This paper explores natural enlargements of real bounded symmetric domains, extending harmonic analysis, boundary orbit relations, and representation theory to broader contexts.
Contribution
It introduces new extensions of real bounded symmetric domains, connecting boundary orbits, crown domains, and representation extensions.
Findings
01
Boundary orbits of G/K relate to those of G_h/K_h.
02
Extensions of K-representations to larger groups are constructed.
03
K-finite matrix coefficients extend analytically to Matsuki cycle spaces.
Abstract
For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally real submanifold in a bounded domain G_h/K_h. We describe the boundary orbits and relate them to the boundary orbits of G_h/K_h. We relate the crown and the split-holomorphic crown of G/K to the crown \Xi_h of G_h/K_h. We identify an extension of a representation of K to a larger group L_c and use that to extend sections of vector bundles over the Borel compactification of G/K to its closure. Also, we show there is an analytic extension of K-finite matrix coefficients of G to a specific Matsuki cycle space.
exp(4πiY1)=21(11−11) and Ad(exp(4πiY1))h=X1.
exp(4πiY1)=21(11−11) and Ad(exp(4πiY1))h=X1.
cI,ϵ:=exp(4πiY(I,ϵ))=κI,ϵ(21(11−11)) and CI,ϵ:=Ad(cI,ϵ),
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Full text
Extensions of real bounded symmetric domains
Gestur Ólafsson
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803
For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of
harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally
real submanifold in a bounded domain Gh/Kh. We describe the boundary orbits and relate them to the
boundary orbits of Gh/Kh. We relate the crown and the split-holomorphic crown of G/K to the
crown Ξh of Gh/Kh. We identify an extension of a representation of K to
a larger group Lc and use that to extend sections of vector bundles over the Borel compactification of G/K to its closure. Also, we show there is an analytic extension of K-finite matrix coefficients of G to a specific Matsuki cycle space.
Key words and phrases:
Structure of semisimple symmetric spaces, bounded domains, duality of symmetric spaces, extension of
representations, crown
2000 Mathematics Subject Classification:
Primary 32M15, 22E46, Secondary: 22E50, 53C35
Introduction
Élie Cartan was the first to prove the existence of a compact real form of a complex semisimple Lie algebra.
This can be considered the introduction of duality into the theory of Riemannian symmetric spaces. Subsequently, even
in the more general context of symmetric spaces, various people have identified several types of
duality. In this paper we explore some of the consequences of a type of duality
involving compactly causal spaces and noncompactly causal spaces or, said
geometrically, involving Hermitian and split-Hermitian
spaces111split-Hermitian
and split-complex are called para-Hermitian in [K85, K87] and elsewhere.
We describe here, in heuristic form, various results to which one is lead
(to conjecture) from this viewpoint. Some of this is, without a doubt, known to experts. Thus,
as we use standard terminology, we relegate precise definitions and careful notation
to subsequent sections, for now we take a more casual approach.
Let Gh be a semisimple Hermitian Lie group of noncompact type with maximal compact subgroup Kh, i.e. Gh/Kh is a Hermitian Riemannian symmetric space. Let τ be an involution commuting with θ and such that, G, the fixed point set of τ has Riemannian symmetric space, G/K, a real form of Gh/Kh. Denote by gh the Lie algebra of Gh and by ghC its complexification. Of course τ induces an involution τ˙ on gh. We let τ˙ also denote the complex linear extension of τ˙ to ghC; while we let η˙ be its complex conjugate−linear extension to ghC. The associated holomorphic (resp. anti-holomorphic) involutions on Gh are denoted τ (resp. η). Then Gh (resp. Kh) is the fixed point set of τ in Gh (resp. Kh), and let Gc (resp. Lc) be the fixed point set of η in Gh (resp. Kh). Then Gc is a semisimple split-Hermitian Lie group, i.e. Gc/Lc is a split-complex pseudo-Riemannian symmetric space with an integrable bi-Lagrangian structure. Now τ restricts to Gc giving an involution τc having fixed point set G such that G/K is a split-real form of Gc/Lc. One could repeat the above with Gc and τ˙c the complex linear extension to gcC. Notice that gcC≅ghC but not equal. Various properties of {Gh,G,η} and {Gc,G,τc} are the main focus of this paper. Detailed discussion about Gc/Lc and its compactification can
be found in the work of Kaneyuki [K85, K87].
We begin with several decompositions involving {Gh,G,η}.
From Harish-Chandra we have the open subset
[TABLE]
then applying η we should obtain similarly the open containment
[TABLE]
From [KS04] we have the complex open neighborhood of Gh
[TABLE]
then with η we should obtain an open neighborhood of G
[TABLE]
Also from [KS04] we have the holomorphic extension of the Iwasawa decomposition
[TABLE]
so that with η we get
[TABLE]
The Akhiezer-Gindikhin crown of Gh is an open subset
[TABLE]
so with η we should get for the ‘ real ’crown of G
[TABLE]
Now in [KS05] and for a real form G/K, the existence of an open subset Ξ0⊂Ξ is shown such that
[TABLE]
A straightforward variation of that argument shows that
[TABLE]
From various sources we have the crown of Gh is biholomorphic to an open subset of flag manifolds
[TABLE]
so that applying η we have for the crown of G an open subset
[TABLE]
In the various parts of the text we will identify several of these fixed point sets for η. The intent of this summary is to motivate several results. Now we give a more careful outline of the paper. The bounded Hermitian domain Gh/Kh has a boundary that is a finite union of Gh orbits whose geometric structure is described in considerable detail in [Sa80]. We summarize this in §1 so that in §2 and §3 using η we may give a similar description of the boundary G orbits for G/K. This geometric description was crucial in [MSIII] to describe the decomposition of a natural holomorphic extension of homogeneous vector bundles to the boundary along these Gh orbits. For the R-form G/K an extension of homogeneous vector bundles over G/K to the boundary will be needed and a geometric description of their decomposition on the orbits. An extension of homogeneous vector bundles over G/K is the content of §4, §5 and §6. In §7 we give a proof of the open neighborhood (c’) using both η and the main result in [Ma03]. Using this, the holomorphic extension of the Iwasawa decomposition (c) from [KS04], together with η we obtain then in §7 an analytic extension of K-finite matrix coefficients of irreducible representations of G to D=LcexpiΩhη˙G.
1. Bounded Symmetric Domains: Complex Case
We recall some facts about bounded symmetric domains in Cn. This goes back
to [KW65a, KW65b, W69, W72], but for structure theory our reference is [Sa80],
although we shall alter his presentation to suit our needs; for analysis see
[KS05], [MSIII].
1.1. Notation
Let Dh be a bounded symmetric domain in Cn. The identity component of the group of holomorphic automorphisms of Dh is a connected noncompact semisimple Lie group that we shall denote by Gh 222The subscript h will be used for objects related to the Hermitian symmetric space.. The group Gh acts transitively, and the isotropy at any base point is a maximal compact subgroup of Gh. We fix one and denote it by Kh, so that Dh≃Gh/Kh. The Lie algebra of Gh (resp. Kh) is denoted by gh (resp. kh), while the superscript C denotes a complexification of the indicated Lie algebra. For a cleaner presentation we assume that Gh is simple, and that it is contained in a simply connected complex Lie group Gh whose Lie algebra is ghC. The analytic subgroup of Gh corresponding to khC is denoted Kh. The reason for requiring Gh to be simply connected comes from the following result, see [He78, Thm. 8.2, p. 320 and p. 351].
Proposition 1.1**.**
Let G be a connected simply connected semisimple Lie group with finite center and σ:G→G an involutive homomorphism. Then
Gσ:={a∈G∣σ(a)=a} is connected.
Proof.
In [He78] G is assumed to be compact; in [Ra74] G is just simply connected. If G is semisimple with finite center here is an easier argument. Let θ be a Cartan involution commuting with σ,
g=k⊕p the associated Cartan decomposition, and K=Gθ. Then K is compact, connected, and simply connected by hypothesis. Furthermore Gσ=Kσexp(pσ). Since
Kσ is connected, the claim follows.
∎
If h is a Lie algebra and if φ˙:gh→h is a Lie algebra homomorphism, then we denote by the same letter the
complex linear extension, i.e. φ˙:ghC→hC. Similarly on the group level, if τ:Gh→H is an analytic homomorphism, and
if H is contained in a complex Lie group H with Lie algebra hC, then
we will denote by the same letter the holomorphic extension, i.e. τ:Gh→H. This extension always
exists as we are assuming
that Gh is simply connected. The same convention will be used for other Lie groups without comment.
Let θh:Gh→Gh be the Cartan involution corresponding to Kh, i.e. θh2=id and Ghθh=Kh. Denote by θ˙h:gh→gh the derived involution. Then kh={X∈gh∣θ˙h(X)=X} and with ph:={X∈gh∣θ˙h(X)=−X}, one has gh=kh⊕ph. The subspace ph can be identified with the tangent space of Dh at eKh. As Dh is a complex domain, there is a complex structure J:ph→ph. Moreover, J extends to a derivation of gh which, as gh is semisimple, must be inner. Since J commutes with adkh∣ph, the derivation is represented by an element Zh in zkh, the center of kh, i.e. J=adZh∣ph. As we also assume that Gh is simple, one knows that zkh is one dimensional, hence J is essentially unique.
As (adZh∣ph)2=−1, adZh has eigenvalues [math], i, and −i. For the respective eigenspaces we have ghC(adZh;0)=khC, and we set ph±:=ghC(adZh;±i). Then ph± is a complex abelian subalgebra of dimension n; Kh acts on ph±; and phC=ph+⊕ph− as a Kh-module. The Kh-modules ph+ and ph− are contragredient and, as the center acts by a different constant, inequivalent.
Denote by Ph+, resp. Ph−, the analytic subgroup of Gh corresponding to the Lie algebra ph+, resp. ph−. Then
Ph± is abelian, simply connected and exp:ph±→Ph± is a holomorphic diffeomorphism and group homomorphism. We denote the inverse of exp∣ph+ by log.
Proposition 1.2**.**
Ph+KhPh−* is open and dense in Gh, and the multiplication map*
[TABLE]
is a holomorphic diffeomorphism. We denote the inverse by
[TABLE]
We consider the usual generalized flag manifold Ph=Gh/KhPh− and a basepoint xo=eKhPh−. The Gh orbit of the basepoint, Gh⋅xo, is Gh/Kh≃Dh. On the other hand, the Bruhat cell Ph+⋅xo is open and dense in Ph. By means of log one obtains a holomorphic isomorphism Ph+⋅xo≃Ph+≃ph+, denoted by g⋅xo↦z(g⋅xo), such that for p∈Ph+⋅xo, k∈Kh and
X∈ph+
(1)
z(k⋅p)=Ad(k)z(p)
2. (2)
z(exp(X)⋅p)=X+z(p).
Restricted to Gh⋅xo the map has image Dh+⊂ph+, the Harish-Chandra bounded realization of Dh.
Theorem 1.3**.**
ph+⊃Dh+≃Dh≃Gh/Kh⊂Ph=Gh/KhPh−.**
In a moment we will discuss the boundary components of Gh/Kh. For that we note that we can
take a closure in ph+ or the closure in Ph. It is a priori not clear that those two closures should
be isomorphic. It is however clear that the closure in Ph is Gh-invariant, but
it is not clear that Gh acts on the closure in ph+. In Lemma 1.7 we
show that c(Dh+), the closure of Dh+ in ph+, viewed as a subset of Ph is
the same as the closure in Ph. In particular, Gh acts on
c(Dh+)333Note that this is not correct for the
unbounded realization of Gh/Kh as the example of the upper half-plane shows..
Let ∂Dh+:=c(Dh+)∖Dh+ be the topological boundary
of Dh+ in ph+. The action of Gh on Dh+ extends to one on ∂Dh+ which then decomposes into a finite disjoint union of Gh-orbits. In a later section we shall give a complete parameterization of the orbits and determination of the isotropy. This is well known, e.g. [Sa80], but we include the proof because of its importance for our treatment of real domains.
1.2. Essential Structure Theory - C forms
Let ch be a Cartan subalgebra of gh containing Zh, hence ch⊂kh. Let Δh be the set of roots of chC in ghC. Since ch⊂kh, θ˙h∣ch=id. Then θ˙h(ghCα)=ghCα, and as dimCghCα=1,
either ghCα⊂khC in which case one calls α a compact root, or ghCα⊂phC and α is called noncompact. Denote by Δhc the set of compact roots, and by Δhn the set of noncompact roots. Then
[TABLE]
We choose the set of positive roots, Δh+, so that {α∣α(Zh)=i}⊂Δh+.
Denote by Wh=W(Δh) the Weyl group generated by
reflections sα, α∈Δh, and denote by Whc the
subgroup generated by sα, α∈Δhc. As α(Zh)=0 for
all α∈Δhc it follows that Δh+ is invariant under Whc.
Recall that α,β∈Δh are called strongly orthogonal if α±β∈Δh. In the usual way one constructs a maximal set {γ1,…,γrh} of strongly orthogonal roots in Δhn+.
Denote by σ˙h:ghC→ghC the conjugation
with respect to gh. For each j=1,…,rh choose Ej∈ghCγj and set Fj=σ˙h(Ej)∈ghC−γj. One can normalize Ej so that with Hj=[Ej,Fj]∈ich one has γj(Hj)=2. Let Zj:=iHj, Xj:=Ej+Fj, and Yj:=i(Ej−Fj). We set
[TABLE]
Then ah is maximal abelian in ph.
More generally, for I⊆{1,…,rh} and ϵ∈{−1,1}#I let Z(I,ϵ):=∑j∈IϵjiHj∈ch, E(I,ϵ):=∑j∈IϵjEj∈ph+, F(I,ϵ):=∑j∈IϵjFj∈ph−. Similarly H(I,ϵ):=−iZ(I,ϵ),
X(I,ϵ):=E(I,ϵ)+F(I,ϵ)∈ph and Y(I,ϵ):=i(E(I,ϵ)−F(I,ϵ))∈kh. We set H0=−iZh. If I={1,…,b} then we write E(b,ϵ) instead of E(I,ϵ), etc., and if
furthermore ϵ=1 then we simply write E(b) etc.
Then Z(I,ϵ), X(I,ϵ) and Y(I,ϵ) generate a subalgebra of gh isomorphic to su(1,1). These determine equivalence classes of holomorphic disks in Dh which, since we are in a homogeneous space, lift to equivalence
classes of compatible homomorphisms κ˙:su(1,1))→gh together with their holomorphic extensions κ˙:sl(2,C)→ghC, here sl(2,C):=su(1,1)C . Amongst these, the
homomorphisms associated to Z(b), X(b) and Y(b) play a critical role and will be referred to as basic homomorphisms.
In passing we note that H(I,ϵ), E(I,ϵ) and F(I,ϵ) generate a subalgebra of ghC isomorphic to sl(2,R) which is the Cayley transform of the su(1,1) described above. Much of this notation is not needed in the complex case but will be needed when we do the real case.
The word ‘Essential’ in the subsection title refers to the fact that we have fixed the structure theory, whereas if, as in [Sa80], one chooses first the geometry of holomorphic disks, one would have a different but equivalent choice of structure theory.
Basic Example**.**
SU(1,1)
The prototype bounded domain is SU(1,1)/U(1)≃{z∈C∣∣z∣<1}. In this case we have Gh=SL(2,C).
The conjugation σ˙1 444The subscript 1 will be used for objects related to this basic case. on sl(2,C)(:=su(1,1)C) with respect to su(1,1), and the holomorphic extension of the
standard Cartan involution θ˙1 of su(1,1) are given by
[TABLE]
Let
[TABLE]
Then [h,e]=2e, [h,f]=−2f, [e,f]=h, σ˙1(h)=−h and σ˙1(e)=f. Taking Z1=ih gives ph+=Ce, khC=Ch, and ph−=Cf.
A computation gives
[TABLE]
Thus
[TABLE]
Identifying Ph+≅ph+≅C, Ph−≅ph−≅C and Kh≅C∗
the maps in (1.1) are given by
[TABLE]
This gives the Harish-Chandra realization of Dh as Dh+=D1={z∈C∣∣z∣<1}. As
[TABLE]
it follows that the action of SU(1,1) on Dh is the usual action g⋅z=cz+daz+b.
To return to the general situation we let in su(1,1)
[TABLE]
For I and ϵ as above,
[TABLE]
defines a Lie algebra homomorphism of sl(2,C) into ghC such that
[TABLE]
It follows, in particular, that
[TABLE]
These are the lifts of holomorphic disks embedded into Dh and are those called standard homomorphisms.
Similarly
[TABLE]
As SL(2,C) is simply connected, there exists a group homomorphism κI,ϵ:SL(2,C)→Gh such that
dκI,ϵ=κ˙I,ϵ. In particular, κI,ϵ(SU(1,1))⊆Gh. We set
κ˙j=κ˙{j},1 and κj=κ{j},1. We also note that if I∩J=∅ then
[κ˙I,ϵ(sl(2,C)),κ˙J,ϵ′(sl(2,C))]={0} and similarly for κI,ϵ and
κJ,ϵ′. In particular, κ1,…,κrn is a maximal family of commuting standard homomorphisms
SL(2,C)→Gh.
A simple matrix calculation shows that
[TABLE]
Thus if we set
[TABLE]
then
[TABLE]
Lemma 1.4**.**
Let C=C{1,…,rh},(1,…,1). Then C(th)=ah.
Theorem 1.5** (Moore).**
Let βj:=[C−1]t(γj∣th).
There are two possibilities for the restricted roots Σh=Σ(gh,ah):
Case I:* Σh=±{βi,21(βj±βk)i=1,…,rh,1≤j<k≤rh},*
Case II:* Σh=±{βi,21βi,21(βj±βk)i=1,…,rh,1≤j<k≤rh}.*
The first case occurs if and only if Dh is a tube type domain.
We will use
[TABLE]
as a positive Weyl chamber. The corresponding set of positive roots are obtained by taking + in front of
the parenthesis in Case I and II above.
1.3. Boundary orbits
Using SU(1,1)-reduction, eq. (1.8) is the main step in the proof that
Dh≃Gh/Kh is diffeomorphic to a bounded domain in ph+. Indeed let
Ωh=∑j=1rh(−1,1)Ej⊂ph+. For t∈Rrh let at:=exp∑j=1rhtjXj. By a calculation in SU(1,1) we have
[TABLE]
In particular,
[TABLE]
Thus we have
[TABLE]
the Harish-Chandra bounded realization of Dh.
Now it is clear that Gh acts on ∂Dh+. For b∈{1,…,rh} recall E(b):=E1+…+Eb
and set Oh(b):=Gh⋅E(b).
Theorem 1.6**.**
Let z∈∂Dh+. Then there exists b∈{1,…,rh} and g∈Gh such that
z=g⋅E(b). In particular,
[TABLE]
Thus, the boundary orbits are parameterized by {1,…,rh}.
Proof.
Let {zn} be a sequence in Dh+ such that zn→z. As ah is maximal abelian in ph, there exists kn∈Kh and tjn∈(−1,1) such that zn=knexp∑j=1rhtjnEj. By applying a Weyl group element we can assume that t1n≥t2n≥…≥trhn≥0. As K and [−1,1] are compact we can assume (by going to subsequences) that kn→k∈K and tjn→tj∈[−1,1]. Let b be such that t1,…,tb=1 and
1>tj≥0 for j>b. Let sj=−tanh−1(tj) for j>b. By (1.9) we have
[TABLE]
We can therefore take g=exp(∑j>bsjXj)k−1.
∎
The closure of Dh in Ph appears to be
bigger than c(Dh+) the closure of Dh+ in ph+. In fact we show,
Lemma 1.7**.**
The closure of Dh in Ph is the same
as the closure, c(Dh+), of Dh+ in exp(ph+)⋅xo.
In particular, the action of Gh extends to c(Dh+).
Proof.
It is clear that the closure in exp(ph+)⋅xo is contained in the
closure in Ph. As in the proof above, let z=limjkjaj⋅xo be in the closure of Dh in Ph. Again let k be a limit of a subsequence of {kj} and recall that kj and k normalize
ph+. As ±Ej∈ph+ it follows that z∈k⋅expph+⋅xo=expph+⋅x0.
∎
Consequently, we do not have to distinguish if we are talking about the closure of Dh in Ph,
or the closure of Dh+ in ph+.
1.4. Isotropy of boundary orbits
We come to the determination of the isotropy of the various orbits in the boundary. Again,
we provide more details than are needed in the complex case, but they will be used later in the real case. Let
[TABLE]
Then the boundary orbit Oh(I,ϵ):=Gh⋅E(I,ϵ) is isomorphic to Gh/Qh(I,ϵ).
If I={1,…,b} and ϵ∈{−1,1}b then we simply write E(b,ϵ),Qh(b,ϵ),Oh(b,ϵ), etc.
If ϵ=(1,…,1) then we do not include it in the notation.
As before, for a standard homomorphism κ˙:sl(2,C)→ghC, i.e. (1.5), we write Eκ for κ˙(e),
Xκ=κ˙(e+f) etc. The corresponding homomorphism SL(2,C)→Gh and the restriction to
SU(1,1) is denoted by κ. The following is valid for an arbitrary standard homomorphism.
To avoid even more burdensome notation we will use subscripts involving κ only when it seems useful. We remark that
[Sa80] (Chapter 2 and 3) refers to a standard homomorphism as one κ˙:sl(2,R)→gh.
There should be no confusion from the terminology herein as the two are related by the Cayley transform introduced
earlier, see [Sa80, p. 107–109] for a detailed discussion.
Given a standard homomorphism let πκ:=ad∘κ˙. Then πκ is a finite dimensional representation of
sl(2,C). As the irreducible representations of sl(2,C) are determined by their dimension with the 1-dimensional
representation being the trivial representation, the 2-dimensional representation being the natural
representation of sl(2,C) acting on C2, and the 3-dimensional representation being the adjoint representation of
sl(2,C) acting on itself. The corresponding highest weights are [math], 1, and 2. According
to [Sa80] Lemma 1.1 p. 90, every irreducible sl(2,C) representation occurring
in πκ has dimension less than or equal to 3. Following [Sa80], for ν∈{0,1,2}
denote by ghC[ν] (resp. gh[ν]) the corresponding isotypic subspace. Then (as sl(2,R) is split)
[TABLE]
and similarly for the complexification ghC. From [Sa80] §1, Chapter 3 we obtain
Lemma 1.8**.**
Let κ:SU(1,1)→Gh be a standard homomorphism. Let
[TABLE]
Then the following conditions are equivalent:
(1)
Zκ(1)=0;
2. (2)
gh[1]={0}* and gh[0] is compact.*
Notice that each of the spaces ghC[ν] is σ˙h and θ˙h stable. As κ˙ is standard, it intertwines the respective Cartan involutions and conjugations. Hence we have similar decompositions for kh, khC, ph, phC, and ph±. In particular, we have
[TABLE]
Also, ghC[0](=zghC(κ˙(sl(2,C))) is a subalgebra, as is
[TABLE]
Since gh[even] is θ˙h-stable, it follows that gh[even] is a reductive subalgebra. Furthermore Zh∈gh[even],
so each non-compact ideal of gh[even] is of Hermitian type.
Next we decompose gh[even] into ideals, gh[even]=⨁jghj, such that gh0 is the maximal compact ideal, while ghj is simple and noncompact for j≥1. It follows that the maximal abelian ideal of gh[even] is contained in gh0, and each ghj, j≥1, is of Hermitian type. Define
[TABLE]
Then gh(1)⊆gh[0],gh[2]⊆ghκ∗ and gh[even]=gh(1)⊕ghκ∗. The corresponding analytic subgroups of Gh will be denoted by the respective upper case Latin letter.
Finally we arrive at the parabolic subalgebra corresponding to κ. Recall that Xκ=κ˙(e+f). Let mhκ0:=gh(adXκ;0), nhκ1:=gh(adXκ;1), nhκ2:=gh(adXκ;2),
nhκ:=nhκ1⊕nhκ2 and qhκ:=mhκ0⊕nhκ555Note that our notation here differs from
[Sa80, p.95] where nhκ1 is denoted by Vκ and nhκ2 is
denoted by Uκ.
Then
qhκ
is a maximal parabolic subalgebra of gh. Denote by Qhκ=Mhκ0Nhκ the
corresponding maximal parabolic subgroup in Gh.
It will be useful to give a more detailed description of Qhκ, the nilradical Nhκ,
the structure of Mhκ0 and the connected component
of Mhκ0.
Let Fκ be the finite abelian group generated γκ=exp(πiXκ). We have the
Levi factor
[TABLE]
We note that by [V77, p. 287] every Ad(m), m∈Mhκ0, is in Ad(mhκ0C) but not necessarily,
as the set Fκ shows, in Ad(mhκ0).
Now Fκ preserves the decomposition
(1.13), and as Fκ⊂Kh it also preserves (1.14).
Finally
[TABLE]
Thus each ph±[ν] is a Kh∩Mhκ0-module.
Consider next the Lie algebras defined in (1.16) and their relationship to the Levi factor mhκ0. Recall cκ=exp(4πiYκ) and Cκ=Ad(cκ). Set ghκ(2)=Cκ−1(khκ∗C)∩gh.
Then mhκ0=mhκ(1)⊕mhκ(2)⊕R(Xκ) where
mhκ(1)=l2⊕gh(1),
l2 is a compact ideal in mhκ0 and
gh(1) is of Hermitian non-compact type having
Zκ(1)
defining the almost complex structure, and mhκ(2)⊕R(Xκ)=ghκ(2).
Let Mκ(i) be the connected
subgroup with Lie algebra mκ(i). Then we have
(Mκ0)o=Mκ(1)Mκ(2) exp R(Xκ) and
Mκ0=FκMκ(1)Mκ(2)exp R(Xκ).
Lemma 1.9**.**
The following holds true:
(1)
adgh(1)∣nhκ1=0* and adghκ(2)∣nhκ1 is faithful. The orbit Gh(2)⋅Eκ is a self-dual cone.*
2. (2)
Let Io=−ad(Yκ)∘θ˙h∣nhκ1=θ˙h∘ad(Yκ)∣nhκ1=2ad(Zκ(1))∣nhκ1. Then Io defines a complex structure on nhκ1. We have
[TABLE]
Now we have all the notation to give a detailed description of the stabilizer of Eκ∈Dh+ and hence the isotropy of the orbit Oh(κ), see §1, Chapter 3 and Proposition 8.5, p. 142 in [Sa80].
Theorem 1.10**.**
Let κ:SU(1,1)→Gh be a standard homomorphism.
Then one has the following:
(1)
Zκ∈kh[2].
2. (2)
Zκ(1)∈kh[0].
3. (3)
gh[even]=gh(adYκ;0)⊕gh(adYκ,2)⊕gh(adYκ;−2)=gh(Cκ4;1)* as a
Mhκ0∩Kh-module.*
4. (4)
adZκ(1)∣gh[0]=adZh∣gh[0]. In particular, the Zh-element in the Hermitian type Lie algebra
gh[0] is Zκ(1).
7. (7)
If Zκ(1)=0 then the stabilizer of Eκ in Gh is ZGh(Xκ,Zκ(1))Nhκ. Hence there is a fibration
[TABLE]
with typical fiber a Hermitian symmetric space.
8. (8)
If Zκ(1)=0 then the stabilizer of Eκ in Gh is ZGh(Xκ)Nhκ=Qhκ. Hence the orbit Oh(κ)≅G/Qhκ≅Kh/Kh∩Mhκ0. In particular,
in this case Oh(κ) is compact.
Next consider the Cartan decomposition of mhκ0 corresponding to the Cartan involution
θ˙∣mhκ0. First we have
gh(1)=kh(1)⊕ph(1),
and (ph(1))C=ph+(1)⊕ph−(1), with
ph±(1) simultaneous ±i eigenspaces of adZκ(1) and adZh. Moreover we have the
identification ph±[0]=ph±(1).
Now consider mhκ(2), the other summand of mhκ0, with Cartan
decomposition kh∩mhκ(2)⊕ph(2). Note that
[TABLE]
Lemma 1.11**.**
Zκ* is in the center of khκ∗ and
Cκ−1(khκ∗C)=mhκ(2)C⊕CXκ.*
Define now
[TABLE]
Lemma 1.12**.**
The Lie algebra lκ of Lκ decomposes into ideals as
[TABLE]
and Zκ defines an almost complex structure on Kh/Lκ.
This gives yet another fibration in Theorem 1.10 (7),(8) here with base Kählerian, namely
Cκ∘θ˙h∘Cκ−1(ph+[1])=khC(adZκ;i)* as Kh∩Mhκ0-modules.*
For convenience we summarize these various identifications in the next
statement.
Proposition 1.14**.**
We have the following Kh∩Mhκ0-isomorphisms:
(1)
ph±[0]≅ph±(1).
2. (2)
ph+[1]≅khC(adZκ;i).
3. (3)
ph±[2]≅ph(2)C⊕CXκ.
2. Bounded Symmetric Domains: Real Case
In this section we consider homogeneous real forms of Dh+, i.e. fixed point sets of anti-holomorphic automorphisms.
We continue to assume that Gh is simple or of the form Gh=G×G where G/K is
a bounded symmetric domain in Cn with G simple. Thus either gh is simple, or gh=(g,g) with
g simple and τ(X,Y)=(Y,X). We use [HÓ96, Ó90, Ó91] as standard references although
the perspective will be slightly different in this section. We will present a parallel presentation for the material
for real domains vis à vis the complex case. The first observation in the real case will be a replacement for Gh.
This will be the Lie group Gc to be described shortly.
2.1. Real Bounded Symmetric Domains and Related Subgroups of Gh
Let τ:Gh→Gh be a non-trivial involution commuting with θh. Let τ˙:gh→gh
be the derived involution which then commutes with θ˙h.
Finally, we let G:=Gh0τ. Then G is a connected, reductive subgroup of Gh having Lie algebra
g:=ghτ˙={X∈gh∣τ˙(X)=X}.
With the usual notation, set qh:={X∈gh∣τ˙(X)=−X}. Then gh=g⊕qh.
As τ˙ and θ˙h commute, it follows that θ˙:=θ˙h∣g defines
a Cartan involution on g and g=k⊕p with k:=g∩kh and p:=g∩ph.
Also, qh=qhk⊕qhp with qhk=qh∩kh and qhp=qh∩ph.
As τ and θh commute, τ induces an involution on Gh/Kh≃Dh denoted τDh:Dh→Dh such that τDh(g⋅z)=τ(g)⋅τDh(z). Via the biholomorphism
Dh≃Dh+, τ induces an involution denoted σh+:Dh+→Dh+ such that
σh+(g⋅z)=τ(g)⋅σh+(z). We assume that σh+ is anti-holomorphic, i.e., defines
a conjugation on Dh+⊂ph+. Then D+:=Dh+σh+ is a totally real submanifold as follows from
Lemma 2.1**.**
τ˙(Zh)=−Zh.
Let K:=G∩Kh=Gθ. Then K is maximal compact in G with Lie algebra k. We have
(see Lemma 1.3 for notation)
[TABLE]
is a realization of the Riemannian symmetric space G/K as a bounded totally real domain in ph+.
We come to the substitute for Gh. Denote by η˙:=σ˙h∘τ˙ the conjugate
linear extension of τ˙ to ghC and, as usual, η the corresponding involution on
Gh. Set gc=(ghC)η˙ and let Gc 666The subscript c will be used for
objects related to this group. be the corresponding analytic subgroup of Gh.
By Lemma 1.1, Gc=Ghη as Gh is assumed simply
connected. gc is a real semisimple subalgebra of ghC which is stable under τ˙ and θ˙h.
Clearly
[TABLE]
and with iqh={X∈gc∣τ˙(X)=−X}, then
[TABLE]
where k=kh∩g and p=ph∩g.
On the
other hand, the involution θ˙c:=τ˙∘θ˙h∣gc defines a Cartan involution on
gc with corresponding decomposition
gc=kc⊕sc
and corresponding Cartan involution θc on Gc (we reserve the notation pc for a parabolic subalgebra).
Then kc=k⊕iqhp and sc=p⊕iqhk. Notice that θ˙c agrees with the conjugate linear extension of θ˙h restricted to gc.
To streamline the notation we let qc:=iqh so that gc=g⊕qc. Then qc=qc∩kc⊕qc∩sc=qck⊕qcp, with
qck=iqhp and qcp=iqhk, i.e., the elliptic and hyperbolic parts have been interchanged.
In the special case that D+ is a bounded complex domain, then Gc=G, the complexification of G.
Lemma 2.2**.**
σh+=η˙∣Dh+. In particular, D+=Dh+∩gc.
Proof.
Recall that H0=−iZh. From Lemma 2.1 and η˙=σ˙h∘τ˙ we get η˙(H0)=H0.
As ph±=ghC(adH0;±1) and khC=ghC(adH0;0) it
follows that η˙(ph±)=ph± and η˙(khC)=khC, similarly η(Ph±)=Ph± and η(Kh)=Kh. For g∈Gh and 0∈ph+ write g⋅0=Z∈Dh+. Then
g=exp(Z)kC(g)p−(g) and
[TABLE]
From this the claim follows.
∎
2.2. Essential Structure Theory - R forms
We shall refine our choice of Cartan subalgebra ch⊂gh to take into account the involution τ˙ and the associated decomposition gh=g⊕qh. We still require ch to contain Zh but now choose the Cartan subalgebra ch such that bh:=ch∩qhk is maximal abelian in qhk. Thus all the notation from §1.2 remains in force here so will be used freely when applicable.
Denote by Σ(ghC,bhC) the set of roots of bhC in ghC. Set ac:=ibh⊂sc.
Recall that σh+ is anti-holomorphic and τ˙(Zh)=−Zh.
Lemma 2.3**.**
bh* is maximal abelian in qhk and qh ; ac is maximal abelian in sc and in qcp=sc∩qc.*
Proof.
The first claim is by construction. Since Zh∈bh one has zghC(bhC)⊂khC=kC⊕qhkC, while qcp=iqhk. Hence sc∩zghC(bhC)⊂iqhk.
∎
Corollary 2.4**.**
Σ(gc,ac), the set of restricted roots of ac in gc, are all restrictions from the complex space Σ(ghC,bhC) to the real form ac.
Let Σc:=Σ(gc,ac). We will view Σc either as the set of roots of ac in gc or the roots of
bhC in ghC without further comment.
Recall that H0=−iZh∈ac. Then ad(H0) has three eigenvalues: 0,±1. We set
[TABLE]
where N− denotes the analytic subgroup of Gc with Lie algebra n− and
Lc:=ZGc(H0). Note that Lc has Lie algebra lc, but that Lc is not
necessarily connected. For future reference we set lc′:=[lc,lc]. We
also note that
[TABLE]
The set Σc of restricted roots decomposes accordingly into two disjoint sets
[TABLE]
and
[TABLE]
If α∈Σcn then α(H0)=±1. We choose the system of positive roots in Σc such that
[TABLE]
and
Σcc+∪{0}=Δhc+∣ac.
From lc=k⊕iqhk notice that K is a maximal compact subgroup of (Lc)o and preserves n±;
ac is maximal abelian in lcp; and Σcc is the set of restricted roots of ac in lc .
Lemma 2.5**.**
Let mc denote the centralizer of ac in kc. Then mc⊂k.
Proof.
Since H0∈ac one has mc⊆zgc(H0)=k⊕iqhk.
∎
Denote by Wcc the Weyl group generated by the roots in Σcc.
Lemma 2.6**.**
Wcc=NK(ac)/ZK(ac)* and Wcc(Σcn+)=Σcn+.*
As Zh∈bh and τ˙(Zh)=−Zh it follows that Δhn, Δhn+ and Δhc are stable under the involution τ˙♯:β↦−β∘τ˙. Moreover τ˙♯∣ich∗=η˙t∣ich∗. Via the identification of Σ(gc,ac) with Σ(ghC,bhC) we extend τ˙♯ to Σc.
The dichotomy present in Moore’s classification of restricted roots in the complex case is reflected in the next several results.
Let γ∈Δhn+. If τ˙♯(γ)=γ then γ and τ˙♯(γ) are strongly orthogonal. In particular, if {γ1,…,γrh} is a set of strongly orthogonal roots in Δhn+, then either
τ˙♯(γj)=γj, or γj and τ˙♯(γj) are strongly orthogonal.
Lemma 2.8**.**
We have either τ˙♯(γj)=γj for all j=1,…,rh, or τ˙♯(γj)=γj for all
j=1,…,rh.
This follows from the classification in A. The classification also shows that τ˙♯γj=γj only for the following five cases:
(1)
Gh=G×G is not simple and G/K is embedded into G/K×G/K diagonally.
2. (2)
gh=so(2,n) and g=so(1,n), n≥3.
3. (3)
gh=su(2p,2q) and g=sp(p,q).
4. (4)
gh=sp(2n,R) and g=sp(n,C).
5. (5)
gh=e6(−14) and g=f4(−20).
As we will see later, the cases τ˙♯(γj)=γj and τ˙♯(γj)=γj are very different from the point
of view of the underlying geometry.
In the case τ˙♯(γj)=γj we set r=rh, while in the case τ˙♯(γj)=γj we set r=rh/2. In the latter case we order the strongly orthogonal roots so that τ˙♯(γ2j−1)=γ2j, j=1,…,r, see [Ó91], Section 3, for more details and discussion.
Lemma 2.9**.**
Assume that r=rh. Then we can choose Ej and Fj=σ˙h(Ej) such that τ˙(Ej)=Fj,
η˙(Ej)=Ej, and η˙(Fj)=Fj. So with Xj=Ej+Fj and Yj=i(Ej−Fj), then η˙(Xj)=τ˙(Xj)=Xj and
η˙(Yj)=τ˙(Yj)=−Yj. In particular,
[TABLE]
[TABLE]
Moreover since RankG/K=RankGh/Kh=r, a is maximal abelian in p and in ph, while ahq is maximal abelian in qhp and in ph.
Proof.
As τ˙♯(γj)=γj it follows that η˙(ghCγj)=ghCγj so
[TABLE]
Thus we can choose Ej∈gcγj such that −Bc(Ej,θ˙c(Ej))=1, where Bc denotes the Killing form on gc. Then [Ej,−θ˙c(Ej)]=Hj. Notice that −θ˙c(Ej)=τ˙(Ej) as Ej∈ph+. Furthermore,
Ej∈gc and hence Ej=σ˙h(τ˙(Ej)) or τ˙(Ej)=σ˙h(Ej)=Fj.
∎
Similarly in the other case we have
Lemma 2.10**.**
Assume that r=rh. Then we can choose Ej and Fj=σ˙h(Ej)
such that τ˙(E2j−1)=F2j, and τ˙(E2j)=F2j−1 for 1≤j≤r, hence η˙(E2j−1)=E2j
and η˙(F2j−1)=F2j. So with Xl=El+Fl and Yl=i(El−Fl), then
τ˙(X2j−1)=X2j=η˙(X2j−1) while τ˙(Y2j−1)=−Y2j=η˙(Y2j−1). One has ah=a⊕ahq with
[TABLE]
Moreover, a is maximal abelian in p and RankG/K=21RankGh/Kh=r.
To allow for uniform treatment of the cases we introduce the notation
Ej′=Ej, Fj′=Fj, Xj′=Xj etc. in case r=rh, and
Ej′=E2j−1+E2j, Fj′=F2j−1+F2j, Xj′=X2j−1+X2j, etc. in
case r=rh. Then in all cases we have τ˙(Ej′)=Fj′ and
[TABLE]
The order in a∗ is obtained from the lexicographic order with respect to the basis {X1′,…,Xr′}.
Similarly we will need an extension of this notation to include subsets and signs. So for I⊂{1,…,r} and ϵ∈{−1,1}#I, if r=rh set I′=I and ϵ′=ϵ;
otherwise, set I′={2j−1,2j∣j∈I}=(2I−1)∪2I with ϵ2j−1′=ϵ2j′=ϵj. Then we will have E′(I′,ϵ′) equal to either E(I,ϵ) in the first case, and to E(2I−1,ϵ)+E(2I,ϵ) in the second case.
Remark 2.11**.**
We mention that all the
classical irreducible Riemannian symmetric
spaces, with a possible extension by the abelian group R+={t∈R∣t>0}, arise in this way as a real form of a bounded
symmetric domain in Cn, see Tables 3 and 4.
The Riemannian symmetric spaces that do not occur this way are those that correspond to the symmetric pairs: (e6(2),su(6)×su(2)), (e6(6),sp(4)), (ϵ7(7),su(8)),
(ϵ7(−5),so(12)×su(2)), (ϵ8(8),so(16)), (ϵ8(−24),ϵ7×su(2)), (f4(4),sp(3)×su(2)),
and (g2(2),su(2)×su(2)), namely, those with a quaternionic Kähler metric or associated to a split exceptional group.
The extra factor R+ occurs in the cases exactly where
Dh+≃Rk+iΩ is a tube type domain and (up to finite coverings) G≃GL(Ω)o is the
automorphism group of the symmetric cone Ω; moreover, here r=rh. These are not all the tube domains, but those for which gh=gc. The simplest case is when Gh=SU(1,1) and
G={exptX1∣t∈R} (see (1.4) for the notation). In this case K is trivial and
exptX1 acts on D+ by
[TABLE]
according to
(1.9). In the general case the R+ factor is expRH0 where
H0=X1+⋯+Xr. The element H0(=−iZh) is centralised by K. Let a=⨁j=1rRXj, which by Lemma 2.9
is maximal abelian in p, and set
A=expa. Then G=KAK. It follows that the action of the R+ is given by
[TABLE]
The Lie algebra g is simple except for the aforementioned tube type cases and the case gh=so(2,p+q), g=so(1,p)×so(1,q), p,q≥2. If G/K itself is a bounded symmetric domain in Cn, then Gh=G×G
and G/K is embedded diagonally into
Dh+×Dh+ (Dh+ the conjugate structure). This is the only case where Gh is not simple.
Non-uniqueness of the bounded realization occurs if g=so(1,p), then one can take gh=su(1,p) or
gh=so(2,p+1); while for g=sp(2,2) one has the choices gh=su(4,4) or e6(−14).
2.3. Boundary Orbits of D
The G orbit of the basepoint, G⋅xo, is D, an open domain in Pc=Gc/LcN−. On the other hand, the Bruhat cell N+⋅xo is open and dense in Pc. As in the complex case, by means of log one obtains an analytic isomorphism N+⋅xo≃N+≅n+,
[TABLE]
the Harish-Chandra bounded realization of D.
There are two possible ways to consider the closure of D and the corresponding boundary
orbits: we can consider the closure in the generalized flag manifold Pc,
or in the open dense set expn+⋅x0≃n+. As for the complex case,
Lemma 1.7, the two agree.
Lemma 2.12**.**
Denote by c(D) the closure of D in Pc. Then c(D) is also the closure of D in
Gh/KhPh− and the closure of expD+⋅x0 in expn+⋅x0. In
particular, the action of G on D+ extends
to the boundary of D+.
Proof.
This follows from the fact that Pc is compact and hence closed in Gh/KhPh−.
∎
Remark 2.13**.**
The above statement is also a consequence of the fact that Pc=(Gh/KhPh−)η.
The rest then follows from Lemma 1.7 by taking η-fixed points.
Denote by ∂D=c(D)∖D the topological boundary of D.
Proposition 2.14**.**
Let I⊆{1,…,r} and let ϵ∈{−1,1}#I.
(1)
If r=rh, then η˙(E(I,ϵ))=E(I,ϵ) and η˙(Oh(I,ϵ))=Oh(I,ϵ).
2. (2)
If r=rh, then η˙(E(2I−1,ϵ))=E(2I,ϵ) and η˙(Oh(2I−1,ϵ))=Oh(2I,ϵ).
3. (3)
Uniformly in all cases we have η˙(E′(I′,ϵ′))=E′(I′,ϵ′) and
η˙(Oh(I′,ϵ′))=Oh(I′,ϵ′).
Proof.
This follows from Lemma 2.9 and Lemma 2.10 as τ(Gh)=Gh.
∎
Clearly ∂D=(∂Dh+)η˙ and each Oh(I′,ϵ′)η˙ is G-invariant although the orbits are yet to be determined. However from Theorem 1.6 we can conclude
Lemma 2.15**.**
(1)
If r=rh, then ∂D=⋃˙b=1rOh(b)η˙.
2. (2)
If r=rh then, ∂D=⋃˙b=1rOh(2b)η˙.
Indeed more can be said in both cases, but we start with some simple observations about the strongly
orthogonal roots γj and the maximal abelian subspace ac.
In the case r=rh we have
γj∘τ=−γj and th=⨁RHj⊂ac (see (1.3) for
notation). Let αj=γj∣ac. Then {α1,…,αr} is a maximal set
of strongly orthogonal roots in Σcn.
In the case r=rh/2 we have dimth∩qc=21dimth and rh is even. We
let αj=γ2j∣ac=γ2j−1∣ac, j=1,…,r. Then the set
{α1,…,αr} is a maximal set of strongly orthogonal roots
in Σcn.
Let {β1,…,βr}⊂Σcn+ be a maximal set of strongly orthogonal roots. Then given a permutation βi→βσ(i) there is an element k∈K that implements it, in particular Ad(k)(Hβi)=Hβσ(i).
Suppose that r=rh. Now γj∈Σcn+ and Ej∈gcγj. Then E(b)=E1+⋯+Eb with Ej∈gcγj. It follows from Lemma 2.16 that dimRgcγj is independent of j, so denote it by a. Also g2γj=0 as can be seen from Lemma 2.16 and Moore’s Theorem. If a>1 then ZKc(ac) acts transitively on the unit sphere in gcγj ( [W73] Theorem 8.11.3, p. 265). But in this case the unit sphere is connected so ZKc(ac)o acts transitively.
We also know from Lemma 2.5 that the Lie algebra of ZKc(ac)o is contained in k. Hence ZKc(ac)o⊂K.
It follows that Ej and −Ej are conjugate under ZK(ac)⊂G.
Now apply this argument to each of the analytic subgroups of Gc corresponding
to the Lie algebra generated by REj⊕RHj⊕RE−j to see that we can find kj∈Gj
such that Ad(kj)Ej=−Ej. But the groups Gi and Gj commute if i=j, thus with k=∏jbkj(1+ϵj)/2 we have Ad(k)E(b,ϵ)=Eb.
Lemma 2.17**.**
The following are equivalent:
(1)
there exists m∈NK(a) such that Ad(m)∣a=−1;
2. (2)
there exists m∈K such that
Ad(m)Ej′=−Ej′, j=1,…,r;
3. (3)
there exists m∈K such that Ad(m)Fj′=−Fj′, j=1,…,r.
Proof.
As noted following Lemma 2.10, τ˙(Ej′)=Fj′. Since τ∣K=id, it follows that (2) and (3) are equivalent.
Assume that there exists m∈K such that Ad(m)∣a=−1. Then Ad(m)(Ej′+Fj′)=−Ej′−Fj′, j=1,…,r. As K⊂Lc we have
Ad(m)n±=n±. Hence Ad(m)Ej′=−Ej′ and Ad(m)Fj′=−Fj′.
On the other hand, if (2) and (3) hold then, as Xj′=Ej′+Fj′, Ad(m)Xj′=−Xj′.
Since a=⨁jRXj′, the claim follows.
∎
Remark 2.18**.**
It follows from Lemma 2.16 that it is
enough to assume that (2) and (3) above hold for one j.
Corollary 2.19**.**
Assume that r=rh. If −1 is not in the Weyl group W=NK(a)/ZK(a), then E(b,ϵ) is
not conjugate to E(b,ϵ′) if
ϵ=ϵ′.
Theorem 2.20**.**
Assume that r=rh and let 1≤b≤r.
(1)
If −1∈W then (Oh(b))η˙=G⋅E(b)=:O(b) is one G-orbit and
[TABLE]
2. (2)
If −1∈W then (Oh(b))η˙=⋃˙ϵ∈{−1,1}bG⋅E(b,ϵ) and
[TABLE]
Proof.
Let z∈(Oh(b))η˙⊆∂D. Using the familiar argument we
can choose kj∈K and aj∈A such that
kjaj⋅0→z. Again, kj has a convergent subsequence, so we can assume that kj→k∈K.
Replace z by w=k−1z
in the same G-orbit. Write aj=exp∑ν=1rtν,jXν. Then
[TABLE]
As aj⋅0→w it follows that there exists a set I such that tanh(tν,j)→ϵν∈{−1,1} for ν∈I and
tanh(tν,j)→xν∈(−1,1) for ν∈I. Hence
[TABLE]
If tν∈R is so that xν=tanh(tν) then exp(−∑ν∈ItνXν)⋅∑ν∈IxνEν=0 so we can assume that w=∑ν∈IϵνEν. As Eν∈gcγν from Lemma 2.16 we can assume that there exists a b and ϵ∈{−1,1}b such that
w=E(b,ϵ). The claim now follows from Lemma 2.17 and Corollary 2.19.
∎
Theorem 2.21**.**
Assume that r=rh. Let 1≤b≤r. Then O(2b)η˙=G⋅E(2b)=G⋅E′(b) is one G-orbit. In particular,
with O(b)=G⋅E′(b) we have
[TABLE]
Proof.
Let z∈(O(2b))η˙. By replacing X2j−1+X2j with Xj′ we see as above that
we can assume that z=∑ν=1bϵνEν′ for some b.
As before let
αi=γ2i−1∣ac. Then αi∈Σcn+ and
[TABLE]
It follows that dimgcαi≥2. We also have 2αi∈Σc. Thus ZK(ac) acts transitively on spheres in
gcαj which implies that Ei′ and −Ei′, which are both in gcαi are conjugate via ZK(ac). Thus we can take ϵi=1 for all i.
The roots αi and αj are conjugate by
s(αi−αj)/2∈Wcc and Eν′∈gαν. It follows that we can assume that J={1,…,b} for some b≤r.
∎
2.4. Isotropy of E(b,ϵ)
In this section we describe the stabilizer in G of E(b,ϵ), respectively E(2b), on the boundary of
D. On the way we give some extra information about the structure
of each part in the stabilizer. Our notation for subgroups of G will be the same as that used
for Gh except we drop the subscript “h”and L=ZG(H0). Our standard homomorphism will
always been assumed to be of the form κI,ϵ for I={1,…,b}⊂{1,…,r}. We
define I′ as in the earlier subsection and then write
κ instead of κI′,ϵ wherever the exact form does not matter. As before,
we write Eκ, Hκ, Xκ, O(κ) etc. for E(b,ϵ), H(b,ϵ),
X(b,ϵ), O(I′,ϵ). We have η˙(Eκ)=Eκ.
Hence if GhEκ is the stabilizer of Eκ in Gh then
the stabilizer GcEκ of Eκ in Gc is GcEκ=(GhEκ)η
and the stabilizer GEκ in G is (GcEκ)τ. Same argument holds also for
the Lie algebra of the stabilizers.
Basic Example**.**
SU(1,1) - cont.
We return to the prototype example, §1.3, and introduce an anti-holomorphic involution. Consider the map τ˙1:sl(2,C)→sl(2,C) given by the matrix multiplication
[TABLE]
Clearly τ˙1 is complex linear, whereas for X∈su(1,1) one has τ˙1(X)=X, in particular τ˙1(su(1,1))=su(1,1). Recall that the conjugate linear extension of τ˙1 from su(1,1) to sl(2,C)(=su(1,1)C) is denoted η˙1, and so on su(1,1) is also given by complex conjugation, as is η1 on SU(1,1). For the involution τ˙1, the Lie subalgebra of sl(2,C) denoted gc is sl(2,R)≅su(1,1). Thus for the subgroup G⊂SU(1,1)∩SL(2,R) we have
[TABLE]
[TABLE]
Since gh≅gc we know (cf. Table 4) that g has an R-factor and that r=rh (cf. Lemma 2.10). As regards compatibility of the involutions,
[TABLE]
consequently777We remark that the results in this subsection are valid for all standard homomorphisms
satisfying (2.8).
[TABLE]
Moreover, with κ˙I,ϵ′:=κ˙I′,ϵ′ we similarly have κ˙I,ϵ′∘τ˙1=τ˙∘κ˙I,ϵ′.
Earlier we recalled the decomposition obtained from πκ:=ad∘κ˙:
[TABLE]
Lemma 2.22**.**
If π is a finite dimensional representation of sl(2,C) then π and π∘τ˙ are equivalent.
Proof.
This is well known. We assume that π is irreducible, then π is uniquely determined by its dimension. As the
dimension of π and π∘τ˙ are equal and π∘τ˙ is irreducible the result follows.
∎
It follows from this Lemma that the decompositions in (2.9) are preserved under τ˙ and
η˙. In particular, where the superscript refers to η˙-fixed points, respectively intersection:
[TABLE]
Remark 2.23**.**
Recall that we have defined κ such that it defines a homomorphism su(1,1)→gh. But as pointed out in 1.7
one can, by extending κ to sl(2,C) and then restrict to sl(2,R), view κ as a
homomorphism sl(2,R) into gc. Then the first decomposition in (2.9) is the isotypic decomposition of
the representation adgc∘κ of sl(2,R). The second decomposition is then obtained
by taking the τ˙-fixed point in each of the spaces gc[j]. We will discuss that in more details
in the next section. Note that the spaces g[j], j=1,2,
are not necessarily κ(sl(2,R))-invariant.
As τ˙(Xκ)=η˙(Xκ)=Xκ it follows that the eigenspaces of
adXκ are τ˙ and η˙ stable and compatible with the decomposition
gh=g⊕qh
and ghC=gc⊕igc. In short, all the essential structure from the previous sections
is invariant under τ˙ and η˙. In particular,
[TABLE]
Let H0 and Zh be as before. Let Hκ=κ(H1) and
Hκ(1)=H0−21Hκ and note that
Hκ,Hκ(1)∈ac⊂qc∩sc.
Complexifying the decomposition in (2.10) and then taking η˙ and τ˙ fixed points
we get
[TABLE]
and
[TABLE]
For the complexification of the Levi factor of the maximal parabolic subalgebra qhκ and
its intersection with gc we also have with lcκ=zgc(Xκ) and with the obvious
notation:
[TABLE]
and
[TABLE]
semidirect products.
Let Lcκ=ZGc(Xκ)=Lcκ(1)Lcκ(2)Aκ where Lcκ(1) is the
analytic subgroup of Gc with Lie algebra lcκ(1),
Lcκ(2)=ZGc(Xκ,H0), and Aκ=expRXκ.
We use analogous notation for g and G
dropping the index c. Up to connected components for Lκ(1), those
Lie algebras, respectively Lie groups,
are obtained by taking τ˙, respectively τ fixed points. Finally we
let
Pcκ* is a maximal parabolic subgroup of Gc.*
2. (2)
If Hκ(1)=0 then Hκ(1) is central in
lκ(1)∩sc and Lcκ(1)/Lκ is, up to compact factors, a split-Hermitian
symmetric space.
3. (3)
Pκ* is a parabolic subgroup in G.*
4. (4)
If H(1)=0 then Lκ(2)/K∩Lκ(2) is the
fixed point set of the conjugation η in the Hermitian symmetric space
Mhκ(1)/Kh∩Mhκ(1) and we have a fibration
[TABLE]
5. (5)
If Hκ(1)=0 then the stabilizer of Eκ in G is Pκ and
O(κ)=G/Pκ=K/K∩L(2) is a compact symmetric R-space.
3. Finer Structure of Qκ
In this section we discuss the finer structure of the stabilizer of Eκ. This material will not be used
in this article but we still think it is worth including.
Recall from §1.5 that gh[0]=zgh(κ˙(su(1,1)) is a subalgebra which has ideal ghκ=⨁ghj⊆gh[0],j≥1ghj.
Lemma 3.1**.**
Let V⊂gh[k] be an irreducible ghκ-module. Then exactly one of the following holds:
(1)
τ˙(V)=V* and τ˙∣V=id. In this case V⊂g and the action of ghκ is trivial.*
2. (2)
τ˙(V)=V* and τ˙∣V=−id. In this case V⊂qh and the action of ghκ is trivial..*
3. (3)
τ˙(V)=V* and τ˙∣V=±id. Then dimV>1 and dimV∩g=1. If dimV=2, then
V∩g⊂n1 or V∩g⊂n−1. If dimV=3, then V∩g⊂gh[0]∩g.*
4. (4)
τ˙(V)=V. Then dimV>1 and V∩τ˙(V)={0} and τ˙∣V⊕τ˙(V):V⊕τ˙(V)→V⊕τ˙(V) is
given by τ˙(X,Y)=(τ˙(Y),τ˙(X)) and (V⊕τ˙(V))∩g={X+τ˙(X)∣X∈V} is three dimensional.
Proof.
It is clear that exactly one of the conditions (1) to (4) must hold. In the case where dimV=2 or dimV=3 the
action of ad∣ghκ and ad∘τ˙∣ghκ on V are different as e and f act differently
on R2 and sl(2,R). Thus, if τ∣V=±id, we must have that the action of ghκ is trivial
as ad∘τ˙1=τ∘ad. Then (1) and (2) follow.
Assume that
τ(V)=V and τ∣V=±id. Then clearly dimV>1. Assume that dimV=2. Then V=Im(id+τ˙)⊕Im(id−τ)=V(τ˙,1)⊕V(τ˙,−1) and each of the eigenspaces is one dimensional. As τ˙1(X1)=X1,
R2=R2(X1,1)⊕R2(X1,−1). Since
κ˙∘τ˙1=τ˙∘κ˙, it follows that adXκ∣V(τ˙,1)=±1. If dimV=3, then the
action is the standard su(1,1) action on its Lie algebra and
[TABLE]
For (4) we note that V∩τ˙(V) is invariant. As V is assumed irreducible, we either have V=τ˙(V) or
V∩τ˙(V)={0}. The rest is now obvious.
∎
Lemma 3.2**.**
Assume that (3) above holds and dimV=2. Then θ(V)∩V={0}. Furthermore,
(1)
θ(V)* is ghκ-stable.*
2. (2)
θ(V(τ˙,±1))=θ(V)(τ˙,±1).
3. (3)
θ(V(adXκ,±1))=θ(V)(adXκ,∓1).
4. (4)
If 0=X∈V(adXκ,±1) then θ(X)∈V(adXκ,∓1)
and [X,θX]∈m0∩p.
Proof.
Fix v∈V(τ˙,1). If [Xκ,v]=v then [Xκ,θ˙(v)]=−θ˙(v),
hence v and θ˙(v) are linearly independent. As dimV(τ˙,1)=1 it follows that θ˙(v)∈V. Similarly, if
[Xκ,v]=−v then [Xκ,θ˙(v)]=v and v∈V. It follows that V∩θ(V)={0}.
∎
The conclusion from this is
Corollary 3.3**.**
If nhκ1={0}, then nκ1={0} and dimnκ1=21dimnhκ1. Furthermore, τ˙∣nhκ1
defines a conjugation on nhκ1 so nκ1 is a totally real subspace.
Remark 3.4**.**
This follows also from the following observation. Lemma 1.9 states that Io=−ad(Yκ)∘θ˙h defines
a complex structure on nhκ1. τ˙ commutes with θ˙h and anti-commutes with ad(Yκ). Hence
Ioτ˙=−τ˙Io which shows that τ˙∣nhκ1 is conjugate linear. Hence nκ1=nhκ1∩g is a real form
for nhκ1 and nhκ1=nκ1⊕Ionκ1.
Lemma 3.5**.**
Let V⊂gh be one dimensional or a simple ideal. Then either τ˙(V)=V, or τ˙(V)∩V={0}
and we have the “group case” where V×τ˙(V) is an ideal, V and τ˙(V) commute, and
(V×τ˙(V))τ˙={(X,τ˙(X))∣X∈V}.
Proof.
If V∩τ˙(V)={0} then V∩τ˙(V) is an ideal in V. As V is either one dimensional or
simple it follows that V=τ˙(V). The rest is obvious.
∎
Lemma 3.6**.**
τ˙(l2)=l2* and l2∩g is an ideal in mκ0. Let L2 be the analytic subgroup of Gh with Lie algebra l2.
Then L2/G∩L2 is a compact symmetric space.*
Proof.
l2 is the maximal compact ideal of mhκ0. As l2+τ˙(l2) is a compact ideal it follows that
τ˙(l2)=l2.
The rest of the Lemma is now obvious.
∎
Lemma 3.7**.**
Assume that gh(1)={0}. We have τ˙(gh(1))=gh(1) and τ˙(Zκ1)=−Zκ1. Let
Gh1 be the analytic subgroup of Gk with Lie algebra gh(1). Then Gh1 is θh and τ invariant. If
Kh1=(Gh1)θh=Kh∩Gh1 then Kh1 is maximal compact in Gh1, Gh1 is a bounded
domain, τ defines a conjugation on Gh1/Kh1 and (G∩Gh1)/(G∩Kh1)=(Gh1/Kh1)τ is
a real form of Gh1/Kh1.
Lemma 3.8**.**
g(1):=g∩gh(1)* is an ideal in mκ0.*
Proof.
We have [mκ0,g(1)]⊂gh(1)∩g=g(1).
∎
The next result follows easily from the above.
Lemma 3.9**.**
τ˙(mhκ(2))=mhκ(2)* and mκ(2)=mhκ(2)∩g is an ideal in mκ0.*
As τ˙(Xκ)=Xκ we have Fκ⊂Ghτ. Let F~κ:=Fκ∩G.
4. Lift from K to (Lc′)0
One of the results in the paper (§6) will be an extension of sections of homogeneous vector bundles over G/K to its closure, and hence the boundary orbits. A key step in the proof will be a lift of irreducible representations of K to Lc. In this section we will do the lift from k to lc′, i.e. from K to (Lc′)o. Subsequently we will treat the full Lc. A glance at Table 5 shows the real forms G divided into three types. In subsequent subsections the proof of the lift will be done for each type.
4.1. The case OCCC
We shall use the terminology of σ-normal system of roots for which a convenient reference is [Wa-I] p. 21-24. For this subsection only we shall denote by G a non-compact connected semisimple Lie group with Lie algebra g, later the results will be applied to (Lc′)o in Table 5. The Killing form on g induces a non-degenerate symmetric bilinear form on g∗ for which we use ⟨⋅,⋅⟩. Let θ be a Cartan involution and write g=k⊕s for the Cartan decomposition of g. Let a be a maximal abelian subspace in s and, as usual, let m=zk(a),
and extend a to a Cartan subalgebra
t=t+⊕a of g. Denote by Δ=Δ(gC,tC) the set of roots of tC in gC. Clearly Δ is a reduced system of roots. Our assumption in this subsection is that all Cartan subalgebras in g are conjugate, to be denoted OCCC.
Lemma 4.1**.**
t+* is a Cartan subalgebra of k and m.*
Proof.
For a Cartan subalgebra c of g let
[TABLE]
and
[TABLE]
Then c=cI⊕cR and the dimensions dimcI and dimcR are constant on each conjugacy class. In particular, for c=t,
tR=a and tI=t+.
If t+ is not a Cartan subalgebra of k, then t+ extends to a Cartan subalgebra t~+ of k which in turn extends to a Cartan subalgebra c~ of g such that t+ is a proper subspace of
t~+, or t+⊊c~I which is not possible by the above discussion.
∎
It follows that t is a fundamental Cartan subalgebra as well as a maximally split Cartan subalgebra. As t=t+⊕a we can restrict roots from Δ to either t+ or a. Denote by Σ=Σ(g,a) the set of (restricted) roots of a in g, i.e. Σ={β∣a∣β∈Δ}∖{0}. For α∈Σ and Δ(α):={β∈Δ∣β∣a=α} we let gα⊂g be the restricted root space, and set
[TABLE]
That g has one conjugacy class of Cartan subalgebra is equivalent to all multiplicities mα, α∈Σ are even.
Next we define the involution that will serve as the σ of the σ-normal system. Let tR=it+⊕a. For λ∈tR∗ let
[TABLE]
and
[TABLE]
We identify λ+ with λ∣t+ and similarly write λ− for λ∣a.
If α∈Δ then αθ,α♯ are in Δ because gαθC=θ(gαC) and gα♯C=θ(g−αC). Also θ and ♯ are isometries for ⟨⋅,⋅⟩.
It is also clear that
[TABLE]
Lemma 4.2**.**
Assume OCCC. Then β♯=β for all β∈Δ. In fact, βθ−β∈/Δ.
Proof.
Let β∈Δ. Suppose that β♯=β. Then β+=0, hence β∈Σ. But then
[TABLE]
Hence mβ is odd which contradicts OCCC.
If βθ=β then βθ−β=0, so is not a root. Assume that βθ=β and that γ=βθ−β∈Δ. Then γθ=−γ so that γ+=0, i.e. γ is a real root. But t is fundamental so there are no real roots.
∎
From this, various properties of the roots will follow. The OCCC condition will impose some additional constraints which we will identify in the next few results.
Lemma 4.4**.**
Let α∈Δ. Then α+∈Δ(kC,t+C).
Proof.
Let Xα=Xα++Xα−∈gαC. Here Xα±=21(Xα±θ(Xα)).
If H∈a then
[TABLE]
It follows that
[TABLE]
Thus Xα±=0. But the same argument shows that for H∈t+ we have [H,Xα±]=α(H)Xα± and therefore
kα+C={0}.
∎
Lemma 4.5**.**
Assume OCCC. Let α∈Δ∖Δ∙. Then α and αθ are strongly orthogonal.
Proof.
We have α−αθ=2α−. By the above 2α−∈Δ. Similarly we have α+αθ=2α+. We just saw that α+∈Δ(kC,t+C). As t+C is a Cartan subalgebra of kC it follows that 2α+∈Δ+.
∎
Corollary 4.6**.**
Assume OCCC. If α∈Δ∖Δ∙={α∈Δ∣α=αθ}. Then ∥α+∥=∥α−∥.
Proof.
This follows from the last lemma which implies that α and αθ are orthogonal or ⟨α,αθ⟩=∥α+∥2−∥α−∥2=0.
∎
Lemma 4.7**.**
Let Δ♯={α+∣αθ=α} (not counted with multiplicities). Then Δ(kC,t+C)=Δ♯⋃˙Δ∙.
Proof.
It is clear that the union is disjoint. Let Σ+ be a set of positive roots in Σ and, as usual, n=⨁γ∈Σ+gγ. Then
[TABLE]
As Σ={α∣a∣αθ=α} the claim follows now using the same argument as in the proof of Lemma 4.4.
∎
The following set of simple roots is adapted from [Wa-I] p. 21-24 with slightly different notation. Let ℓ+:=dimt+, ℓ2:=dima and ℓ=ℓ++ℓ2=dimtR. We choose a lexicographical ordering in tR∗ with respect to a basis H1,…Hℓ so that H1,…,Hℓ+ is a basis for
it+. Let Δ+ be the corresponding set of positive roots and Π the set of simple roots. Then by Lemma 4.2 and [Wa-I] there exists ℓ1 such that the following holds:
(1)
Π∙={α1,…,αℓ1} is a set of simple roots for Δ∙ (contained in Δ∙+=Δ∙∩Δ+). Furthermore Π∙={α∈Π∣αθ=α}.
2. (2)
ℓ=ℓ1+2ℓ2.
3. (3)
If 1≤ν≤ℓ2 then αℓ1+νθ=αℓ1+ℓ2+ν and
αℓ1+ℓ2+νθ=αℓ1+ν.
Lemma 4.8**.**
Πc={α1,…,αℓ1,αℓ1+1+,…,αℓ1+ℓ2+}* is a
simple system in Δ+(kC,t+C).*
Let Ψ={μ1,…,μℓ} denote the set of fundamental weights for Π.
Lemma 4.9**.**
Let Ψc:={μj+∣j=1,…,ℓ1+ℓ2} (where we identify
μj with μj+ for j=1,…,ℓ1). Then
Ψc is the set of fundamental weights corresponding to the simple system Πc.
Proof.
We have to show that
[TABLE]
This is clear for ν=1,…,ℓ1 as in this case μν=μν+. Assume now that ℓ1+1≤ν≤ℓ1+ℓ2.
Then for 1≤σ≤ℓ1 we have
[TABLE]
Assume ℓ1+1≤σ≤ℓ1+ℓ2 and write σ=ℓ1+j, 1≤j≤ℓ2. then
[TABLE]
because θ is an involution. As ασθ=αℓ1+ℓ2+j and
∥ασ+∥2=21∥ασ∥2=21∥ασθ∥2 we get
[TABLE]
∎
Denote by Λ+(K) the set of highest weights of irreducible representations of K and similarly by Λ+(G) the space of highest weights of
irreducible finite-dimensional representations of G. If μ∈Λ+(K) then we denote the corresponding irreducible representation of
K by σμ. If μ∈Λ+(G) then the corresponding irreducible representation of G with highest weight μ is denoted by
τμ.
Let G be the universal covering of G and let K denote the
analytic subgroup of G corresponding to the Lie algebra k. Then
K is simply connected and locally isomorphic to K. Furthermore,
the center of G, Z(G), is contained in K.
Theorem 4.10**.**
Let μ=∑j=1ℓ1+ℓ2kjμj+∈Λ+(K). Set μ:=∑j=1ℓ1+ℓ2kjμj. Then μ∈Λ+(G),
descends to be in Λ+(G). Moreover σμ is contained in
τμ∣K with multiplicity one.
Proof.
It is clear that μ∈Λ+(G) and μ∈Λ+(K).
Denote by
τμ respectively
σμ the corresponding representation of G, respectively
K. Clearly σμ is contained in
τμ∣K. Let Z be the kernel of the canonical projection
G→G. Then Z⊂K and K≃K/Z.
Since μ∈Λ+(K) it follows that σμ∣Z=id.
As Z is central in G and τμ is
irreducible one has τμ∣Z is a scalar.
But σμ is contained in τμ∣K, it follows that
τμ∣Z=id. Hence τμ defines
a representation of G and μ∈Λ+(G). The multiplicity one assertion is
clear because there is no way to write μ as a non-trivial linear combination (μ,0)−∑nα(α+,α−)∣t+ of positive roots (α+,α−) and nα≥0
(and at least one =0). The rest is now obvious.
∎
4.2. The special cases
We turn to the third type in Table 5. The technique is a variation of σ-systems from
the previous subsection. Here we use some results from [Kn96] on Vogan diagrams. The
procedure parallels that followed in the OCCC case. One begins with t=t+⊕a a
fundamental Cartan subalgebra of g but here not a maximal split Cartan. Hence again there
are no real roots. Of course t determines a parabolic subalgebra which will play no direct role.
We have Δ=Δ(gC,tC) the set of roots of tC in gC, W the Weyl group
of Δ; let Δ(kC,t+C) be the set of roots of t+C in kC and WK its
Weyl group. We choose a lexicographical ordering in tR∗ with respect to a basis H1,…Hℓ
so that H1,…,Hℓ+ is a basis for
it+. Let Δ+ be the corresponding set of positive roots and Π={α1,⋯,αl}
the set of simple roots. Denote by Ψ={μ1,…,μℓ} the set of fundamental weights for Π.
As before, for λ∈tR∗ let
λθ:=λ∘θ. Then we have the restriction to
t+,λ+:=21(λ+λθ), and the
restriction to a, λ−:=21(λ−λθ). A different
but important feature arises here in that imaginary roots can be compact or noncompact.
Thus we must examine Σ+, the restrictions of Δ+ to t+. Also we
make a choice of simple roots for Δ(kC,t+C) compatible with Δ+.
To us it seemed easiest to continue with the remaining details in each case separately.
Example 4.11**.**
We start with g=so(5,5) and k=sp(2)×sp(2)=so(5)×so(5). Using standard notation and as presented in [Kn96] p. 359 we have Π={α1=e1−e2,α2=e2−e4,α3=e4−e5,α4=e5−e3,α5=e5+e3}, where t+=<e1,e2,e4,e5> and a=<e3>. Clearly θ:Π→Π interchanges α4 and α5, so relative to the involution θ we have a normal σ-system with a σ order for which Π is a σ-fundamental system. A computation using the Cartan on [Kn96, p. 359]
determines the set of restrictions, Σ+, of Δ+ to t+ which, from [Wa-I], is a (non-reduced) root system and contains the positive roots of the Levi subalgebra so(4,4)⊕a. We let WΣ+ be its
Weyl group. Now set Wθ={w∈W∣w∘θ=θ∘w}. Then Wθ induces
a map on t+ and, from [Wa-I] p. 24, as there are no real roots we have Wθ∣t+=WΣ+. Finally with regard to Weyl groups (following [H10] p. 1016 and others) we will take a distinguished set
of representatives for Wθ/WK, viz. let Dt++ be the positive Weyl chamber for Δ+(kC,t+C) and let Dg+ be the projection of the positive chamber for Δ+ in t(=t+⊕a) to t+.
Then set W1={w∈Wθ∣w∣t+(Dg+)⊂Dt++}. Then W1 gives the required coset representatives.
Yet another computation is necessary to obtain Δ+(kC,t+C)={e1,α1+,α2++α3++α4+,α1++2α2++2α3++2α4+}∪{e4,α3+,α4+,α3++2α4+}, with compatible basis of simple roots {α1+=e1−e2,α2++α3++α4+=e2}∪{α3+=e4−e5,α4+=e5}. As for the fundamental weights for g, one gets
[TABLE]
[TABLE]
while for k,
[TABLE]
In terms of the ei, for a highest weight μ we have μ=∑15miμi=(m1+m2+m3+2m4+m5)e1+(m2+m3+2m4+m5)e2+(2m4−m5)e3+(m3+2m4+m5)e4+(2m4+m5)e5. So, similar to the procedure in the Theorem above, to obtain μ as a natural lift from t+ we take m4=m5 giving μ+=μ=M1e1+M2e2+M4e4+M5e5 with M1≥M2≥M4≥M5≥0.
Now take a candidate highest weight μ=∑14niμi+ of k to lift to μ. In terms of the ei we have μ=(n1+n2)e1+n2e2+(n3+n4)e4+n4e5=N1e1+N2e2+N4e4+N5e5 and N1≥N2≥0,N4≥N5≥0. Clearly when N2=m2+N4, i.e. N2≥N4, we have a μ to lift to μ . However this determines a chamber in t+ for the action of W1. Now
g=so(5,5) is type D5 so the Weyl group contains all permutations of the ei. We summarize in Table 1
Case 4.11 various possibilities for the chamber and an element of W1 that maps the chamber to the original one. We use the abbreviation i⟷Ni and ei−ej⟷sei−ej∈W1.
So given μ∈Λ+(K) one finds it in the first column, applies w−1 to it obtaining a highest weight of the form 1≥2≥4≥5 which can be lifted to a natural μ∈Λ+(G). It is clear the w−1 belongs to W1 as it takes a chamber of dominant K-weights to another. The result then follows from the multiplicity one Theorem in [H10] which says the K-type w(μ)∣t+ occurs with multiplicity 1 in Vμ.
An alternative approach to the existence of the K-submodule is to use the generalization of the PRV conjecture ([MPR11]), but this does not yet give multiplicity 1.
Example 4.12**.**
Next we consider g=e6(6) and k=sp(4). We shall use the notation of [Bo68] so that we have a basis
{α1,α2,α3,α4,α5,α6} for Δ+(gC,tC). We use Table C p. 532 in [Kn96] for a compatible basis of the simple roots for k=sp(4)⊂e6(6). In particular,
node 2 is black, and under θ, nodes 3↭5 and 1↭6.
This suggests the following basis for Δ+(kC,t+C): {γ1=α2+α4+2α3+α5,γ2=2α1+α6,γ3=2α3+α5,γ4=α4}. Note that we use γ because these are not always the projections to t+, e.g. γ2=α2+.
From [Bo68] one computes that ⟨αi,αi⟩=2, and since the fundamental weights satisfy 2⟨αj,αj⟩⟨μi,αj⟩=δi,j we have that the fundamental weights μi are the dual basis to the simple roots αi. Similarly one obtains that 1=⟨γ1,γ1⟩=⟨γ2,γ2⟩=⟨γ3,γ3⟩ while ⟨γ4,γ4⟩=2. Then for the fundamental weights of Δ+(kC,t+C) we can take ω1=2μ2,ω2=2μ1+μ6,ω3=2μ3+μ5−μ2,ω4=μ4−μ2.
Let μ∈Λ+(K). Then μ=∑14niωi with ni≥0 and integers. In terms of the μi we have
[TABLE]
Here we must make the assumption that n1−n3 is an even integer. Then, as before, we are left with a few cases which will be handled using the Weyl group, i.e. W1. We begin with the case n1−n3−2n4≥0. Here we lift μ to μ=n2μ1+n3μ3+n4μ4+(2n1−n3−n4)μ2. Then μ+=μ so we have a valid lift. In Table 2 Case 4.12, similar to that above, the first column contains the various cases for μ, the second the sequence of roots whose reflections give w, and the third the lift to Λ+(G) to which you apply w−1 and the restriction gives μ.
So here, given μ=∑14niωi∈Λ+(K) (n1−n3 an even integer) one finds it in the first column, applies w−1 to it obtaining a highest weight μ∈Λ+(G). It is clear the w−1 belongs to W1 as it takes a chamber of dominant K-weights to another. The result then follows from the multiplicity one Theorem in [H10] which says the K-type w−1(μ)∣t+ occurs with multiplicity 1 in Vμ.
Life would be easier if one knew more about the action of W1 on the chambers Dg+; unfortunately, we were unable to obtain the result we needed which necessitated the lengthy computations. These computations were facilitated by having the expressions of the simple roots of e6(6) expressed in terms of the fundamental weights.
Example 4.13**.**
The next case g=so(1,n−1) and k=so(n−1) is elementary and surely in several places in the literature. Assume that n−1≥3 to avoid the Abelian case. Base extend the Lie algebras to C. The fundamental representations of k are either exterior powers of the standard representation or spin. All these are known to occur with multiplicity one in the similar representation of g. Then define a length function on highest weights in the usual way: l(μ)=l(∑1lniωi)=∑1lni. Induction and using Cartan composition provides a natural lift.
One can be more precise using standard material on highest weights and branching, e.g. as in [GW98] p. 351. Say relative to a suitable Cartan subalgebra the highest weight of gC is given by a decreasing sequence Λi while the highest weight of kC is given by a similar sequence μi. Then depending on the parity of n−1, i.e. n−1=2k or n−1=2k−1, either one takes Λi=μi,i<kandΛk=∣μk∣, or Λi=μi,i<kandΛk=0.
For g=so(1,2)≅sl(2,R) and k=so(2) the procedure is the same as in the previous examples, viz., each irreducible unitary representation of so(2) occurs as the highest (lowest) weight of an irreducible finite dimensional representation of sl(2,R).
Example 4.14**.**
The remaining special case is g=so(p,q)k=so(p)×so(q) and p,q=2. If at least one of the factors in k has even parity then we have an equal rank situation. Then we have in W1 all reflections generated by noncompact roots, in particular we have transpositions between ei,1≤i≤p, and ej,p+1≤j≤q. By means of these we can arrange the highest weights of the factors to be in decreasing order for g and thus obtain a lift for any highest weight of k. If both p,q are odd then we are not equal rank but g is still of type Dl whose Weyl group contains enough reflections to accomplish the same goal.
4.3. The Isometries
We turn to the remaining type in Table 5. Here k is the Lie algebra of the isometries of a standard representation on a finite dimensional vector space while g is the Lie algebra of all automorphisms of the vector space. For the cases at hand we will have no need of the spin representation of k. It is classical that all other such representations are obtainable from exterior powers of the standard representation together with Cartan composition, all of which have natural lifts to g.
5. Extension from (Lc′)0 to Lc
In the previous section we considered the extension of representations from K to (Lc′)0. In this
section first we discuss the extension from the connected group
(Lc′)0 to Lc′. For that we need more information about
Lc/(Lc)0. Let Pmin0=Mmin0Ac0N0 be a minimal parabolic subgroup in (Lc′)0, where
ac=ac0⊕RH0, so that ac0 is maximal abelian in lc′∩sc. Then Pmin=MminAcNmin is a minimal parabolic subgroup in Gc where
Nmin=N0N−, A=expRH0, Ac=Ac0A and Mmin=ZKc(ac). Note that
Mmin has the same Lie algebra as Mmin0 and hence (Mmin)0=(Mmin0)0.
We now use well known results about the connected components of Mmin to describe the
connected components of Lc. As ac=ac0⊕RH0 where
ac0 is maximal abelian in lc′, the roots Σcc can be identified with
Σ(lc′,ac0) via restriction.
Let F1:=exp(iac)∩Kc. We note that
if f∈F1 then
[TABLE]
Thus f2=e and F1≃Z2s for some s. We remark that were F1 cyclic then the desired extension can be found in [Kn86] Lemma 14.22.
Choose generators f1,…,fu∈F1 so that with F=∏{e,fj} we have
Mmin=F(Mmin)0≃F×(Mmin)0, see [He78, Ch. VII] for details, in particular Theorem 8.5. But we
will not need the exact form of F1. The following lemma now follows:
Lemma 5.2**.**
Let F be as above. Then Lc′=F(Lc′)0.
Lemma 5.3**.**
Let μ~∈Λ+((Lc′)0) and denote by (τμ~,Vμ~) the
corresponding irreducible representation. Let f∈F. Then the representations τμ~ and
f⋅τμ~:m↦τμ~(fmf) are equivalent.
Proof.
Clearly f⋅τμ~ is an irreducible representation of (Lc′)0. Let t=t+⊕ac0
be a Cartan subalgebra of lc′. Then for H∈t we have
[TABLE]
Thus f⋅τμ~ and τμ~ have exactly the same weights. In particular the highest
weights are the same. Hence f⋅τμ~≃τμ~.
∎
It follows that for each f∈F there exists Tf∈GL(Vμ~) such that for all m∈(Lc′)0,
Tfτμ~(fmf)=τμ~(m)Tf. If f=e we take Tf=id.
Note that Tf is unique up to a scalar λ∈T. Let Vμ~(μ~) be the highest-weight space. Then
dimVμ~(μ~)=1. Hence there exists 0=vμ~ such that Vμ~(μ~)=Cvμ~.
Lemma 5.4**.**
For f∈F let Tf be as above. Then we can choice Tf such that
(1)
Tf2=id,
2. (2)
Tf(vμ~)=vμ~.
Tf* is uniquely determined by (1) and (2).*
Proof.
We have for m∈(Lc′)0 by repeating the definition twice that
[TABLE]
As τμ~ is irreducible there exists cf∈T such that Tf2=cfid. (1) now follows
by replacing Tf by cf−1/2Tf. As dimVμ~(μ~)=1 and Tf leaves the weight spaces
invariant, it follows that Tf∣Vμ~(μ~) is scalar, say multiplication by df=0. By (1) it follows
that df2=1. Hence we can replace Tf by df−1Tf to obtain (2) and (1). If
Tf and Sf satisfy (1) and (2) then Sf−1=Sf and SfTf=cid for some c∈C. But by
(2) it follows that SfTf(vμ~)=vμ~=cvμ~. Hence c=1.
∎
From now on we always assume that Tf, ∈F, satisfies (1) and (2).
Lemma 5.5**.**
Let f,g∈F. Then TfTg=TgTf.
Proof.
As fg=gf it follows that fgfg=f2g2=e. As above this implies that
S=TfTgTfTg=(TfTg)2 is an τμ~-intertwining operators. Hence there
exists d∈C∗ such that S=did. But Tf∣Vμ~(μ~)=Tg∣Vμ~(μ~)=1. Hence
[TABLE]
Thus d=1 and S=id. As Tf2=Tg2=id it follows, by multiplying S first by
Tf and then by Tg that TfTg=TgTf. Hence, the claim.
∎
Corollary 5.6**.**
Let f1,…,fu be generators for F and let f=f1i1⋯fuiu, ij∈{0,1}.
Then Tf=Tf1i1⋯Tfuiu.
Proof.
The operator Sf=Tf1i1⋯Tfuiu satisfies Sτμ~(fmf)=τμ~(m)S as well as (1) and (2) in Lemma 5.4. Hence Sf=Tf.
∎
Theorem 5.7**.**
Let F be as above, let μ~∈Λ+((Lc′)0) and let Tf, f∈F, as
in Lemma 5.4. Define
[TABLE]
Then τμ~ is an irreducible representation of Lc′.
Proof.
We need only show that τμ~:Lc′→GL(Vμ~)
is a homomorphism, τμ~(fmgn)=τμ~(fm)τμ~(gn), f,g∈F and m,n∈(Lc′)0.
But we have
[TABLE]
∎
The final step, the extension to all of Lc is now easy. We use that Lc≃Lc′×A. Hence we can
take any character χ on A and define
[TABLE]
Remark 5.8**.**
If one needs to extend τμ~ to the complexification LcC of Lc, a common compatibility issue arises. LcC is not the direct
product Lc′C×AC, one needs to be more careful with the choice of χ. Then the requirement is
that each Tf has to be scalar c(f) and c(f)=eiχ˙(H) where f=expiH. For that one needs to use the
exact form of F to determine possible choices of χ.
On the other hand, since lc⊗C≅kh⊗C and we work with finite dimensional representations, a lift from k to lc gives a lift from k to kh.
6. Extension of sections of homogeneous vector bundles
We return to the notation of §2. We consider the generalized flag manifold Pc=Gc/LcN− and a basepoint xo=eLcN−. The G orbit of the basepoint, G⋅xo, is D≅G/K, an open domain in Gc/LcN−.
For a unitary representation (σ,V) of K on the complex vector space V we let
V denote the associated homogeneous vector bundle over D. Without loss of generality, we can assume that σ is
irreducible, in which case we shall denote by μ a
highest weight and, as before, by Vμ its representation space. In [Br07] and [Ka05] homogeneous vector bundles over certain complex homogeneous spaces were shown to have an extension to natural compactifications, e.g. the wonderful compactification. In [MSIII] again in the complex setting in somewhat greater generality homogeneous holomorphic vector bundles over Hermitian (locally) symmetric manifolds were extended to the Borel compactification and a detailed analysis of their restriction to the boundary orbits was obtained.
We shall give a version of this for the real domain D⊂Gc/LcN−. Here, we just give the extension of V to V over the
compactification D⊂Gc/LcN−, subsequently we shall analyze the restriction to the boundary orbits.
In the previous section for such (σμ,Vμ) we produced a natural lift (τμ,Vμ) from K to Lc (with some minor exceptions). Then extending the representation trivially on N− we have an irreducible finite dimensional representation of LcN−. Denote the associated homogeneous bundle over Pc=Gc/LcN− by Vμ. Since τμ contains σμ with multiplicity one we have that V is a subbundle of Vμ. In particular, Vμ is defined over ∂D and gives an extension of V to the boundary of D≅G/K.
7. Analytic extension of K-finite matrix coefficients of G to Gc
The first task is to construct a G-invariant domain in Gc that will serve as the domain of
‘para’-analytic (or split-holomorphic) extension of K-finite matrix coefficients of G.
In [Ma03] he provides a general setup for cycle spaces. We shall show that this also gives the target domain in Gc.
To prepare for this we recall some previous notation related to various involutions that have played
a role here; to simplify the notation we will omit the dot on involutions on the Lie algebra as it will
always be clear whether we are discussing the Lie algebra or the group. Then we
recall some facts about the crown of a semisimple Lie group, in particular for Gh whose crown will
be denoted by Ξh, see
[AG90, KS04, KS05] and especially [KS05, Sec. 7]. Once we recall the construction from
[Ma03] of a real analytic cycle domain ΞM for G/K inside
Gc/Lc, we then show that ΞM=(Ξhη)o is a totally real submanifold
of Ξh. We also discuss the connection between the crowns
of Gh/Kh and G/K, in particular in the case r=rh/2 we show
that Ξ=(Ξhτ)o. We
conclude the section by proving analytic extension of orbit maps of representations
to the real analytic cycle space thereby justifying the name real analytic crown.
The involution basic to this paper is τ:gh→gh, giving the real form
g=ghτ and G=(Ghτ)o. The eigenspace decomposition w.r.t. τ
is gh=g⊕qh. Recall that the complex linear extension of τ (or θh) is still denoted τ (or θh), while the conjugate linear extension of τ to ghC is η=σh∘τ=τ∘σh. Then
gc=(ghC)η while Gc=Ghη. gc is a semisimple Lie algebra stable under τ and θh.
The resulting eigenspace decompositions are gc=g⊕qc=lc⊕gc−θh, where
lc=k⊕iqhk and gc−θh=p⊕iqhp (see also the
discussion after Lemma 2.1).
We
have G=(Gh∩Gc)o.
For the restrictions to gc, resp. Gc, we will still use the notation τ but introduce τa=θh∣gc.
Notice that τ and τa commute (because τ and θh do).
The involution
θc=τ∘θh∣gc defines a Cartan involution on gc with corresponding Cartan decomposition
gc=kc⊕sc. We have kc=k⊕iqhp and sc=p⊕iqhk showing
that θc agrees with the conjugate linear
extension of θh restricted to gc. Then it is consistent to denote this on gh by τa=τ∘θh. It should always be clear which involution is being discussed.
We have τa=θc∘τ so our notation
agrees with the standard notation for the involution on gc associated with τ.
As is standard in this R-form setup lcC=khC, lc=zgc(H0) and τa=Ad(exp(πiH0)).
Let, as before, ah be a maximal abelian subgroup of ph. Let Σh=Σ(gh,ah),
let Σh+ be a positive system, and take the basepoint to be xo=eKh∈Gh/Kh. Define
[TABLE]
The Gh-invariant set Ξh was dubbed by Gindikhin the crown of Gh/Kh. Motivated by the results in [KS04] we call
Ξh the crown of Gh.
The set Ξh is an open Gh-invariant complex submanifold of Gh.
Similarly, Ξh is a Gh-invariant complex domain in Gh/Kh. Ξh and
Ξh are independent of the choice of ah as any two such are Kh conjugate.
Write Ω, Ξ and
Ξ for the corresponding sets obtained by this construction for G and G/K.
We denote by ∂Ξh, resp. ∂Ωh, the topological boundary of
Ξh, resp. Ωh. Set
[TABLE]
Then Ξh+ is an open Gh-invariant subset of Ξh such that Ξh=(NKh(ah)Ξh+)o=(WhΞh+)o.
For restricted roots we keep the
notation from Lemma 2.10 and (2.6). Thus
β1,…,βr∈Σ(gh,ah) are strongly orthogonal roots
(up to sign they are the Cayley transform of the strongly orthogonal roots αj, per
the discussion after Theorem 2.21).
We denote by Xj, j=1,…,r, the dual basis and as usual we have ah=⨁RXj.
We also define Yj∈qh∩ph as in Lemma 2.9 and let then ahq=⨁RYj.
If r=rh
then ah=a is maximal abelian in ph and p, and ahq is maximal abelian
in ph and qh∩ph.
If r=rh/2 we choose the ordering so that β2j=β2j−1∘τ=τtβ2j−1 and assume, as we may, that τX2j−1=X2j. Let Xj′=X2j−1+X2j,
Xj−=X2j−1−X2j, a=⨁j=1rRXj′
and ahq=⨁j=1rRXj−. Then a is maximal abelian in p and ahq
is maximal abelian in ph∩qh.
We let
[TABLE]
and note, that according to Moore’s theorem γk∈Σ+(gh,ah).
Note that previously the notation γj was used
for strongly orthogonal roots in Δ. We
note that γ2j−1∣a=β2j−1∣a=β2j∣a=0 and
γ2j∣ahq=β2j−1∣ahq=−β2j∣ahq=0.
Let’s recall that θh is inner, in particular θh=Ad(expπZh). Then
[TABLE]
Thus g and ghτa, resp. qh and gh−τa, are conjugate. Statements
that are formulated for τ and its eigenspaces are therefore also valid for τa and
its eigenspaces.
Let the notation be as above. Then the following holds true:
(a)
Ωh={∑j=1rtjXj∣(∀j∈{1,…,rh})∣tj∣<π/2}.
(b)
We have g1expiX1⋅xo=g2expiX2⋅xo for some g1,g2∈Gh and Y1,Y2∈Ωh, if and only if there exists
k∈ZKh(Y1) and w∈NKk(ah) such
that g1=g2wk and Y1=Ad(w−1)Y2.
(c)
If g1expiX1⋅xo=g2expiX2⋅xo∈Ξh+ then X1=X2 and there exists m∈ZKh(ah) such that
g1=g2m.
(d)
If xn=gnexpYn⋅xo∈Ξh is a sequence
such that xn→∂Ξh⊂Gh/Kh then Yn→Y∈∂Ωh.
(e)
Ξh⊂NhAhKh.
Proof.
(a) is the comment after [KS05, Lem. 7.4] and follows
easily from Moore’s Theorem; (b) is [KS05, Prop. 3.1]; (c) is [KS05, Cor. 4.2];
(d) is [KS05, Lem. 2.3] and (e) is (1.1).
∎
Next we recall the construction of the cycle space or Matsuki crown of
G/K in Gc/Lc, per [Ma03].
Remark 7.2**.**
To assist the reader we give the correspondence between the notation in [Ma03] with our setup. Here the left hand side lists Matsuki’s notation and the right hand side the corresponding
object in this article: g↔gc, h↔g, h′↔lc, k↔kc, m↔sc, q↔qc. Similarly for the groups. In particular
G↔Gc and
H↔G. As Lc=ZGc(H0) might be disconnected, so H′↔Lc0.
Let t be a maximal abelian subspace of
kc∩qc=iqhp. Denote by Σ(gcC,tC)
the roots of tC in ghC≅gcC. As θc∣tC=idtC, given a root space
gcCα=gcC(t,α) we have θc(gcC(t,α))=gcC(t,α) and one
decomposes it according to the
eigenvalues of θc getting gcC(t,α)=kcC(t,α)⊕scC(t,α). Let Σc(scC,t)={α∈it∗∣scC(t,α)={0}}. Finally set
[TABLE]
As before we define ΩM+ as the intersection of ΩM with a positive Weyl chamber.
Let T(ΩM)=expΩM⊂T=expt, T(Ωh)=expiΩh, and define
[TABLE]
Theorem 7.3** (Matsuki).**
ΞM* is open in Gc and ΞM is connected and open in Gc/Lc.*
Proof.
This will follow from [Ma03, Prop. 1] using the dictionary above. For that we need some material about
τ=τa∘θc-stable parabolic subalgebras in gc. Let, as
before, a⊂p be maximal abelian. Let Σ=Σ(gc,a) denote the set
of roots of a in gc and let Σ+ be a set of positive roots. Define
n=⨁α∈Σ+(gc,a)gcα and
m= the orthogonal complement of a in zgc(a).
Then pc=m⊕a⊕n is
a minimal θc∘τa stable parabolic subalgebra in gc, see [Ma79] or [vdB88].
Let Pc=McANc be the corresponding minimal θc∘τa stable
parabolic subgroup.
That LcPc is open in Gc follows from [Ma79]. Hence by Matsuki duality [Ma79] GPc
is closed. Now compare this with the assumption on [Ma03, p. 565] and we see that we can take Pc for
the parabolic P in [Ma03] or [vdB88], i.e. P↔Pc.
∎
Set S=S(GcPc;LcPc)={x∈Gc∣x−1GPc⊂LcPc}. Then S is open and if S0 denotes a connected
component, we have
ΞM⊆S0.
We mention a slightly different interpretation of ΩM. We refer to [HÓ96, Chap. 5] for a
more detailed discussion. The abelian Lie algebra t is a maximal abelian subspace of
kc∩qc=iqhp. But the generalized flag manifold Gc/LcN− is diffeomorphic to
Kc/Kc∩Lc≅Kc/FK which is a Riemannian symmetric space. We have
even more, LcN− is a maximal parabolic subgroup with abelian nilradical, n−. Hence
Kc/Kc∩Lc≃Gc/LcN− is a symmetric R-space. Note that Ad(F)
normalizes k and hence FK is a group. Furthermore, (FK)∩G=K. Since kc=k⊕iqhp
we have the tangent space at eFK is given by iqhp. Thus t is the Lie algebra of a maximal torus
(an Iwasawa torus) in the tangent space. But we have the open embedding D≃G/K⊂Gc/LcN−
so the tangent space at eK can be identified with p which has maximal abelian subalgebra a. On
the other hand, Gc/LcN−≃Kc/FK is, up to covering, the compact dual symmetric space to G/K.
Thus within gC there is an R-isomorphism ϕ:ac⊕t→acC, i.e. between the split-complexification and the complexification.
As η(Gh)=Gh, η(Kh)=Kh and we can choose ah so that η(ah)=ah it
follows that η(Ξh)=Ξh and η(Ξh)=Ξh. As
Gc/Lc=Gc⋅xo it follows that we can view Gc/Lc as a real
form of Gh/Kh. We note that ΞM is not connected unless Lc is, but
(ΞM)o=GcexpΩM(Lc)o.
Remark 7.5**.**
The Matsuki crown is defined with respect to Gc. To connect the
notation to gh we make some additional observations. First notice that if a1 is maximal abelian in
qh∩ph=phτa if and only if t=ia1 is maximal abelian in qc∩kc=i(qh∩ph).
As sc=i(qh∩kh)⊕p we have Σ(scC,tC)=Σ(gh−τa,a1) and
[TABLE]
quite analogous to the construction in the group case. This shows that there is
a fundamental difference between the case r=rh/2 and r=rh. In the first case we can
take a=⨁RXj′ as before, and a1=ahq=⨁RXj−. In
particular a and t commute as in the group case. For r=rh the space a is already maximal
abelian so there is no way to chose t so that a and t commute.
If r=rh we always have Σ(g,a)⊆Σ(gh,a) and
Σ(gh−τa,ahq)⊆Σ(gh,ahq) which implies that
Ωh⊆Ω.
So if we define Ωh as a
subset of ahq, Ωh⊆Ωhq. Similarly, if r=rh/2,
as Ω and Ωhq are defined by via restriction of roots in Σ(gh,ah) to
a, resp. ahq, and because Ωh is invariant under τ and −τ it follows that
Ωh∩a=prg(Ωh)⊆Ω and
Ωh∩ahq=prqh(Ωh)⊆Ωhq. Here prg
is the projection along qh onto g and
prqh is the projection along g onto qh. This clearly implies that we always have
Ξ⊆(Ξhτ)o and ΞM⊆(Ξhη)o.
Lemma 7.6**.**
Let the notation be as above.
(a)
Assume that r=rh and Ωh=Ω. Then
Ξ=(Ξhτ)o.
(b)
Assume that r=rh and Ωh=Ωhq⊂ahq. Then
ΞM=(Ξhη)o.
(c)
Assume that r=rh/2 and Ωh∩a=Ω. Then
Ξ=(Ξhτ)o.
(d)
Assume that r=rh/2 and Ωh∩ahq=Ωhq. Then
ΞM=(Ξhη)o.
Proof.
We prove only (c) and (d). The proofs of (a) and (b) are simpler following the same line
of argument.
(c): We have Ξ=Gexp(Ω)⋅xo⊂GhexpΩh⋅xo=Ξh⊂Gh/Kh.
Taking τ fixed points implies the inclusion Ξ⊆(Ξhτ)o. As Gh is simply connected it
follows that G=Ghτ. Hence (Ξhτ)o=(Ξh∩G/K)o. As Ξ is open in
G/K it follows that Ξ is open in (Ξhτ)o. Assume that
Ξ is not closed in (Ξhτ)o. Then there
exists a sequence ξj=gjexpYj⋅xo, gj∈G, Yj∈Ω, such
that ξj→ξ∈∂Ξ∩(Ξhτ)o. According to Theorem 7.1 part (d)
there exists Y∈∂Ω such that Yj→Y. Hence there exists α∈Σ(g,a)
such that ∣α(Yj)∣→π/2. Let
[TABLE]
Then ghα is ad(ah) invariant. It follows that there exists
β∈Σ(gh,ah) such that βa=α.
Thus Y∈∂Ξh contradicting the assumption that ξ∈Ξh. Thus Ξ is
closed in (Ξhτ)o.
Part (d) follows in the same way replacing τ by η and in the last argument replacing a by
ahq.
∎
Lemma 7.7**.**
Assume that r=rh/2. Write ah=a⊕ahq and let β∈Σ(gh,ah). If
β∣ahq=0 and H∈ahq is so that β(H)=1 then adH:prg(ghβ)→prqh(ghβ) is an isomorphism. In particular, if β∣a=0, then
{0}=prg(ghβ)⊆gβ∣a and {0}=prqh(ghβ)⊆(gh−τa)β∣ahq.
Proof.
Let X=Xg+Xq∈ghα with Xg=prg(ghβ) and
Xq=prqh(ghβ). Then adH(X)=X=[H,Xg]+[H,Xq]. As [H,Xg]∈q and [H,Xq]∈g
it follows that [H,Xg]=Xq and [H,Xq]=Xg. The last part follows by replacing τ by τa which
interchanges the role of a and ahq.
∎
Lemma 7.8**.**
We have the following.
(a)
Assume that r=rh then we have:
(a-i)
If βj∈Σ(g,a) for all j=1,…,r then Ω=Ωh.
(a-ii)
If βj∈Σ(g−τa,ahq) then ΩM=iΩhq.
(b)
If r=rh/2 then we have:
(b-i)
If γ2j∣a=β2j∣a∈Σ(g,α), j=1,…,r then
Ω=Ωh∩a.
(b-ii)
If γ2j−1∣ahq=β2j∣ahq∈Σ(gh−τa,ahq) then
ΩM=iΩhq.
Proof.
This follows directly from Moore’s Theorem.
For example consider (b-ii). We
only have to show that Ωh∩ahq⊆Ωhq. Let X=∑j=1rhtjXj∈Ωh∩ahq. Then ∣tj∣<π/2 for all j. Furthermore
X=−τX. Hence X=∑j=1rt2j−1(X2j−1−X2j) and hence ∣β2j−1(X)∣<π/2. The claim now follows from Moore’s
Theorem as all the roots in Σ(gh−τa,ahq) are restrictions of
roots in Σ(gh,ah)
∎
Finally we come to the relationship of various crowns.
Theorem 7.9**.**
Let the notation be as above. Then the following holds:
(a)
If r=rh then Ωh=Ωhq and
ΞM=(Ξhη)o.
(b)
If r=rh/2. Then Ω=Ωh∩a and
ΩM=Ωh∩ahq, Furthermore ΞM=(Ξhη)o and
Ξ=(Ξhτ)o.
Proof.
Lemma 7.6 and Lemma 7.8 imply that we have to show that βj∣a∈Σ(g,a) respectively
βj∣ahq∈Σ(gh−τa,ahq), for
j=1,…,rh. For (a) we use
su(1,1)-reduction for τa to show that βj∈Σ(gh−τa,ahq).
For (b) this follows from Lemma 7.7 as each βj has a non-zero restriction to
a and ahq.
∎
Basic Example**.**
SU(1,1) - cont.
Recall that gh=su(1,1)=kh⊕ph=g⊕qh, where
[TABLE]
while g=R⋅X and qh=kh⊕R⋅Y. As before, SU(1,1)
has two natural choices of Iwasawa: A=expRX or Ahq=expRY.
From [KS04] we know that either choice gives, with the obvious notation,
[TABLE]
[TABLE]
Also from before we have T(Ω)=expiΩ=expiΩhq. Since a and
ahq are conjugate via Kh we have
[TABLE]
Taking fixed points of the conjugate linear η gives
[TABLE]
where
[TABLE]
Now GT(Ω)(Lc)0 is connected and contains the identity.
Take nak∈NAKh. Then
[TABLE]
Multiplication gives
[TABLE]
On the other hand, if gtl∈GT(Ω)(Lc)0 then
[TABLE]
Since in this example G=A and Lc⊂Kh it suffices to express t in terms of nak. An elementary, though tedious, computation gives the following solutions provided 0≤∣θ∣<4π:
[TABLE]
With these substitutions it is straightforward to verify that t=nak with n∈N∩Gc,a∈A,k∈(Lc)0. Also notice that [Y,(ii−i−i)]=2(ii−i−i), thus ∣θ∣<4π is the full range to describe
T(Ω). Thus GT(Ω)Lc⊂(N∩Gc)ALc.
Example 7.10** (Cayley Type Spaces).**
There are examples where r=rh and
Ω=Ωh. The simplest case is the rank one case (so(1,n),so(1,n−1)) with n≥3.
But the following example shows that we have no general statement in this case.
Assume that g=g′⊕RH0 is
not simple with g′=[g,g] simple. Then ah=a=a′⊕RH0 with a′=a∩g′. We have by
Moore’s Theorem, Theorem 1.5, we have
[TABLE]
Let again X1,…,Xr be so that αi(Xj)=δij and use those as coordinate axes. Then
Ωh=(−π/2,π/2)r.
On the other hand the condition for Ω is 21∣xi−xj∣<π/2. Thus Ωh⫋Ω.
Interchanging τ and τa we see that Ωh=Ω which again leads to
Ξ=Ξhη.
Given (π,E) an irreducible Banach representation of G and a K-finite vector v∈E
Theorem 3.1 in [KS04] states that the orbit map g→π(g)v has a holomorphic extension to the domain
Ξ⊂G. There is an analogous result here with the domain ΞM just constructed and the group Gc in place of G.
Theorem 7.11**.**
Let (π,E) be an irreducible Banach representation of G, and let v∈E be
a K-finite vector. Then the map g→π(g)v has an analytic extension to (ΞM)0=GT(ΩM)(Lc)0⊂Gc.
Proof.
The key to the result is that (Lc)0 and G have the same maximal compact subgroup K. First we consider the case r=rh. Then ah and ahq are Kh conjugate, so Ghexp(iΩh)Kh=GhexpiΩhqKh=GhT(Ω)Kh. From [KS04], GhT(Ωh)Kh⊂NhAhKh is open and the projection maps to Ah and Kh are holomorphic. Now ΞM0=GT(ΩM)Lc0⊂S0⊂[(GhT(ΩhKh))η]0⊂Gc.
The restriction of the projection maps to Ah and Kh gives analytic maps to A=(Ah)η and Lc0=(Kh)η but as ΞM is connected, to Lc0.
Since r=rh, a≅ah is also an Iwasawa for G, Denote the map to Lc0 by ℓ.
Since both Lc0 and G have the same maximal compact subgroup, K, composition of ℓ with the usual κ projection of Lc0 to K gives an analytic map from ΞM0=GT(ΩM)Lc0⊂[(GhT(Ω)Kh)η]0 to K. With these analytic maps from ΞM0
to A and K we are now in the position of the proof of Theorem 3.1 in [KS04] and can continue it verbatim to obtain the result.
If r=rh then as we have seen r=2rh. As in Lemma 2.12 ah=a⊕aqh as a Lie algebra direct sum, i.e. a, aqh are abelian and [a,aqh]=0. Also from the Lemma we have η, restricted to ah, is one on a and −1 on aqh. Then the conjugate linear extension η is one on a⊕iaqh, i.e. Ahη=Aexpiaqh with A⊂G. Thus ΞM0=GT(ΩM)Lc0⊂[(GhT(ΩhKh)η]0⊂Gc Again has the restriction of the holomorphic projection maps taking values in K and expiaqh with the latter isomorphic to expia. Thus here to we are in the position of Theorem 3.1 of [KS04].
∎
Remark 7.12**.**
In the Basic ExampleG≅R∗, the representations of G are just characters, so from the above expression the continuation of the characters to GT+Lc as just translation in the variable by −21 log cos(2θ).
Example 7.13** (The case of SU(m,1)).**
We will show that the computations for SU(m,1) reduce to those of the Basic Example. Here gh=su(m,1)=kh⊕ph=g⊕qh, where
[TABLE]
and g=so(m,1). Using obvious block matrices let
[TABLE]
As before, SU(m,1) has two natural choices of Iwasawa:
[TABLE]
We will do the computations for SU(3,1) for then the procedure for SU(m,1) will be clear. Either choice of Iwasawa gives
[TABLE]
[TABLE]
[TABLE]
where T(Ω)=expiΩ=expiΩhq. Taking fixed points of
the conjugate linear η gives
[TABLE]
where
[TABLE]
(Zi∈C,v∈R) while Lc=(GL(3,R)001/det). Again in GT+Lc it suffices to consider only the t term. Now in (GhT+Kh)η the right action by Lc has the effect of multiplying the last column by det−1, but
[TABLE]
Consequently we must have Z1=0=Z2, reducing the computations to the case SU(1,1) thus obtaining essentially the same formulae for t=nak as before. In particular, GT(Ω)Lc⊂(N∩Gc)ALc⊂(NAKh)η.
Appendix A The Classification
In the following tables we set gl+(n,C)=sl(n,C)⊕Rid and t=iR= the Lie algebra of the torus T={z∈C∣∣z∣=1}.
In Table 4 the items listed below the line are those where
Gh/Kh is a tube type domain and gc≅gh. That happens if and only if g≃lc if and only if g has a one-dimensional center. We denote the compact real form of E6 by e6. We also note that sl(n,R)×R=gl(n,R) but we
write it using sl(n,R)×R so that it fits better into the general picture. Same comments hold for u(n) and su(n)×t.
In Table 4 and Table 5 we can assume the q≥p because interchanging the role of p and q leads to
isomorphic cases. The case gh=so(2,q)⊃g=so(1,q) corresponds to the case p=0,
and the case p=1 corresponds to the case gh=so(2,n)⊃g=so(1,n−1)×R.
The case gh=so(2,2) is excluded because so(2,2) is not simple.
In Table 5 we have reorganized Table 4 into three groups. The first group consists of those g for which the lc has
one conjugacy class of Cartan subalgebra (denoted OCCC). The second group consists of those g for which
lc consists of automorphisms of a vector space while the maximal compact, k, of g corresponds to
isometries of the space. The third group consists of exceptions that will be treated individually. Of course there
are ways, say using the octonions, to incorporate some of the third group into the second but
we prefer this way. Notice that in all groups k is the maximal compact for both g and lc.
Acknowledgements
The research by Ólafsson was partially supported by NSF grants DMS-0801010
and DMS-1101337; both authors are grateful for support provided by the Max-Planck-Institut für Mathematik, Bonn
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