (cf. Notation). Since, $\,H^{2}(G,\tau)$ is a reproducing kernel space, we have that $\,H^{2}(G,\tau)[V_{\mu}^{H}]$ is a reproducing kernel space. Thus, the orthogonal projector $P_{\lambda,\mu}$ onto $\,H^{2}(G,\tau)[V_{\mu}^{H}]$ is a represented by a Carleman matrix kernel $K_{\lambda,\mu}$. In this section, under certain hypothesis, we express the matrix kernel $K_{\lambda,\mu}$ in terms of the matrix kernel $K_{\lambda}$ and the distribution character $\Theta_{\pi_{\mu}^{H}}$ of the representation $(\pi_{\mu}^{H},V_{\mu}^{H}).$ We are quite convinced the formula is true under more general hypothesis. The proposed formula is

Proposition 6.8**.**

Assume the restriction to $H$ of $\pi_{\lambda}$ is an $H-$admissible representation. Then, $P_{\lambda,\mu}$ is equal to the Carleman operator given by the kernel

[TABLE]

In order to avoid cumbersome notation, for this subsection, sometimes, we write $xv:=\pi_{\lambda}(x)v$, $x\in G,v\in\,H^{2}(G,\tau).$ A proof of Proposition 6.8 is given at the end of this section. For the time being, we show Proposition 6.8 under the hypothesis: $G$ is any Lie group and $K,H$ are compact subgroups of $G$. Thus, $V_{\mu}^{H}$ is a finite dimensional vector space. We fix $N$ a reproducing kernel $G-$invariant subspace of $L^{2}(G\times_{\tau}W).$ Under these hypotheses, the orthogonal projector $P_{N}$ onto $N$ is represented by different kernels. For any kernel $K_{N}$ that represents the orthogonal projector $P_{N}$, we want to show $P_{N[V_{\mu}^{H}]}$ is represented by $K_{1}(y,x):=\int_{H}d_{\mu}\bar{\chi}_{\pi_{\mu}^{H}}(h)K_{N}(h^{-1}y,x)dh$. According to a classical result, the orthogonal projector onto $L^{2}(G\times_{\tau}W)[V_{\mu}^{H}]$ is the linear operator $"\pi(d_{\mu}\bar{\chi}_{\pi_{\mu}^{H}})".$ Therefore, for $f\in N$ and $x\in G$, we have

[TABLE]

Thus, $K_{1}$ is a kernel that represents the orthogonal projector onto $N[V_{\mu}^{H}].$ This concludes the verification of Proposition 6.8 for a compact subgroup $H$.

Our proof of Proposition 6.8 is based on an expression for the orthogonal projector onto a closed subspace $E$ of $\,H^{2}(G,\tau)$. For this, we fix a representation $(\pi,V)$ equivalent to $(\pi_{\lambda},\,H^{2}(G,\tau))$ and we assume $W\subset V.$ Then, the map $V\ni v\mapsto(G\ni x\mapsto f_{v}(x):=P_{W}(\pi(x^{-1})v)$ is $G-$equivariant, continuous and bijective from $V$ onto $\,H^{2}(G,\tau)$. We also fix an orthonormal basis $\{v_{i}\}$ for $W$, the equality

xxxxxxxxxxxxxx $P_{W}(\pi(y^{-1})v)=\sum_{1\leq j\leq dimW}(\pi(y^{-1})v,v_{j})_{V}v_{j}$

shows any element of $\,H^{2}(G,\tau)$ is a finite sum of matrix coefficients for $V$.

We notice, for $z,v\in V$, $(f_{v},f_{z})_{L^{2}(G)}=\frac{\dim W}{d_{\lambda}}(v,z)_{V}$. Hence, an unitary equivalence $i$ from $W$ onto $\,H^{2}(G,\tau)[W]$ is given by $i(v)=\sqrt{\frac{d_{\lambda}}{\dim W}}f_{v},$ we have $e_{1}(i(v))=i(v)(e)=\sqrt{\frac{d_{\lambda}}{\dim W}}v.$ Hence $f_{j}:=i(v_{j})=\sqrt{\frac{d_{\lambda}}{\dim W}}f_{v_{j}}$ is an orthonormal basis for $\,H^{2}(G,\tau)[W].$ As usual, $\{v_{j}^{\star}\}$ denotes the dual basis to the basis $\{v_{j}\}.$

For $f\in\,H^{2}(G,\tau)$ the integral below is absolutely convergent, because the product of two $L^{2}$ functions gives an integrable function. We define

xx $C_{E}(f)(x):=\frac{d_{\lambda}}{\dim W}\int_{G}\sum_{j,k}(P_{E}(yf_{j}),xf_{k})_{L^{2}(G)}(v_{k}\otimes v_{j}^{\star})(f(y))dy.$

The indexes in the sum run from 1 to $\dim W$. A straightforward computation, shows that, for $f\in L^{2}(G\times_{\tau}W)$, the function $C_{E}(f)$ belongs to $\Gamma(G\times_{\tau}W).$ We want to show,

Lemma 6.9**.**

$H^{2}(G,\tau)$* is as in Section 2, $E$ is a closed subspace of $H^{2}(G,\tau)$. Then, $C_{E}$ is equal to the orthogonal projector from $\,H^{2}(G,\tau)$ onto $E.$*

Proof.

For $f_{v}\in E\,(\text{resp.}\,\,f_{v}\in E^{\perp}),$ we show that $C_{E}(f_{v})=f_{v}(\text{resp.}\\ C_{E}(f_{v})=0).$ In fact, $f_{v}(y)=\sum_{1\leq r\leq\dim W}(\pi(y^{-1})v,v_{r})_{V}v_{r}.$ Hence,

[TABLE]

Whence, for $f_{v}\in E^{\perp}$, we have $C_{E}(f_{v})=0$, whereas for $f_{v}\in E$, since $P_{E}(f_{v})=f_{v}$, we obtain

[TABLE]

And, $\sum_{j}(f_{j},f_{v_{j}})_{L^{2}(G)}=\sum_{j}\frac{\sqrt{d_{\lambda}}}{\sqrt{\dim W}}(f_{v_{j}},f_{v_{j}})_{L^{2}(G)}=\frac{\sqrt{\dim W}}{\sqrt{d_{\lambda}}}\dim W.$ Thus, $\frac{\dim W}{d_{\lambda}}C_{E}(f_{v})=\frac{\dim W}{d_{\lambda}}f_{v},$ and we have shown Lemma 6.9. ∎

Note. A consequence of Lemma 6.9 is that $C_{E}$ is a continuous linear operator in $\,H^{2}(G,\tau)$. Further, it readily follows that $C_{E}$ is a continuous linear operator on $L^{2}(G\times_{\tau}W)_{disc}$.

Proof of Proposition 6.8.

After we recall the equality

xxxxxxx $K_{\lambda}(y,x)=e_{1}\circ P_{\,H^{2}(G,\tau)[W]}\pi_{\lambda}(x^{-1}y)P_{\,H^{2}(G,\tau)[W]}\circ i$

and Lemma 6.9, we conclude: to show that the matrix kernel for the orthogonal projector onto $\,H^{2}(G,\tau)[V_{\mu}^{H}]$ is equal to the function $(y,x)\mapsto d_{\mu}\Theta_{V_{\mu}^{H}}(h\mapsto K_{\lambda}(h^{-1}y,x)),$ is equivalent to show the equality

[TABLE]

The right hand side of the above equality obviously is a well defined function, we now show the left hand side defines a function. For this, we show that for fixed $y,x$ the function $H\ni h\mapsto K_{\lambda}(hy,x)$ is tempered in the sense of Harish-Chandra. Indeed, the $f_{j}^{\prime}s$ are $K-$finite vectors, hence $\pi_{\lambda}(x)f_{j},\pi_{\lambda}(y)f_{j}$ are smooth vectors for $\,H^{2}(G,\tau)$. Thus, they are tempered functions in the sense of Harish-Chandra. Since, the inner product $(\pi_{\lambda}(h)\pi_{\lambda}(x)f_{i},\pi_{\lambda}(y)f_{j})_{L^{2}(G)}$ can be rewritten as a convolution of tempered functions, we obtain that when we let $h$ varies in $G,$ the matrix coefficient $(\pi_{\lambda}(\cdot)\pi_{\lambda}(x)f_{i},\pi_{\lambda}(y)f_{j})_{L^{2}(G)}$ is a tempered function on $G$. In [HHO, Proposition 2.2], it is shown that the restriction to $H$ of $(\pi_{\lambda}(\cdot)\pi_{\lambda}(x)f_{i},\pi_{\lambda}(y)f_{j})_{L^{2}(G)}$ is a tempered function. Since the character of a discrete series representation is a tempered distribution, we obtain the left hand side defines a function of $x,y.$ To follow, we fix $x,y\,\text{in}\,G$, smooth vectors $v,w\,\text{in}\,\,H^{2}(G,\tau).$ We verify

[TABLE]

Scholium: We have that a $H-$smooth vector in $V_{\lambda}[V_{\nu}^{H}]$ is $G-$smooth. In fact, by hypothesis, $V_{\lambda}[V_{\nu}^{H}]$ is a finite sum of irreducible unitary representations for $H.$ Thus, [Wa2, Theorem 11.8.2], the subspace $(V_{\lambda}[V_{\nu}^{H}])^{H-smooth}$ is finitely generated module over the algebra of rapidly decreasing functions $\mathscr{S}(H)$ on $H.$ Hence, each element of $(V_{\lambda}[V_{\nu}^{H}])^{H-smooth}$ is a finite sum of functions $(\pi_{\lambda})_{|_{H}}(g)(f_{1})$, where $g$ is a rapidly decreasing function on $H$ and $f_{1}$ is an $L-$finite element in $V_{\lambda}[V_{\nu}^{H}])$. Now, from [DV] [K3, Proposition 1.6] it follows that the hypothesis of being $H-$admissible yields that $L-$finite elements are $K-$finite. Thus, any vector in $(V_{\lambda}[V_{\nu}^{H}])^{H-smooth}$ is a finite sum of functions of the type $(\pi_{\lambda})_{|_{H}}(g)(f_{1})$ with $g$ rapidly decreasing on $H$ and $f_{1}$ a smooth vector for $G$. Further, $(V_{\lambda})^{\infty}$ endowed with the smooth topology is a Frechet representation for $G$. Whence, $(V_{\lambda})^{\infty}$ is a Frechet representation for $H$. Therefore, since $f_{1}$ is a $G-$smooth vector we conclude, $(\pi_{\lambda})_{|_{H}}(g)(f_{1})$ is an element of $(V_{\lambda})^{\infty}.$ Thus, every $H-$smooth vector in $V_{\lambda}[V_{\nu}^{H}]$ is $G-$smooth.

Our hypothesis is that $res_{H}(\pi_{\lambda})$ is an $H$-admissible representation. Thus, there exists a subset $Spec(res_{H}(\pi_{\lambda}))$ of the set of Harish-Chandra parameters for $H$ so that we have the Hilbert sum $V_{\lambda}^{G}=\oplus_{\nu\in Spec(res_{H}(\pi_{\lambda}))}V_{\lambda}[V_{\nu}^{H}]$ and $V_{\lambda}[V_{\nu}^{H}]\not=\{0\}$ if and only if $\nu\in Spec(res_{H}(\pi_{\lambda}))$.

Next, we write for $z\in G,u\in(V_{\lambda}^{G})^{\infty}$

[TABLE]

We claim the convergence of the series above is absolutely in both, $L^{2}-$topology and the topology for $(V_{\lambda})^{\infty}$. Moreover, the series converges in the topology of uniform convergence on compact sets for functions as well for any derivatives.

Our hypothesis shows that $L^{2}-$convergence is obvious. Since $\pi_{\lambda}(z)u$ is a smooth vector, we have that for every $\nu$, the vector $P_{\lambda,\nu}(\pi_{\lambda}(z)u)$ is $H-$smooth, the previous claim shows $P_{\lambda,\nu}(\pi_{\lambda}(z)u)$ is $G-$smooth. We recall a result of Harish-Chandra [HC2, Lemma 5] which asserts: the Fourier series of a smooth vector converges absolutely in smooth topology. Therefore, the Fourier series of $\pi_{\lambda}(z)$ as well as the Fourier series for $P_{\lambda,\nu}(\pi_{\lambda}(z)u)$ converges absolutely in smooth topology. Since, $H-$admissible implies $L-$admissible, and that the subspace of $L-$finite vectors is equal to the subspace of $K-$finite vectors, we have that the series $P_{\lambda,\nu}(\pi_{\lambda}(z)u)$ is a rearrangement of a subseries of the Fourier series for $\pi_{\lambda}(z)u$. Thus, the series $\sum_{\nu}P_{\lambda,\nu}(\pi_{\lambda}(z)u)$ converges absolutely in smooth topology. The third affirmation follows from [At].

We recall, in [HC2], it is shown the smooth vectors in $\,H^{2}(G,\tau)$ are tempered functions and convergence in smooth topology implies convergence in the space of tempered functions. Therefore, the series of functions

[TABLE]

converges in the topology for the space of tempered functions. The equality follows from the series (a) applied to $z=hyv$ and the continuity of the inner product on each variable. Applying the equality $(P_{\lambda,\nu}(hyv),xw)_{L^{2}(G)}=(P_{\lambda,\nu}(hyv),P_{\lambda,\nu}xw)_{L^{2}(G)}$ in (b), and, applying to the resulting series the character of $(\pi_{\mu}^{H})^{\star}$, we obtain

[TABLE]

Next, the function $h\mapsto(hP_{\lambda,\nu}(yv),P_{\lambda,\nu}(xw))_{L^{2}(G)}$ is a matrix coefficient for $V_{\lambda}[V_{\nu}^{H}],$ besides, $P_{\lambda,\nu}(yv),P_{\lambda,\nu}(xw)$ are smooth vectors for $H$, hence, the orthogonality relations as written in [HC2, Lemma 84] gives us

[TABLE]

Thus,

[TABLE]

After we apply the above equality to $v=f_{j},w=f_{i}$ and add up, we conclude a proof of Proposition 6.8. ∎

7. Criteria for discretely decomposable restriction

As in previous sections, we keep the hypothesis and notation of Section 2. The objects are $G,K,(\tau,W),\,H^{2}(G,\tau),H,L.$ We recall the orthogonal projector $P_{\lambda}$ onto $\,H^{2}(G,\tau)$, is given by a smooth matrix kernel $K_{\lambda}(y,x)=K_{\lambda}(x^{-1}y,e)=\Phi_{0}(x^{-1}y)$ (cf. Appendix 11.6). Here, $\Phi_{0}$ is the spherical function associated to the lowest $K-$type $(\tau,W)$ of $\pi_{\lambda}^{G}$. Harish-Chandra showed that $\Phi_{0}$, and hence, $tr(\Phi_{0})$ are tempered functions for the definition of Harish-Chandra, [HC2][Wa1, 8.5.1]. In [HHO], we find a proof that the tempered functions on $G$ restricted to $H$ are tempered functions. A tempered function is called a cusp form if the integral along the unipotent radical of any proper parabolic subgroup of any left translate of the function is equal to zero [Wa1, 7.2.2]. Let $r_{n}:H^{2}(G,\tau)\rightarrow L^{2}(H\times_{\tau_{n}}(\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)}\otimes W)$ be as in Example 4.1. The notation $r_{n}(\Phi_{0}^{\star})$ means the family of functions $r_{n}(K_{\lambda}(\cdot,e)^{\star}w)=r_{n}(\Phi_{0}^{\star}(\cdot)w),w\in W.$ The previous considerations yield the family $r_{n}(\Phi_{0}^{\star})$ consists of tempered functions. The purpose of this section is to show:

Theorem 7.1**.**

Let $\pi_{\lambda}$ be a discrete series for $G$, let $\Phi_{0}$ be its lowest $K-$type spherical function. Then, $r_{n}(\Phi_{0}^{\star})$ is a cusp form on $H$, for every $n=0,1,\dots$, if and only if $\pi_{\lambda}^{G}$ restricted to $H$ is discretely decomposable. In turn, this is equivalent to: for each $y\in G$, the restriction of $K_{\lambda}(\cdot,y)$ to $H$ is a cusp form.

Remark 7.2*.*

T. Kobayashi [KO2, Theorem 2.8] has shown that for a symmetric pair $(G,H)$, $\pi_{\lambda}$ restricted to $H$ is algebraically discretely decomposable if and only if $res_{H}(\pi_{\lambda})$ is $H-$admissible. Whence, coupling the previously quoted result of Kobayashi, with Theorem 7.1 and Proposition 7.3 we may state: for a symmetric pair $(G,H),$ the restriction of $\pi_{\lambda}$ to $H$ is admissible if and only if for every $n=0,1,\dots$ $r_{n}(\Phi_{0}^{\star})$ is a cusp form if and only if

$\Phi_{0}^{\star}$ is left $\mathfrak{z}(U(\mathfrak{h}))-$finite.

For a symmetric pair $(G,H),$ another criterion for $H-$admissibility has been obtained by [HHO]. For this, they write $H=K_{0}\times H_{1}$, with $K_{0}$ a compact subgroup and $H_{1}$ a noncompact semisimple subgroup. Let $H^{\sigma\theta}$ be the dual subgroup. Then, $H^{\sigma\theta}=K_{0}\times H_{2}$. Let $M_{i}$ denote the centralizer in $L\cap H_{1}=L\cap H_{2}$ of respective Cartan subspaces. Harris-He-Olafsson show that if $M_{1}M_{2}=L\cap H_{1}$, then, the representation $\pi_{\lambda}$ restricted to $H$ is admissible, and $\dim Hom_{H}(\pi_{\mu}^{H},res_{H}(\pi_{\lambda}))$ is computed via a formula that involves $r_{n}$, the Harish-Chandra character for $\pi_{\mu}^{H}$, the lowest $L-$type for $\pi_{\mu}^{H}$ and the limit of a sequence.

In [DV], for any pair $(G,H)$ and $\pi_{\lambda}$ that satisfies Condition C, it is shown: $\pi_{\lambda}$ is a $H-$admissible representation; a "Blattner-Kostant" type formula for $\dim Hom_{H}(\pi_{\mu}^{H},res_{H}(\pi_{\lambda}))$. For a symmetric pair $(G,H)$, Condition C is equivalent to $H-$admissibility.

In order to show Theorem 7.1 we first show

Proposition 7.3**.**

We let $G,H,\pi_{\lambda},K_{\lambda},\Phi_{0}$ be as in Theorem 7.1. Then, $\pi_{\lambda}$ restricted to $H$ is a discretely decomposable representation for $H$ if and only if the function $y\mapsto K_{\lambda}(y,e)^{\star}=\check{\Phi}_{0}(y)$ is left $\mathfrak{z}(U(\mathfrak{h}))-$finite.

Proof.

For the direct implication, we proceed as follows. Our hypothesis is $\pi_{\lambda}$ is discretely decomposable; this allows us to write $V_{\lambda}$ as the Hilbert sum of the $H-$isotypic components; hence, there exists a family $(P_{i})_{i\in{\mathbb{Z}}_{\geq 0}}$ of orthogonal projectors on $V_{\lambda}$ which are $H-$equivariant so that we have the orthogonal direct sum decomposition $V_{\lambda}=\oplus_{i}P_{i}(V_{\lambda}),$ and for every $i$, $P_{i}(V_{\lambda})$ is equal to the isotypic component of an irreducible $H-$module. Next, we fix $w\in W$, we recall the function $y\mapsto K_{\lambda}(y,e)^{\star}(w)=:k_{w}(y)$ is a $K-$finite element of $\,H^{2}(G,\tau)$ and $k_{w}$ belongs to $\,H^{2}(G,\tau)[W]\equiv W.$ After we decompose $\,H^{2}(G,\tau)[W]$ as a sum of irreducible $L-$submodules, we write $k_{w}=f_{1}+\dots+f_{s}$, where $f_{j}$ is so that the linear subspace spanned by $\pi_{\lambda}(L)f_{j}$ is an irreducible $L-$submodule of $\,H^{2}(G,\tau)[W].$ To continue, we set $f_{1}:=f_{j}.$ A result of Harish-Chandra, [HC2, Lemma 70], states that an irreducible representation of $L$ is the $L-$type of at most finitely many discrete series representations for $H.$ Thus, the representation of $L$ in the subspace spanned by $\pi(L)f_{1}$ is an $L-$type of at most finitely many discrete series representations for $H.$ Therefore, $P_{i}(f_{1})=0$ for all but finitely many indices $i$. Let us say $P_{i}(f_{1})\not=0$ for $i=1,\dots,N.$ Since $f_{1}$ is a $K-$finite vector in $V_{\lambda}$, we have that $f_{1}$ is a smooth vector for $\pi_{\lambda},$ therefore, $P_{i}(f_{1})$ is a $H-$smooth vectors in $P_{i}(V_{\lambda}).$ Owing to $P_{i}(V_{\lambda})$ is an isotypic representation, we have that $\mathfrak{z}(U(\mathfrak{h}))$ applied to $P_{i}(f_{1})$ is contained in the one dimensional vector subspace spanned by $P_{i}(f_{1})$. Hence, $f_{1}$ is a finite sum of $\mathfrak{z}(U(\mathfrak{h})-$finite vectors. Thus, $k_{w}$ is a $\mathfrak{z}(U(\mathfrak{h}))-$finite vector. $W$ is a finite dimensional vector space, let us conclude that the map $K_{\lambda}(\cdot,e)^{\star}=\Phi_{0}(\cdot)^{\star}$ is left $\mathfrak{z}(U(\mathfrak{h}))-$finite.

For the converse statement, owing to our hypothesis, for each $w\in W$ we have that $k_{w}$ is $\mathfrak{z}(U(\mathfrak{h}))-$finite element of $\,H^{2}(G,\tau)_{K-fin}.$ A result of Harish-Chandra, [Wa1, Corollary 3.4.7 and Theorem 4.2.1], asserts that a $U(\mathfrak{h})-$finitely generated, $\mathfrak{z}(U(\mathfrak{h}))-$finite, $(\mathfrak{h},L)-$module has a finite composition series, whence, we conclude that the representation of $U(\mathfrak{h})$ in $U(\mathfrak{h})k_{w}$ has a finite composition series. Thus, $\,H^{2}(G,\tau)_{K-fin}$ contains an irreducible subrepresentation for $U(\mathfrak{h}).$ Hence, [K3, Lemma 1.5] yields that $H^{2}(G,\tau)_{K-fin}$ is infinitesimally discretely decomposable as $\mathfrak{h}-$module. Finally, since $\pi_{\lambda}$ is unitary, in [K4, Theorem 4.2.6], we find a proof that an algebraically (infinitesimally) discretely decomposable unitary representation is Hilbert discrete decomposable; hence, $\pi_{\lambda}$ is discretely decomposable. ∎

Now, we are ready to show Theorem 7.1.

Proof of Theorem 7.1.

For the direct implication, the hypothesis is that $\pi_{\lambda}$ restricted to $H$ is discretely decomposable. Thus, Proposition 7.3 yields

$K_{\lambda}(\cdot,e)^{\star}w$ is $\mathfrak{z}(U(\mathfrak{h}))-$finite. Since in [OV], it is shown $r_{n}$ is a continuous intertwining map for $H$, and $K_{\lambda}(\cdot,e)^{\star}w$ is a tempered function, a result of [HHO] previously quoted, lets us conclude: $r_{n}(K_{\lambda}(\cdot,e)^{\star}w)$ is a tempered, $\mathfrak{z}(U(\mathfrak{h}))-$finite function on $H.$ A result of Harish-Chandra, [HC2][Wa1, 7.2.2], implies $r_{n}(K_{\lambda}(\cdot,e)^{\star}w)$ is a cusp form. For the converse statement, results of Harish-Chandra asserts $L^{2}(H\times_{\tau_{n}}(\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)}\otimes W)_{disc}$ is a finite sum of discrete series representations for $H$ and the space of cusp forms in $L^{2}(H\times_{\tau_{n}}(\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)}\otimes W)$ is contained in $L^{2}(H\times_{\tau_{n}}(\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)}\otimes W)_{disc}$. Therefore, owing to our hypothesis, for every $n$, $r_{n}(K_{\lambda}(\cdot,e)^{\star}w)$ belongs to $L^{2}(H\times_{\tau_{n}}(\mathfrak{p}/\mathfrak{p}^{\prime})^{(n)}\otimes W)_{disc}$. The $L^{2}-$continuity of $r_{n}$ yields $r_{n}(closure(\pi_{\lambda}(H)k_{w}))$ is contained in a finite sum of discrete series representations. Whence, $\oplus_{n}r_{n}$ maps continuously the closure of $\pi_{\lambda}(H)k_{w}$ into a discrete Hilbert sum of irreducible representations. Besides, the map $\oplus_{n}r_{n}$ is injective (the elements of $\,H^{2}(G,\tau)$ are real analytic functions). Hence, the closure of $\pi_{\lambda}(H)k_{w}$ is a discrete Hilbert sum of discrete series representations. We now proceed as in the direct proof Proposition 7.3 and obtain $k_{w}$ is a left $\mathfrak{z}(U(\mathfrak{h}))-$finite function. Whence, Proposition 7.3 lets us conclude $res_{H}(\pi_{\lambda})$ is Hilbert discretely decomposable. The second equivalence follows from a simple computation. ∎

Remark 7.4*.*

A simple application of Theorem 7.1 is that the tensor product representation in $\,H^{2}(G,\tau)\boxtimes\,H^{2}(G,\tau)^{\star}$ restricted to diagonal of $G$ is never discretely decomposable, because the lowest $K-$type trace spherical function for this particular tensor product is $\phi_{0}(x)\overline{\phi_{0}(y)},$ hence, restricted to $G$ it is not a cusp form.

8. Reproducing kernels and existence of discrete factors

As usual $G,H,\,H^{2}(G,\tau)=V_{\lambda},K_{\lambda},H^{2}(H,\sigma)=V_{\mu}^{H},K_{\mu}$ are as in Section 2. The purpose of this section is to begin an analysis of the relation between the matrix kernel $K_{\lambda}$ of a discrete series for $G$, the matrix kernel $K_{\mu}$ of a discrete series for $H$ and the existence of a nonzero $H-$intertwining linear map from one representation into the other. To begin with, since $K_{\lambda}$ is given by the spherical function associated to the lowest $K-$type of $\pi_{\lambda}^{G}$, the restriction of $y\mapsto tr(K_{\lambda}(y,x))=:k_{\lambda}(y,x)$ to $H$ is a square integrable function [OV]. Let $k_{\lambda}:=trK_{\lambda},k_{\mu}:=trK_{\mu}.$ We recall [Va2],

Proposition 8.1**.**

$(k_{\lambda}(\cdot,e),k_{\mu}(\cdot,e))_{L^{2}(H)}$* nonzero implies $V_{\mu}^{H}$ is a discrete factor for $res_{H}(\pi_{\lambda}^{G}).$*

The Proposition is an immediate consequence of the following

Fact 8.2*.*

[Va2] Let $G,H,\pi_{\lambda}$ be as usual. Let $(\rho,V_{1})$ an irreducible square integrable for $H$. We assume there exists smooth vector $w_{1},w_{2}$ in $V_{\lambda}$ and a $L-$finite vector $z$ in $V_{1}$ so that

[TABLE]

Then, there exists a nonzero, continuous linear $H-$map from $V_{1}$ into $V_{\lambda}.$

Remark 8.3*.*

The converse to Proposition 8.1 is false, as it shows the following example. We consider arbitrary $G,H$ and we assume $res_{H}(\pi_{\lambda}^{G})$ is an $H-$admissible representation. Since $\,H^{2}(G,\tau)[W]$ consists of $L-$finite vectors, the subspace $\,H^{2}(G,\tau)[W]$ is contained in a finite sum of irreducible $H-$subrepresentations. Let $\pi_{\mu}^{H}$ be a representation for $H$ whose corresponding isotypic component is nonzero and it is contained in the orthogonal to the subspace spanned by the $H-$isotypic components whose intersection with $\,H^{2}(G,\tau)[W]$ is nonzero. Since, for $w\in W$, $K_{\lambda}(\cdot,e)^{\star}w\in\,H^{2}(G,\tau)[W]$, we have that both $tr(K_{\mu}(\cdot,e))=k_{\mu}(\cdot,e)$ and $tr(K_{\lambda}(\cdot,e))=k_{\lambda}(\cdot,e)$ are linear combination of matrix coefficients of nonequivalent representations for $H,$ the orthogonality relations implies $(k_{\lambda}(\cdot,e),k_{\mu}(\cdot,e))_{L^{2}(H)}=0.$

A way to show a converse to Proposition 8.1 is to consider the functions

[TABLE]

[TABLE]

Both integrals converge absolutely for every $x,y\in G$ owing to in [OV] it is shown both integrands are product of square integrable functions. Moreover, the functions $\Upsilon,$ $\upsilon$ are continuous because each integral can be rewritten as a convolution product and we recall the fact: convolution of two square integrable functions is defined and yields a continuous function.

Under the supposition the representation $\pi_{\mu}^{H}$ is integrable it readily follows the smoothness of both functions as well as that each function is a Carleman kernel. In fact, we notice $z\mapsto K_{\lambda}(zy,x))$ belongs to either the Frechet space $(L^{2}(G)\otimes Hom_{\mathbb{C}}(W,W))^{\infty}$ or to the Hilbert space $L^{2}(G)\otimes Hom_{\mathbb{C}}(W,W)$ and $k_{\mu}(\cdot,e)\in L^{1}(H)$. Next, we write $\upsilon(y,x)=res_{H}(L)_{(k_{\mu}(\cdot,e))}(z\mapsto K_{\lambda}(zy,x))$ Therefore, both functions $\upsilon(\cdot,x),\upsilon(y,\cdot)$ are smooth and square integrable. For the function $\Upsilon$, we write $\Upsilon(y,x)=res_{H}(L)_{(K_{\mu}(\cdot,e)^{\star})}(z\mapsto k_{\lambda}(zy,x))$ and proceed as for $\upsilon.$ Among properties of the functions are:

0. $tr(\Upsilon(e,e))=(k_{\lambda}(\cdot,e),k_{\mu}(\cdot,e))_{L^{2}(H)},$ $tr(\Upsilon(y,x))=\overline{tr(\upsilon(y,x))}$.

1. $\Upsilon(s_{1}yk,s_{2}xk)=\sigma(s_{1})\Upsilon_{\mu,\lambda}(y,x)\sigma(s_{2}^{-1}),\text{for}\,x\in G,s_{j}\in L,k\in K.$

2. The function $\Upsilon_{\mu,\lambda}$ is real analytic.

This is because the distribution on $G$ defined by either the function $y\mapsto\Upsilon_{\mu,\lambda}(y,x)$ or $x\mapsto\Upsilon_{\mu,\lambda}(y,x)$ is a real analytic $L-$spherical function. Indeed, both distributions are eigenfunctions of the elliptic differential operator $R_{\Omega_{G}}+2R_{\Omega_{K}}$, the regularity Theorem leads us to the real analyticity.

3. $\Upsilon_{\mu,\lambda}(y,x)=(\Upsilon_{\mu,\lambda}(x,y))^{\star}.$

4. Remark 8.3 shows that some times $\Upsilon(e,e)=0$.

5. $\upsilon(syk_{1},sxk_{2})=\tau(k_{2}^{-1})\upsilon(y,x)\tau(k_{1}),k_{1},k_{2}\in K,s\in L$.

The main result of this subsection is:

Theorem 8.4**.**

Let $G,H,(\pi_{\lambda},\,H^{2}(G,\tau))$, $(\pi_{\mu}^{H},V_{\mu}^{H}=H^{2}(H,\sigma))$ as in Section 2. The following four conditions are equivalent

a) The isotypic component $\,H^{2}(G,\tau)[V_{\mu}^{H}]$ is not zero.

b) The function $\Upsilon_{\mu,\lambda}$ is nonzero.

c) The function $tr(\Upsilon_{\mu,\lambda})$ is nonzero.

d) There exists $D\in U(\mathfrak{g}\times\mathfrak{g})$ so that $[L_{D}(\Upsilon)](e,e)\not=0.$

e) The function $\upsilon_{\lambda,\mu}$ is nonzero.

Proof of Theorem 8.4.

We show $a)\Rightarrow b)\Leftrightarrow c)\Rightarrow e)\Rightarrow a)$ and $b)\Leftrightarrow d).$ Some of the implications are obvious. a) implies b). By hypothesis, there exists a nonzero intertwining map $T:H^{2}(H,\sigma)\rightarrow\,H^{2}(G,\tau)$. Owing to Schur’s Lemma $T$ is an injective map. Let $K_{T}:H\times G\rightarrow Hom_{\mathbb{C}}(Z,W)$ be the kernel that represents $T$. We know, for $z\in Z,$ the function $h\mapsto K_{\mu}(h,e)^{\star}z$ belongs to $H^{2}(H,\sigma)$ and it is nonzero. Thus, $T(K_{\mu}(\cdot,e)^{\star}z)$ is a nonzero function for each $z\in Z.$ Since $k_{\lambda}$ represents $P_{\lambda}$ we obtain

[TABLE]

Now $K_{T}(h,y)=K_{T}(e,h^{-1}y).$ Thus, after we replace $y$ by $hy$, we recall Haar measure is left invariant and $k_{\lambda}$ is a complex valued function, the left hand side becomes

[TABLE]

Thus, $\Upsilon$ is not the zero function. We now show $b)\Leftrightarrow c).$ The equality 1) forces the linear span of the image of $\Upsilon_{\mu,\lambda}$ is an $L\times L$ invariant subspace of $End_{\mathbb{C}}(Z).$ Thus, owing to $End_{\mathbb{C}}(Z)$ is an $L\times L-$module irreducible, we have there exists $(y,x)$ so that $tr(\Upsilon_{\mu,\lambda}(y,x))\not=0$ unless $\Upsilon_{\mu,\lambda}$ is equal to the zero map. $c)\Rightarrow e)$ follows from $tr(\upsilon_{\lambda,\mu})=\overline{tr(\Upsilon_{\mu,\lambda})}.$ To show $e)\Rightarrow a)$. We notice that the hypothesis implies we may apply Fact 8.2. Therefore the isotypic component is nonzero. The two functions are real analytic, thus, we conclude the proof. ∎

8.1. Examples of discrete factors in $res_{H}(\pi_{\lambda})$

As before, we fix $G,H,\\ K,L$ and the representation $(\pi_{\lambda},H^{2}(G,\tau))$. A consequence of Proposition 8.1 is the following result shown in [Va],

Fact 8.5*.*

We suppose $(\sigma,Z)$ is an $L-$subrepresentation of the lowest $K-$type $(\tau,W)$. We further assume there exists a discrete series representation $H^{2}(H,\sigma)$ with lowest $L-$type $(\sigma,Z).$ Then, there exists a nonzero intertwining map from $H^{2}(H,\sigma)$ into $\,H^{2}(G,\tau)$.

The proof for this fact is based on an explicit integral formulae for $k_{\mu},k_{\lambda}$ obtained by Flensted-Jensen, these formulae let us to apply Proposition 8.2. We present one application of fact 8.5 to the analysis of the tensor product of two representations. For other applications (cf. [Va2]).

Example 8.6*.*

Notation is as in Section 2. We produce some irreducible subrepresentation of the restriction to the diagonal subgroup $H=G_{0}$ of $G:=G_{0}\times G_{0}$ ($G_{0}$ a semisimple Lie group) for a tensor product $\pi_{\lambda}\boxtimes\pi_{\lambda_{1}}.$ We choose both Harish-Chandra parameters to be dominant with respect to $\Psi$ and so that their sum is far away from the kernel of any noncompact simple root. It readily follows that $\pi_{\lambda+\lambda_{1}+2\rho_{n}-\rho_{c}}^{K_{0}}$ is a subrepresentation of the lowest $K_{0}\times K_{0}-$type for $\pi_{\lambda}\boxtimes\pi_{\lambda_{1}}.$ Due to our choice, the parameter $\lambda+\lambda_{1}+2\rho_{n}-\rho_{c}$ is far from the noncompact walls for the Weyl chamber for $\Psi.$ It follows (cf. [DHV]) that $\pi_{\lambda+\lambda_{1}+2\rho_{n}-\rho_{c}}^{K_{0}}$ is the lowest $K_{0}-$type of the discrete series $\pi_{\lambda+\lambda_{1}+\rho_{n}-\rho_{c}}^{G_{0}}$. Therefore, Fact 8.5 implies $res_{G_{0}}(\pi_{\lambda}\boxtimes\pi_{\lambda_{1}})$ contains the irreducible representation $\pi_{\lambda+\lambda_{1}+\rho_{n}-\rho_{c}}^{G_{0}}.$

9. Proof of Proposition 3.7 and Theorem 3.5

Proof of Proposition 3.7.

A straightforward computation based on the invariance of Haar measure gives $f)$. To show $b)$, we notice $K_{T}(s^{-1}h,e)^{\star}w=K_{T}(h,s)^{\star}w=(\tau(s^{-1})K_{S}(h,e))^{\star}w=K_{T}(h,e)^{\star}\tau(s)w.$ Thus, dimension of the span of $\{L_{s}(h\mapsto K_{T}(h,e)^{\star}w):s\in L\}$ is less or equal than $dimW$. Next, we show $a)$. Let $g\in L^{2}(H\times_{\nu}E)$ arbitrary, we have

[TABLE]

The first equality is justified by 3.1, the second by $P_{\lambda}(Tg)=Tg$ and (2.1), the third by definition of adjoint linear map. Since the functions $v\mapsto K_{T}(v,y)^{\star}(w),v\mapsto[T^{\star}(t\mapsto K_{\lambda}(t,y)^{\star}(w))](v)$ belong to $L^{2}(H\times_{\nu}E)$ and $g$ is arbitrary in $L^{2}(H\times_{\nu}E)$ we obtain a).

For further use we denote the subspace of smooth vectors in a representation $V$ by $V^{\infty}$.

Next, we verify the map $K_{T}$ is smooth and that $K_{T}(\cdot,x)^{\star}w$ is a smooth vector. The smoothness of $K_{T}$ follows from $(h,x)\mapsto K_{T}(h,x)^{\star}w$ is smooth for each $w\in W.$ We first notice $K_{\lambda}(y,x)=K_{\lambda}(e,y^{-1}x)$ and $x\mapsto K_{\lambda}(x,e)^{\star}w$ is a $K-$finite vector in $\,H^{2}(G,\tau)$. Thus, $y\mapsto K_{\lambda}(y,x)^{\star}w=L_{x}(y\mapsto K_{\lambda}(y,e)^{\star}w)$ is a smooth vector in $\,H^{2}(G,\tau)$ and hence the map $G\ni x\mapsto K_{\lambda}(\cdot,x)^{\star}w)\in\,H^{2}(G,\tau)$ is smooth. Since $T^{\star}$ is a continuous linear $H-$map, we have that $K_{T}(\cdot,x)^{\star}w=T^{\star}(y\mapsto K_{\lambda}(y,x)^{\star}w)(\cdot)\in L^{2}(H\times_{\nu}E)^{\infty}$, hence $G\ni x\mapsto T^{\star}(y\mapsto K_{\lambda}(y,x)^{\star}w)\in L^{2}(H\times_{\nu}E)^{\infty}$ is a smooth map. Next, [Wa1, section 1.6]), we endow the space of smooth vectors in a representation $(\pi,V)$ with the topology that a sequence of smooth vectors $v_{n}$ converges to a smooth vector $v$ iff $\pi(D)v_{n}$ converges to $\pi(D)v$ in norm for every $D\in U(\mathfrak{h})$. Poulsen [Po, Proposition 5.1] has shown that the space of smooth vectors $L^{2}(H\times_{\nu}E)^{\infty}$ in $L^{2}(H\times_{\nu}E)$ is the subspace of smooth functions $f$ so that $L_{D}(f)$ is square integrable for every $D\in U(\mathfrak{h}).$ Further, Poulsen showed point evaluation from $L^{2}(H\times_{\nu}E)^{\infty}$ into $E$ is a continuous linear map. Whence, the the following composition gives a smooth map $(h,x)\mapsto h^{-1}x\mapsto T^{\star}(v\mapsto K_{\lambda}(v,h^{-1}x)^{\star}w)(e).$ Now $a)$ and the equality $h_{T}(e,h^{-1}x)=h_{T}(h,x)$ concludes the proof of $c)$.

$d)$ follows from Sobolev’s inequality $\|f(e)\|_{E}\leq\sum_{1\leq b\leq N}\|L_{X_{b}}(f)\|_{L^{2}(H)}$, as shown by Poulsen, [Po, Lemma 5.1], applied to $f(h):=K_{T}(h,x)^{\star}w$ at $h=e$. In fact, owing to c), $K_{T}(\cdot,x)^{\star}w$ is an smooth vector and we have $L_{X_{b}}^{(1)}K_{T}(h,x)^{\star}w=L_{X_{b}}(T^{\star}(K_{\lambda}(\cdot,x)^{\star}w)(h)=T^{\star}(L_{X_{b}}^{(1)}K_{\lambda}(\cdot,x)^{\star}w)(h)$. Thus, $\|L_{X_{b}}^{(1)}K_{T}(\cdot,x)^{\star}w\|_{L^{2}(H)}\leq\|T^{\star}\|\,\|L_{X_{b}}K_{\lambda}(\cdot,x)^{\star}w\|_{L^{2}(G)}.$

The equality $K_{\lambda}(y,x)=K_{\lambda}(x^{-1}y,e)$ together with the left invariance of Haar measure yields $\|L_{X_{b}}K_{\lambda}(\cdot,x)^{\star}w\|_{L^{2}(G)}=\|L_{Ad(x^{-1})X_{b}}K_{\lambda}(\cdot,e)^{\star}w\|_{L^{2}(G)}.$ Next, we fix a basis $\{Y_{a}\}_{1\leq a\leq M}$ for the subspace of elements of $U(\mathfrak{g})$ of degree less or equal to $\dim\mathfrak{h}.$ We write $Ad(x^{-1})X_{b}=\sum_{a}\phi_{a,b}(x)Y_{a}$. Let $C$ denote an upper bound for the numbers $\|L_{Y_{a}}K_{\lambda}(\cdot,e)^{\star}\|_{L^{2}(G)}$ for $1\leq a\leq M.$ Then, Sobolev’s inequality applied to $K_{T}(\cdot,x)^{\star}w$ at $h=e$, together with the previous inequalities gives $\|K_{T}(e,x)^{\star}w\|_{W}\leq\|T^{\star}\|\sum_{1\leq b\leq N}\|L_{X_{b}}K_{\lambda}(\cdot,x)^{\star}w\|_{L^{2}(G)}\\ =\|T^{\star}\|\sum_{b}\|L_{Ad(x^{-1})X_{b}}K_{\lambda}(\cdot,e)^{\star}w\|_{L^{2}(G)}\leq C\|T^{\star}\|\|w\|_{W}\sum_{a,b}|\phi_{a,b}(x)|,$ and

$\|K_{T}(e,x)^{\star}\|_{Hom(W,Z)}=\sup_{|w\|\leq 1}\{\|K_{T}(e,x)^{\star}w\|_{W}\}\leq C\|T^{\star}\|\sum_{a,b}|\phi_{a,b}(x)|.$ Thus, $d)$ follows.

Now, we verify $e)$. That is, the function $x\mapsto\|K_{T}(\cdot,x)^{\star}\|_{L^{2}(H)}$ is bounded, for this, we verify the inequality

xxxxxxxxx $\|K_{T}(\cdot,x)^{\star}\|_{L^{2}(H)}<<\|K_{\lambda}(\cdot,e)\|_{L^{2}(G)}\|\,\|T\|$.

To begin with, for $x\in G,w\in W$, we have

xxxxxxx $\|K_{T}(\cdot,x)^{\star}w\|_{L^{2}(H)}\leq\|w\|\|K_{\lambda}(\cdot,e)\|_{L^{2}(G)}\|\,\|T\|$. Indeed,

[TABLE]

Therefore, after we simplify, we obtain

[TABLE]

Thus,

[TABLE]

To continue, we fix an orthonormal basis $\{z_{j}\}$ (resp. $\{w_{i}\}$) for $E$ (resp. for $W$). Then, there exists a constant $C_{1}=C_{1}(E,W)$, so that for each linear map $R:W\rightarrow E$ we have, $\|R\|_{Hom(W,E)}\leq C_{1}\sqrt{\sum_{i,j}|(Rw_{i},z_{j})_{E}|^{2}}.$ Therefore,

[TABLE]

Whence, for every $x\in G$, we obtain

[TABLE]

This concludes the proof of Proposition 3.7. ∎

Proof of Theorem 3.5 .

In order to show $a)$, we recall that Schur’s Lemma implies that for a closed $H-$irreducible subspace $N$ of $\,H^{2}(G,\tau)$ either $S(N)=\{0\}$ or $S(N)$ is a closed $H-$irreducible subspace. Thus, our hypothesis forces the image of $S$ is equal to a sum of $H-$irreducible subspaces of $L^{2}(H\times_{\nu}E)$. Plancherel’s Theorem of Harish-Chandra shows that closure of the sum of the $H-$irreducible subspaces in $L^{2}(H\times_{\nu}E)$, is equal to a finite sum of irreducible subspaces; this follows from Frobenius reciprocity and [HC2, Lemma 72]. From now on, $L^{2}(H\times_{\nu}E)_{disc}$ denotes the sum of $H-$invariant irreducible subspaces in $L^{2}(H\times_{\nu}E)$. On the smooth vectors of a $H-$irreducible subspace $N$ the Casimir operator acts by a constant since smooth vectors are smooth functions, the Casimir operator acts by the same constant on the whole of $N.$ In [At], we find a proof that a $L^{2}-$eigenspace of an elliptic operator on a fiber bundle is a reproducing kernel subspace. Finally, the Casimir operator of $\mathfrak{h}$ acts as an elliptic operator on $L^{2}(H\times_{\nu}E)$. Thus, we conclude that $L^{2}(H\times_{\nu}E)_{disc}$ is a finite sum of reproducing kernel subspaces; hence, $L^{2}(H\times_{\nu}E)_{disc}$ is a reproducing kernel subspace. Whence, the image of $S$ is contained in a reproducing kernel subspace, which lets us conclude $S$ is a kernel map. It readily follows kernel $K_{S}$ of $S$ is equal to $K_{S^{\star}}^{\star}.$

Subsequently, we show $b)$. The hypothesis $S$ is a continuous $H-$map and Proposition 3.7 yields that $S^{\star}$ is an integral map. Thus, at least formally, we think $S$ as the adjoint of an integral map. For this, we formally define

[TABLE]

and we consider the subspace

[TABLE]

It readily follows that $S_{0}$ restricted to $\mathcal{D}_{K_{S^{\star}}^{\star}}$ is an integral operator with kernel $K_{S_{0}}(x,h)=K_{S^{\star}}(h,x)^{\star}.$ Besides, $S$ is an integral map when restricted to $\,H^{2}(G,\tau)\cap\mathcal{D}_{K_{S^{\star}}^{\star}}$. To follow, we construct a subspace $\mathcal{D}$ of $\mathcal{D}_{K_{S^{\star}}^{\star}},$

[TABLE]

For this, we show that for $f\in\mathcal{D}$, the integral that formally defines the function $h\mapsto S_{0}(h)$ is absolutely convergent almost everywhere in $h$ and the resulting function belongs to $L^{2}(H\times_{\nu}E).$ We apply the integral version of Minkowski’s inequality for $p=2$ and we obtain

[TABLE]

The right hand side of the previous inequality is a finite number because $f$ belongs to $\mathcal{D}.$ Hence, $\int_{G}K_{S^{\star}}(h,x)^{\star}f(x)dx$ is absolutely convergent almost everywhere in $h$ and the resulting function belongs to $L^{2}(H\times_{\nu}E).$

As a consequence, we obtain that for $g\in L^{2}(H\times_{\nu}E)$ and $f\in\mathcal{D}$ the following two iterated integrals are absolutely convergent

[TABLE]

We are ready to conclude the proof of Theorem 3.5 b). Whenever $\pi_{\lambda}$ is an integrable discrete series representation, it follows from the work of Harish-Chandra, [HC1, Lemma 76], that any $K-$finite vector $f$ in $\,H^{2}(G,\tau)$ is integrable with respect to Haar measure on $G$. We claim that any smooth vector in $\,H^{2}(G,\tau)$ is an integrable function. For this we recall the space of rapidly decreasing functions $\mathscr{S}(G)$ on $G$, defined by [Wa1, page 230]. Owing to [Wa1, Lemma 2.A.2.4], any rapidly decreasing function is integrable. Next, in [Wa2, Theorem 11.8.2], it is shown that the subspace $\,H^{2}(G,\tau)^{\infty}$ of smooth vectors in $\,H^{2}(G,\tau)$ is an algebraically irreducible module over $\mathscr{S}(G)$. Therefore, any smooth vector in $\,H^{2}(G,\tau)$ is equal to the convolution of a rapidly decreasing function on $G$ times a $K-$finite vectors. Hence, any smooth vector is convolution of two integrable functions on G. Classical harmonic analysis yields any smooth vector in $\,H^{2}(G,\tau)$ is an integrable function. Therefore, for a smooth vector $f$ in $\,H^{2}(G,\tau)$, Proposition 3.7 $e)$, forces the integral

[TABLE]

is absolutely convergent. Hence, $\mathcal{D}$ contains the smooth vectors in $H^{2}(G,\tau)$, and we have verified $b)$. Thus, we have shown Theorem 3.5. ∎

Example 9.1*.*

The following example shows that statement in Theorem 3.5 b) might not be sharp. Details for some of the statements in this example are found in [DaOZ] and references therein. We set $G=SL_{2}(\mathbb{R})$ and $H$ equal to the subgroup of diagonal matrices in $G.$ We fix as compact Cartan subgroup $T$ the subgroup of orthogonal matrices in $G.$ We fix as positive root $\alpha$ one of the roots in $\Phi(\mathfrak{g},{\mathfrak{t}})$ and $\rho:=\frac{1}{2}\alpha.$ Then, the set of Harish-Chandra parameters for $G$ is $\{n\rho,n\in\mathbb{Z}\backslash\{0\}\}.$ The lowest $K-$type for $\pi_{p\rho}^{G}$ ($p\geq 1$) is $(\pi_{(p+1)\rho}^{K},{\mathbb{C}}).$ After we identify $G/K$ with the upper half plane $\mathcal{H}_{+}$ and we trivialize the vector bundle $G\times_{\pi_{(p+1)\rho}^{K}}\mathbb{C}\rightarrow G/K$ we have that $H^{2}(G,\pi_{(p+1)\rho}^{K})$ can be identified with the space

[TABLE]

The bundle $H\times_{\pi_{(p+1)\rho}^{K}}\mathbb{C}\rightarrow H/L$ is also trivial and $L^{2}(H\times_{res_{L}(\tau)}W)$ is identified with $L^{2}(i\mathbb{R}_{>0},t^{p}dt).$ The restriction map $r:\,H^{2}(G,\tau)\rightarrow L^{2}(H\times_{res_{L}(\tau)}W)$ becomes $r(f)(iy)=f(iy),y\in\mathbb{R}_{>0}.$ After we write the polar decomposition $r^{\star}=VQ$ we have

[TABLE]

Owing to Theorem 3.5, for an integrable discrete series, equivalently $|p|\geq 2,$ the linear map $V^{\star}$ restricted to the subspace of smooth vectors in

$H^{2}(G,\pi_{(p+1)\rho}^{K})$ is equal to the integral linear map

[TABLE]

For the non-integrable discrete series $H^{2}(G,\pi_{2\rho}^{K})$ the Lebesgue integral on the right of $({\ddagger})$ does exists for any $K-$finite vector. In fact,

[TABLE]

Whence, the following integral converges absolutely,

[TABLE]

The inner integral is computed by means of Cauchy integral formula applied to compute the first derivative of $z\mapsto e^{-i(z-iy)t}$ at the point $-(iy+i)$. The path of integration is the half circumference of diameter $[-c,c]$ with $c$ positive and very large so that $-(iy+i)$ is an interior point of the path. We now have that for each $n\geq 0$, the resulting function defined by the right hand side of $({\ddagger})$ belongs to $L^{2}(i\mathbb{R}_{>0},tdt)$. Next, the set of functions $(\frac{z-i}{z+i})^{n}\frac{1}{(z+i)^{2}},n\geq 0$ span the subspace of $K-$finite vectors. Therefore, functional analysis implies the right hand of $({\ddagger})$ evaluated in each of this set of generators of $(H^{2}(SL_{2}(\mathbb{R}),\pi_{2\rho}^{K}))_{K-fin}$ is equal to $V^{\star}$ evaluated at each of these generators. This concludes the verification that $V^{\star}$ restricted to the subspace of $K-$finite vectors is an integral map.

10. Other model to realize discrete series representations

We refer to the article [Hi] for this section. Let $d_{\star}m$ denote a fixed $G-$invariant Radon measure on $G/K$, after we normalize Haar measure on $K$ so that $K$ has volume one, and normalize Haar measure on $G$, we have the equality

[TABLE]

Owing to the Iwasawa decomposition for $G=ANK$ the principal bundle $K\rightarrow G\rightarrow G/K$ is equivalent to the trivial bundle. Therefore, for each representation $\tau:K\rightarrow U(W)$, the vector bundle $G\times_{\tau}W\rightarrow G/K$ is parallelizable. We fix a section $\sigma$ for $G\rightarrow G/K$ so that $\sigma(eK)=1,$ we obtain a cocycle $c:G\times G/K\rightarrow Gl(W),\text{by}\,c(g,x)=\tau(\sigma(g\cdot x)^{-1}g\sigma(x))$. That is, $c$ is a continuous map which satisfies

[TABLE]

We now recall how the above datum gives rise a unitary representation of $G$ equivalent to $L^{2}(G\times_{\tau}W).$ Let $o=eK$. We consider the inner product on $W-$valued functions on $G/K$ defined by

[TABLE]

The group $G$ acts unitarily on the corresponding space of square integrable functions $L_{c,\tau}^{2}(G/K,W)$ by the formula

[TABLE]

We claim the following map determines an unitary equivalence

[TABLE]

The inverse map is

[TABLE]

The unitary equivalence is due to that for $F,H\in L_{c,\tau}^{2}(G/K,W)$ the following equality holds

[TABLE]

For a convenient function $\tilde{K}:G/K\times G/K\rightarrow End_{\mathbb{C}}(W),$ we consider the integral operator

[TABLE]

and we define for $g,h\in G,\tilde{k}(g,h):=c(h,o)^{-1}\tilde{K}(g\cdot o,h\cdot o)c(g,o),$ then, we have the commutative diagram

[TABLE]

The kernel $\tilde{k}$ defines an intertwining linear operator for the left action of $G$ on $L^{2}(G\times_{\tau}W)$ if and only if $\tilde{k}$ is an invariant kernel, that is, $\tilde{k}(ga,gb)=\tilde{k}(a,b),a,b,g\in G$ and $\tilde{k}(gk,hk_{1})=\tau(k_{1}^{-1})\tilde{k}(g,h)\tau(k),g,h\in G,k,k_{1}\in K.$ To an invariant kernel $\tilde{k},$ the corresponding kernel $\tilde{K}$ satisfies

[TABLE]

Next we consider the kernels $K_{1},K_{2}$ which defines respective intertwining linear operators on $L_{c,\tau}^{2}(G/K,W).$ Then readily follows the equality

[TABLE]

In particular, whenever $W$ is unidimensional, the kernel $K_{1}$ is equal to $K_{2}$ times a $G-$invariant function.

For each Iwasawa decomposition $G=ANK$ a section of the principal bundle $G\rightarrow G/K$ is $\sigma(anK)=an.$ Therefore, from the preceding considerations we obtain a unitary equivalence between $L^{2}(G\times_{\tau}W)$ and $L_{c,\tau}^{2}(G/K,W)$ by mean of the cocycle $c$ associated to $\sigma$ and $\tau.$ When $G/K$ is a Hermitian symmetric space, there is another section $\sigma_{+}$ by means of the subgroups $P_{-},P_{+},K_{\mathbb{C}}$ of the complex Lie group $G_{\mathbb{C}}$. We have that $G\subset P_{+}K_{\mathbb{C}}P_{-}$ and $G/K$ is realized as a bounded domain in $\mathfrak{p}_{+}.$ Every $g\in G$ is uniquely written as $g=y_{+}(g)\mu(g)y_{-}(g)$, with $y_{\pm}(g)\in P_{\pm},\,\mu(g)\in K_{\mathbb{C}}.$ Next, we fix $(\tau,W)$ irreducible representations of $K$ so that it is the lowest $K-$type of a holomorphic discrete series $\,H^{2}(G,\tau)$. By means of $\sigma_{+}$ and the representations $\tau$ we define $c_{+,\tau}=\tau(\mu(g))$ and we have that the maps $f\mapsto F=c_{+,\tau}(g,o)f(g)$ carries $\,H^{2}(G,\tau)$ onto the subspace of holomorphic functions in $L_{c_{+,\tau}}^{2}(G/K,W)$. Based on this model for the holomorphic discrete series, many authors have contributed to the study of branching problems and harmonic analysis. Other authors have chosen different sections of the principal bundle $G\rightarrow G/K$. Their choice, allowed them to analyze other discrete series, branching problems, harmonic analysis. It is out of our knowledge to explicit all the work is done on the subject. We would like to call the attention of work, on holomorphic discrete series, of Jacobsen-Vergne, T. Kobayashi with his collaborators and the work of G. Zhang on quaternionic discrete series [LZ].

We would like to point out that after we fix respective smooth section for the principal bundle $G\rightarrow G/K$, (resp. $H\rightarrow H/L$), we may translate the reproducing kernel $K_{\lambda}$ (resp. $K_{\mu}$) to a reproducing kernel $\tilde{K}_{\lambda}$ (resp. $\tilde{K}_{\mu}$) in $L_{c,\tau}^{2}(G/K,W)$ (resp. $L_{c,\sigma}^{2}(H/L,Z)$), the eigenspaces of the Casimir operator to eigenspaces of the Casimir operator, and as we already indicated kernel maps goes to kernel maps, it is a simple matter to verify differential operators correspond to differential operators and so on. We may say that each statement in this paper has a correlative in the language of the spaces $L_{c,\tau}^{2}$. For example, a function that corresponds to an element of $\,H^{2}(G,\tau)$ growth at most as the function $\|\tau(c(g,o))\|.$ Differential operators intertwining $\,H^{2}(G,\tau)$ with $H^{2}(H,\sigma)$ corresponds to differential operators between the corresponding spaces.

10.1. An example of discrete decomposition

We consider the groups $G=U(1,n),H=U(1,n-1)\times U(1)$, $K=U(1)\times U(n),$ $L=U(1)\times U(n-1)\times U(1)$ and we fix an integer $\alpha$ positive and large. Then, $\tau_{\alpha}(k)=det(k)^{\alpha},k\in K$ is a character of $K.$ To follow, we write the decomposition of $\,H^{2}(G,\tau_{\alpha})$ as an $H-$representation and we compute reproducing kernels, immersions, projections, etc. Let $\mathcal{D}_{n}=\{z\in{\mathbb{C}}^{n},|z|<1\}$ denote the unit ball in ${\mathbb{C}}^{n}.$ The group $G$ acts means of fractional transformations, the action is transitive on $\mathcal{D}_{n}$ and the isotropy subgroup at the origin of ${\mathbb{C}}^{n}$ is $K.$ We fix a Lebesgue measure $dm_{n}$ on ${\mathbb{C}}^{n}$ and define the measure $d\mu_{\alpha}=(1-|z|^{2})^{\alpha-(n+1)}dm_{n}.$ Then, in [DOZ] we find an explicit isomorphism between the Hilbert spaces $H^{2}(G,\tau_{\alpha})$ and $V_{n,\alpha}:=\mathcal{O}(\mathcal{D}_{n})\cap L^{2}(\mathcal{D}_{n},d\mu_{\alpha}).$ The unitary representation of $G$ on $V_{n,\alpha}$ is by means of the action:

[TABLE]

For $f\in\mathcal{O}(\mathcal{D}_{n})$ we write the convergent power series in $\mathcal{D}_{n},$

[TABLE]

For an integer $m\geq 0$, we consider the linear subspace

[TABLE]

Then, $\mathcal{H}_{m}:=\tilde{\mathcal{H}}_{m}\cap V_{n,\alpha}$ is a closed subspace in $V_{n,\alpha}.$ We denote $V_{m}$ for the orthogonal complement of $\mathcal{H}_{m+1}$ in $\mathcal{H}_{m}.$ Thus, a typical element of $\mathcal{H}_{m}$ (resp. $V_{m}$) is

[TABLE]

It readily follows that the action of $H$ on the a polynomial (resp. holomorphic function) in $z_{1},\cdots,z_{n-1}$ (resp. in $z_{n}$) is again a polynomial (holomorphic function) in the same variables. Whence, the subspaces $V_{m}$ are invariant for the action of $H.$ Therefore, we have the orthogonal decomposition $V_{n,\alpha}=V_{0}+V_{1}+V_{2}+V_{3}+\cdots$

The orthogonal projector $P_{\alpha,m}$ onto $V_{m}$ is given by

[TABLE]

Here, $K_{\alpha,m}$ is the reproducing kernel for the subspace $V_{m}.$ Hence, $K_{\lambda}=\sum_{m\geq 0}K_{\lambda,m}.$ Let’s write $z^{\prime},w^{\prime}$ for vectors in ${\mathbb{C}}^{n-1}.$ Then, up to a constant

[TABLE]

Hence, the $m-$th summand is equal to $K_{\alpha,m}.$

The representation of $H$ on $V_{m}$ is equivalent to $H^{2}(H,\tau_{\alpha+m}).$ An equivariant map $T_{m}$ from $H^{2}(H,\tau_{\alpha+m})\equiv V_{n-1,\alpha+m}$ onto $V_{m}$ is given by

[TABLE]

A expression for $T_{m}$ as integral map is

[TABLE]

This is due to that $\frac{1}{(1-(\,^{t}\!w^{\prime})^{\star}z^{\prime}))^{\alpha+m}}$ is the reproducing kernel for $V_{n-1,\alpha+m}.$ Finally, an intertwining map $S:V_{n,\alpha}\rightarrow H^{2}(H,\tau_{\alpha+m})$ is

[TABLE]

11. Appendix

11.1. Kernel linear maps

In this subsection we recall definitions and facts on linear maps defined by kernels.

Definition 11.1**.**

For two measure spaces $(Y,\mu),(X,\nu)$, a linear transformation $T$ from $L^{2}(Y)$ into $L^{2}(X)$ is called an integral map, kernel map or an integral operator, if there exists a function $K_{T}:Y\times X\rightarrow{\mathbb{C}}$ so that, for almost everywhere in $x\in X$, the function $y\mapsto K_{T}(y,x)g(y)$ belongs to $L^{1}(Y)$ for each function $g$ in the domain of $T$, and the equality $Tg(x)=\int_{Y}K_{T}(y,x)g(y)dy$ holds almost everywhere in $x\in X.$

In an obvious way, the definition generalizes to linear maps between vector bundles. For a measurable kernel $k:Y\times X\rightarrow{\mathbb{C}}$, the domain $\mathcal{D}_{k}$ of the linear map defined by $k$ is is the set of $f\in L^{2}(Y)$ so that $\int_{Y}k(y,x)f(y)d\mu(y)$ converges for almost every $x\in X$ and the resulting function is square integrable with respect to $\nu.$ A kernel $k$ is a Carleman kernel if the function $y\mapsto k(y,x)$ is square integrable for almost every $x\in X$. An integral is a Carleman map if it is realized via a Carleman kernel. For a Carleman kernel $k$, the linear map defined on $\mathcal{D}_{k}$ is a closed linear map. As usual, $F^{\star}$ denotes the adjoint of a linear map $F.$ Formally, the adjoint of $T$ is an integral operator with kernel $K_{T^{\star}}(x,y):=K_{T}(y,x)^{\star},$ however, even though when $T$ is continuous, the linear map $T^{\star}$ might not be equal to an integral operator on the whole dual space, as example Appendix 11.3 shows.

11.2. Adjoint to a kernel linear map

For an integral map $T$, if the adjoint linear map is an integral map of kernel $K_{T^{\star}},$ then $K_{T^{\star}}(x,y)=K_{T}(y,x)^{\star}.$

11.3. Example

For an example of an integral map whose adjoint is not an integral map, we consider $X={\mathbb{Z}}$ with discrete topology and usual Haar measure and $Y=S^{1}=\mathbb{C}/2\pi i\mathbb{Z}$ with Haar measure. Point evaluation from $\ell^{2}({\mathbb{Z}})$ into ${\mathbb{C}}$ is continuous because $|a_{n}|\leq\|(a_{k})_{k}\|_{2}.$ Thus, any continuous linear map from $L^{2}(S^{1})$ into $\ell^{2}({\mathbb{Z}})$ is a Carleman kernel map. Next, we consider $T:L^{2}(S^{1})\rightarrow\ell^{2}({\mathbb{Z}})$ given by the kernel $k_{T}(z,n)=\bar{z}^{n}$ (Fourier transform). To follow, we verify $T^{\star}$ restricted to the subspace of absolutely convergent series is an integral map, and, $T^{\star}$ is not an integral map. It readily follows that the adjoint of $T$ is the linear map

xxxxxxxxxxxx $\ell^{2}({\mathbb{Z}})\ni(f_{n})_{n}\mapsto\sum_{n}f_{n}z^{n}\in L^{2}(S^{1}).$

We notice that the adjoint kernel $k_{T}^{\star}(n,z)=z^{n}$ defines an integral map $(T^{\star})_{0}:\ell^{2}({\mathbb{Z}})\rightarrow L^{2}(S^{1})$ with domain the linear subspace

xxxxxxxxxxxxxx $\mathcal{D}_{k_{T}^{\star}}=\{(a_{n})_{n}:\,\,\,\sum_{n}|a_{n}|<\infty\}.$

In fact, Lebesgue integral is absolutely convergent, whence $f$ belongs to $\mathcal{D}_{K_{T}^{\star}}$ iff $\int_{X}|k_{T}^{\star}(x,y)f(x)|dx<\infty\,\forall y\in Y,$ and the resulting function belongs to $L^{2}(S^{1})$. In our case, $\int_{X}\cdots=\int_{\mathbb{Z}}|f(n)K_{T}^{\star}(n,z)|dn=\sum_{n}|f_{n}|.$ Thus, the domain of $K_{T}^{\star}$ is as we have claimed. Now, if $T^{\star}$ where an integral operator, Appendix 11.2, would give the kernel that should represent $T^{\star}$ ought to be $z^{n}$, hence, $\mathcal{D}_{K_{T}^{\star}}$ would be equal to $\ell^{2}({\mathbb{Z}}).$ A contradiction.

11.4. Reproducing kernel subspace

For the purpose of this paper, a closed subspace $V$ of $L^{2}(G\times_{\tau}W)$ is a reproducing kernel subspace, if $V$ consists of smooth functions and evaluation at each $x\in G$ is a continuous linear map. Hence, for each $w\in W,x\in G$ the linear functional on $V$

[TABLE]

is represented by a function $k_{x}(\cdot)^{\star}(w)\in V.$ Thus, the function $y\mapsto k_{x}(y)^{\star}(w)$ is square integrable, smooth and the following equality hold:

[TABLE]

The function

[TABLE]

is linear. We define $K_{V}:G\times G\rightarrow End_{\mathbb{C}}(W)$ to be $K_{V}(y,x)(w)=k_{x}(y)(w).$ Since the product of two square integrable functions gives an integrable function, we have, for $f\in L^{2}(G\times_{\tau}W)$ the integral below is absolutely convergent

[TABLE]

In [Hi], [OO1], [OO2] we find a proof that the map $f\mapsto P_{V}(f)$ is the orthogonal projector onto $V.$ Let $j_{W}:W\rightarrow W^{\star}$ the conjugate linear map determined by the inner product $(...,...)_{W}$. For each orthonormal basis $\{f_{j},j=1,2,\dots\}$ for $V.$ Then, in [At] we find a proof of

[TABLE]

We analyze the convergence of the series (A-4.2) in Appendix 11.5.

The main examples of reproducing kernel subspaces come from the statements below, in [At] we find proofs of the stated facts.

It readily follows that a continuous linear map $T$ from $L^{2}(H\times_{\nu}E)$ into a reproducing kernel space $V$ is a Carleman map.

11.5. $L^{2}$-kernel of an elliptic operator

Let $D$ be elliptic operator which maps sections of vector bundle into sections of perhaps another vector bundle over $G/K.$ Define $Ker_{2}(D)$ equal to the totality of $L^{2}-$sections which are in the kernel of $D$ as distributions. Then, $Ker_{2}(D)$ is closed in $L^{2}$ and owing to the regularity Theorem $Ker_{2}(D)$ consists of smooth functions, moreover, $L^{2}$ convergence of a sequence, implies uniform convergence on compact sets of the sequence as well as any derivative of the sequence, hence, $Ker_{2}(D)$ is an example of reproducing kernel Hilbert space. In [At, Prop 2.4], we find a proof of: the matrix kernel $K_{Ker_{2}(D)}:G\times G\rightarrow End_{\mathbb{C}}(W)$ determined by $Ker_{2}(D)$ is a smooth function and the convergence of the sequence in (A-4.2), as well as any derivative, is uniform on compact sets. In [At], we find a proof that $tr(K_{Ker_{2}(D)}(x,x))=\sum_{j}\|f_{j}(x)\|^{2},x\in G/K$. Whence, Schwarz inequality yields the convergence of (A-4.2) is absolute. As a Corollary, we obtain. Let $N$ be $L^{2}-$closed subspace of $Ker_{2}(D)$. Then, $N$ is a reproducing kernel subspace and the matrix kernel for $N$ is a smooth function. This is so, because the series that represents the matrix kernel for $N$ is a sub-series of the absolutely convergent series (A-4.2). For the same reason as before, the matrix kernel for $N$ is a real analytic function. In particular, the matrix kernel $K_{\lambda}$ for $\,H^{2}(G,\tau)$ is a real analytic function and the matrix kernel for any closed subspace in $\,H^{2}(G,\tau)$ is also given by a real analytic function.

11.6. Kernel for $P_{\lambda}$

The linear operator $P_{\lambda}$ has two representations as an integral operator. One matrix kernel that represents $P_{\lambda}$ is $K_{\lambda}(y,x)=\Phi_{0}(x^{-1}y)$, where, $z\mapsto\Phi_{0}(z):=d(\pi_{\lambda})P_{W}\pi_{\lambda}^{G}(z)P_{W}$ is the spherical function associated to the lowest $K-$type $W.$ Let $\phi_{0}(z)=d(\pi_{\lambda})tr(\Phi_{0}(z)),$ then, the the trace kernel which represents $P_{\lambda}$ is

[TABLE]

where $f_{1},\cdots$ is an orthonormal basis for the lowest $K-$type $V_{\lambda}[W].$ Of course, there are some identifications we have avoided to explicit. For a proof of the explicit representation of $P_{\lambda}$ by $K_{\lambda},k_{\lambda}$ we refer to [OO2], [WW]. A sketch of proof is presented below. Certainly, we recover $\Phi_{0}$ from $\phi_{0}$ by the formula $\Phi_{0}(z)=\int_{K}\tau(k)\phi_{0}(k^{-1}z)dk.$

To follow, we sketch of proof that the kernel’s $K_{\lambda},k_{\lambda}$ represent $P_{\lambda}$ as an integral operator. The orthogonal projector onto $\,H^{2}(G,\tau)$ is represented by the matrix kernel $K_{\lambda}(y,x)=d(\pi_{\lambda})P_{W}\pi_{\lambda}(x^{-1}y)P_{W}.$ In fact, let $f_{v}$ be as in Proposition 6.8, then

[TABLE]

A trace kernel that represents the orthogonal projector onto $H^{2}(G,\tau)$

is $k_{\lambda}(y,x)=d(\pi_{\lambda})\phi_{0}(x^{-1}y)$. Indeed,

[TABLE]

12. Notation

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

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

• $d_{\lambda}=d(\pi_{\lambda})$ dimension of $\pi_{\lambda},$ $P_{\lambda},P_{\mu},K_{\lambda},k_{\lambda},K_{\mu},k_{\mu},$ (cf. Section 2).

-$\Phi_{0},\phi_{0}$ (cf. Appendix 11.6).

-$M_{K-fin}(resp.\,M^{\infty})$ $K-$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.

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

-For a module $M$ (resp. a simple submodule $N$)over a ring, $M[N]$ 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.

-$K_{\lambda,\mu}$ matrix kernel for the orthogonal projector $P_{\lambda,\mu}$ onto $\,H^{2}(G,\tau)[V_{\mu}^{H}].$

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

-${\mathbb{N}}:=\{1,2,\cdots\}.$

Acknowledgements The authors gratefully thank the referees for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.

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. The second author thanks Aarhus University for generous support, its hospitality and excellent working conditions during the preparation of this paper.