Horn problem for quasi-hermitian Lie groups

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Horn problem for quasi-hermitian Lie groups

Paul-Emile Paradan

To cite this version:

Paul-Emile Paradan. Horn problem for quasi-hermitian Lie groups. 2020. �hal-02867106�

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Horn problem for quasi-hermitian Lie groups

Paul-Emile Paradan^∗ June 13, 2020

Abstract

In this paper, we prove some convexity results associated to orbit projection of non-compact real reductive Lie groups.

1 Introduction

This paper is concerned with convexity properties associated to orbit projection.

Let us consider two Lie groups G ⊂ G˜ with Lie algebras g ⊂ ˜g and corresponding projectionπ_g,˜_g : ˜g^∗ →g^∗. A longstanding problem has been to understand how a coadjoint orbit ˜O ⊂˜g^∗ decomposes under the projectionπ_g,˜_g. For this purpose, we may define

∆_G( ˜O) ={O ∈g^∗/G; O ⊂π_g,˜_g( ˜O)}.

When the Lie group G is compact and connected, the set g^∗/G admits a natural identification with a Weyl chambert^∗_≥0. In this context, we have the well-known convexity theorem [12, 1, 10, 16, 13, 34, 22].

Theorem 1.1 Suppose that Gis compact connected and that the projectionπ_g,˜_g is proper when restricted toO. Then˜ ∆_G( ˜O) ={ξ ∈t^∗_≥0; Gξ ⊂π_g,˜_g( ˜O)} is a closed convex locally polyhedral subset oft^∗.

When the Lie group ˜Gis also compact and connected, we may consider (1) Π( ˜G, G) :=n

(˜ξ, ξ)∈˜t^∗_≥0×t^∗_≥0; Gξ ⊂ πg,˜g G˜ξ˜o . Here is another convexity theorem [14, 17, 4, 2, 3, 25, 20, 35].

Theorem 1.2 Suppose that G⊂G˜ are compact connected Lie groups. Then Π( ˜G, G) is a closed convex polyhedral cone and we can parametrize its facets by cohomological means (i.e. Schubert calculus).

In this article we obtain an extension of Theorems 1.1 and 1.2 in the case where the two groupsG and ˜Gare not compact.

Let us explain in which framework we work. We suppose that ˜G/K˜ is a Hermitian symmetric space of a non-compact type. Among the elliptic coadjoint orbits of ˜G, some of them are naturally K¨ahler ˜K-manifolds. These orbits are called the holomorphic coadjoint orbits of ˜G. They are the strongly elliptic coadjoint orbits closely related to the holomorphic discrete series of Harish-Chandra. These orbits intersect the Weyl chamber ˜t^∗_≥0 of ˜K

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into a sub-chamber ˜C_hol called the holomorphic chamber. The basic fact here is that the union

C⁰_G/_˜ _K_˜ := [

˜ a∈C˜_hol

G˜˜a

is an open invariant convex cone of ˜g^∗. See §2.1 for more details.

LetG/K ⊂G/˜ K˜ be a sub-Hermitian symmetric space of a non-compact type. As the projection π_g,˜_g : ˜g^∗ → g^∗ sends the convex cone C_G/⁰_˜ _K_˜ inside the convex cone C_G/K⁰ , it is natural to study the following object reminiscent of (1) :

(2) Π_hol( ˜G, G) :=n

(˜ξ, ξ)∈C˜_hol× C_hol; Gξ ⊂ πg,˜g G˜ξ˜o .

Let ˜µ∈C˜_hol. We will also give a particular attention to the intersection of Π_hol( ˜G, G) with the hyperplane ˜ξ = ˜µ, that is to say

(3) ∆_G( ˜G˜µ) :=n

ξ ∈ Chol; Gξ ⊂ π_g,˜_g G˜˜µo .

Consider the case where Gis embedded diagonally in ˜G:= G^s for s≥2. The corresponding set Π_hol(G^s, G) is called the holomorphic Horn cone, and it is defined as follows

Horn^s_hol(G) :=n

(ξ₁,· · · , ξ_s+1)∈ C_hol^s+1; Gξ_s+1 ⊂ Xs j=1

Gξ_jo .

The first result of this article is the following Theorem.

Theorem A.

• Π_hol( ˜G, G) is a closed convex cone of ˜C_hol× C_hol.

• Horn^s_hol(G) is a closed convex cone of C_hol^s+1 for any s≥2.

We obtain the following corollary which corresponds to a result of Weinstein [37].

Corollary. For anyµ˜∈C˜_hol,∆G( ˜G˜µ) is a closed and convex subset of C_hol.

A first description of the closed convex cone Π_hol( ˜G, G) goes as follows. The quotient q of the tangent spaces T_eG/K and T_eG/˜ K˜ has a natural structure of a Hermitian K- vector space. The symmetric algebra Sym(q) of q defines an admissible K-module. The irreducible representations of K (resp. ˜K) are parametrized by a semi-group ∧^∗₊ : (resp.

∧˜^∗₊). For any λ ∈ ∧^∗₊ (resp. ˜λ ∈ ∧˜^∗₊), we denote by V_λ^K (resp. V_˜^K^˜

λ ) the irreducible representation ofK (resp. ˜K) with highest weight λ(resp. ˜λ). If E is a representation of K, we denote by

V_λ^K :E

the multiplicity of V_λ^K inE.

Definition 1.3 1. Π^Z( ˜G, G) is the semigroup of ∧˜^∗₊× ∧^∗₊ defined by the conditions:

(˜λ, λ)∈Π^Z( ˜G, G)⇐⇒h

V_λ^K : V_λ_˜^K^˜ ⊗Sym(q)i 6= 0.

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2. Π( ˜G, G) is the convex cone defined as the closure of Q^>0·Π^Z( ˜G, G).

The second result of this article is the following Theorem.

Theorem B. We have the equality

(4) Π_hol( ˜G, G) = Π( ˜G, G) \ C˜hol×Chol.

A natural question is the description of the facets of the convex cone Π_hol( ˜G, G). In order to do that, we consider the group ˜K endowed with the following ˜K ×K-action : (˜k, k)·˜a = ˜k˜ak⁻¹. The cotangent space T^∗K˜ is then a symplectic manifold equipped with a Hamiltonian action of ˜K×K. We consider now the Hamiltonian ˜K×K-manifold T^∗K˜ ×q, and we denote by ∆(T^∗K˜ ×q) the corresponding Kirwan polyhedron.

Let W =N(T)/T be the Weyl group of (K, T) and let w₀ be the longest Weyl group element. Define an involution ∗ : t^∗ → t^∗ by ξ^∗ = −w0ξ. A standard result permits to affirm that (˜ξ, ξ)∈Π( ˜G, G) is an only if (˜ξ, ξ^∗)∈∆(T^∗K˜ ×q) (see §4.2).

We obtain then another version of Theorem B.

Theorem B, second version. An element (˜ξ, ξ) belongs to Π_hol( ˜G, G) if and only if (˜ξ, ξ)∈C˜hol× Chol and (˜ξ, ξ^∗)∈∆(T^∗K˜ ×q).

Thanks to the second version of Theorem B, a natural way to describe the facets of the cone Π_hol( ˜G, G) is to exhibit those of the Kirwan polyhedron ∆(T^∗K˜ ×q). In this later case, it can be done using Ressayre’s data (see§5).

The second version of Theorem B permits also the following description of the convex subsets ∆_G( ˜G˜µ), ˜µ ∈ C˜hol. Let ∆_K( ˜Kµ˜ ×q) be the Kirwan polyhedron associated to the Hamiltonian action of K on ˜Kµ˜×q. Here q denotes the K-module q with opposite complex structure.

Theorem C.For any µ˜∈C˜hol, we have ∆_G( ˜G˜µ) = ∆_K( ˜Kµ˜×q).

Let us detail Theorem C in the case whereGis embedded in ˜G=G×Gdiagonally. We denote byptheK-Hermitian space T_eG/K. Letκ be the Killing form of the Lie algebra g. The vector space p is equipped with the symplectic 2-form Ω¯p(X, Y) = −κ(z,[X, Y]) and the compatible complex structure−ad(z).

Let us denote by ∆K(Kµ1×Kµ2×p) and by ∆K(p) the Kirwan polyhedrons relative to the Hamiltonian actions of K on Kµ₁ ×Kµ₂ ×p and on p. Theorem C says that for any µ₁, µ₂ ∈ Chol, the convex set ∆_G(Gµ₁×Gµ₂) is equal to the Kirwan polyhedron

∆K(Kµ1×Kµ2×p).

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To any non-empty subsetC of a real vector spaceE, we may associate its asymptotic cone As(C) ⊂E which is the set formed by the limits y = lim_k→∞t_ky_k, where (t_k) is a sequence of non-negative reals converging to 0 andy_k∈C.

We finally get the following description of the asymptotic cone of ∆_G(Gµ₁×Gµ₂).

Corollary D.For any µ1, µ2 ∈ C_hol, the asymptotic cone of ∆G(Gµ1×Gµ2) is equal to

∆_K(p).

In [28]§5, we explained how to describe the cone ∆K(p) in terms of strongly orthogonal roots.

Let us finish this introduction with few remarks on related works:

− When Gis compact, equal to the maximal compact subgroup ˜K of ˜G, the results of Theorems B and C were already obtained by G. Deltour in his thesis [6, 7].

− In [9], A. Eshmatov and P. Foth proposed a description of the set ∆G(Gµ1×Gµ2).

But their computations give not the same result as ours. From their main result (Theorem 3.2) it follows that the asymptotic cone of ∆_G(Gµ₁×Gµ₂) is equal to the intersection of the Kirwan polyhedron ∆T(p) with the Weyl chambert^∗_≥0. But since ∆_K(p)6= ∆T(p)∩t^∗_≥0 in general, it is in contradiction with Corollary D.

Notations

In this paper,Gdenotes a connected real reductive Lie group : we take here the convention of Knapp [18]. We have a Cartan involution Θ onG, such that the fixed point setK:=G^Θ is a connected maximal compact subgroup. We have Cartan decompositions at the level of Lie algebrasg=k⊕pand at the level of the groupG≃K×exp(p). We denote byba G-invariant non-degenerate bilinear form on g that is equal to the Killing form on [g,g], and that defines a K-invariant scalar product (X, Y) := −b(X,Θ(Y)). We will use the K-equivariant identification ξ 7→ ξ,˜ g^∗ ≃ g defined by (˜ξ, X) := hξ, Xi for ξ ∈ g^∗ and X∈g.

When a Lie groupHacts on a manifoldN, the stabilizer subgroup ofn∈N is denoted byH_n={g∈G, gn=n}, and its Lie algebra byh_n. Let us define

(5) dim_H(X) = min

n∈Xdim(h_n) for any subsetX ⊂N.

2 The cone Π

hol

( ˜ G, G ) : first properties

We assume here that G/K is a Hermitian symmetric space, that is to say, there exists a G-invariant complex structure on the manifold G/K, or equivalently there exists a K- invariant elementz ∈k such that ad(z)|p defines a complex structure onp : (ad(z)|p)² =

−Idp. This condition imposes that the ranks ofGand K are equal.

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We are interested in the following closed invariant convex cone of g^∗: C_G/K ={ξ ∈g^∗,hξ, gzi ≥0, ∀g∈G}.

2.1 The holomorphic chamber

LetT be a maximal torus ofK, with Lie algebrat. Its dualt^∗ can be seen as the subspace of g^∗ fixed by T. Let us denote by g^∗_e the set formed by the elliptic elements: in other wordsg^∗_e:= Ad^∗(G)·t^∗.

Following [37], we consider the invariant open subsetg^∗_se={ξ ∈g^∗|G_ξ is compact}of strongly ellipticelements. It is non-empty since the groups Gand K have the same rank.

We start with the following basic facts.

Lemma 2.1 • g^∗_se is contained in g^∗_e.

• The interior C_G/K⁰ of the cone C_G/K is contained in g^∗_se.

Proof : The first point is due to the fact that every compact subgroup of G is conjugate to a subgroup ofK.

Letξ ∈ C_G/K⁰ . There exists ǫ >0 so that

hξ+η, gzi ≥0, ∀g∈G, ∀kηk ≤ǫ.

It implies that|hη, gzi| ≤ hξ, zi,∀g∈G_ξ and ∀kηk ≤ǫ. In other words, the adjoint orbit G_ξ·z⊂gis bounded. For any g=e^Xk, with (X, k)∈p×K, a direct computation shows that kgzk = ke^Xzk ≥ k[z, X]k = kXk. Then, there exists ρ > 0 such that kXk ≤ ρ if e^Xk∈G_ξ for somek∈K. It follows that the stabilizer subgroupG_ξ is compact. ✷

Let ∧^∗ ⊂t^∗ be the weight lattice : α∈ ∧^∗ if iα is the differential of a character ofT. LetR⊂ ∧^∗ be the set of roots for the action ofT on g⊗C. We haveR=R_c∪R_n where R_c and R_n are respectively the set of roots for the action ofT on k⊗C and p⊗C. We fix a system of positive roots R⁺_c in R_c, and we denote by t^∗_≥0 the corresponding Weyl chamber.

We havep⊗C=p⁺⊕p⁻ where theK-modulep^± is equal to ker(ad(z)∓i). LetR^±,z_n be the set of roots for the action ofT on p^±. The union

(6) R⁺=R⁺_c ∪R^+,z_n

defines then a system of positive roots inR. We notice thatR^+,z_n is the set of rootsβ∈R satisfying hβ, zi = 1. Hence R^+,zn is invariant relatively to the action of the Weyl group W =N(T)/T.

Let us recall the following classical fact concerning the parametrization of theG-orbits inC_G/K⁰ via the holomorphic chamber

C_hol :={ξ∈t^∗_≥0,(ξ, β)>0, ∀β ∈R^+,z_n }.

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Proposition 2.2 • The setC_G/K⁰ ∩t^∗_≥0 is contained in C_hol.

• For any compact subset K of Chol, there exists c_K > 0 such that hξ, gzi ≥ c_Kkgzk,

∀g∈G, ∀ξ∈ K.

• The map O 7→ O ∩t^∗_≥0 defines a bijective map between the set of G-orbits in C_G/K⁰ and the holomorphic chamber C_hol.

Proof : Let ξ ∈ t^∗_≥0 ∩ C_G/K⁰ . The first point is proved if we check that (ξ, β) > 0 for any β ∈ R^+,z_n . LetX_β, Y_β ∈ p such that X_β +iY_β ∈ (p⊗C)_β. We choose the following normalization : the vector h_β := [X_β, Y_β] satisfies hβ, h_βi = 1. We see then that (ξ, β) =

1

khβk²hξ, hβi for any ξ ∈g^∗. Standard computation [27] gives

e^t^ad(X^β⁾z=z+ (cosh(t)−1)h_β+ sinh(t)Y_β, ∀t∈R.

By definition, we must havehξ+η, e^t^ad(X^β⁾zi ≥0,∀t∈R, for anyη ∈t^∗ small enough. It imposes that hξ, h_βi>0. The first point is settled.

Now choose some maximal strongly orthogonal system Σ ⊂ R^+,zn . The real span a of the X_β, β ∈Σ is a maximal abelian subspace of p. Hence any element g ∈ G can be writteng=ke^X(t)k^′ withX(t) =P

β∈Σt_βX_β andk, k^′ ∈K. We get

(7) gz =k



z+X

β∈Σ

(cosh(t_β)−1)h_β+X

β∈Σ

sinh(t_β)Y_β



 and

hξ, gzi=hk⁻¹ξ, zi+X

β∈Σ

(cosh(t_β)−1)hk⁻¹ξ, h_βi.

For any ξ ∈ C_hol, we define c_ξ := min_β∈R^+,z

n hξ, h_βi > 0. Let π : k^∗ → t^∗ be the projection. We havehk⁻¹ξ, zi =hπ(k⁻¹ξ), zi and hk⁻¹ξ, h_βi=hπ(k⁻¹ξ), h_βi. The convexity theorem of Kostant tell us that π(k⁻¹ξ) belongs to the convex hull of {wξ, w ∈ W}. It follows that hk⁻¹ξ, zi ≥ hξ, zi > 0 and hk⁻¹ξ, h_βi ≥ c_ξ > 0. We obtain then that hξ, gzi ≥ ¹₂min(hξ, zi, cξ)e^kX(t)k for anyξ ∈ Chol, where kX(t)k= sup|tβ|. From (7), it is not difficult to see that there existsC >0 such thatkgzk ≤Ce^kX(t)k for anyg=ke^X(t)k^′. Let K be a compact subset of Chol. Take c_K = _2C¹ min(min_ξ∈Khξ, zi,min_ξ∈Kc_ξ) > 0.

The previous computations show that hξ, gzi ≥ c_Kkgzk, ∀g ∈ G, ∀ξ ∈ K. The second point is proved.

The first two points show that t^∗_≥0∩ C_G/K⁰ = Chol. The third point follows then from the fact that C_G/K⁰ is contained in g^∗_e (see Lemma 2.1). ✷

We have a canonical map p :C_G/K⁰ → Chol defined by the relations Gξ∩t^∗_≥0={p(ξ)},

∀ξ∈ C_G/K⁰ . The following result is needed in§4.1.

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Lemma 2.3 The map p is continuous.

Proof : Let (ξ_n) be a sequence of C_G/K⁰ converging to ξ_∞ ∈ C_G/K⁰ . Let ξ_n^′ = p(ξ_n) and ξ_∞^′ = p(ξ_∞): we have to prove that the sequence (ξ_n^′) converges toξ^′_∞. We choose elements g_n, g_∞∈G such thatξ_n=g_nξ_n^′,∀n, andξ_∞=g_∞ξ_∞^′ .

First we notice that −b(ξn, ξn) =kξ^′_nk², hence the sequence (ξ_n^′) is bounded. We will now prove that the sequence (g_n) is bounded.

Let ǫ >0 such that hξ∞+η, gzi ≥ 0,∀g∈G,∀kηk ≤ ǫ. If kξ−ξ_∞k ≤ǫ/2, we write ξ= ¹₂(ξ∞+ 2(ξ−ξ∞)) + ¹₂ξ∞, and then

hξ, gzi= 1

2hξ_∞+ 2(ξ−ξ_∞), gzi+1

2hξ_∞, gzi ≥ 1

2hξ_∞, gzi, ∀g∈G.

Now we have hξ_n^′, zi = hξn, g_nzi ≥ ¹₂hξ∞, g_nzi if n is large enough. This shows that the sequence hξ∞, g_nzi is bounded. If we use the second point of Proposition 2.2, we can conclude that the sequence (gn) is bounded.

Let (ξ_φ(n)^′ ) be a subsequence converging to ℓ ∈ t^∗_≥0. Since (g_φ(n)) is bounded, there exists a subsequence (g_φ◦ψ(n)) converging to h ∈ G. From the relations ξ_φ◦ψ(n) = g_φ◦ψ(n)ξ^′_φ◦ψ(n),∀n ∈ N, we obtain ξ_∞ = hℓ. Then ℓ = p(ξ_∞) = ξ^′_∞. Since every subsequence of (ξ_n^′) has a subsequential limit toξ_∞^′ , then the sequence (ξ_n^′) converges to ξ^′_∞.

✷

2.2 The cone Π_hol( ˜G, G) is closed

We suppose thatG/K is a complex submanifold of a Hermitian symmetric space ˜G/K. In˜ other words, ˜Gis a reductive real Lie group such thatG⊂G˜ is a closed subgroup preserved by the Cartan involution, and ˜K is a maximal compact subgroup of ˜Gcontaining K. We denote by ˜g and ˜k the Lie algebras of ˜G and ˜K respectively. We suppose that there exists a ˜K-invariant element z ∈ k such that ad(z)|˜p defines a complex structure on ˜p : (ad(z)|˜p)² =−Id˜p.

Let CG/˜ K˜ ⊂ ˜g^∗ be the closed invariant cone associated to the Hermitian symmetric space ˜G/K˜. We start with the following key fact.

Lemma 2.4 The projection π_g,˜_g: ˜g^∗→g^∗ sends C_G/⁰_˜ _K_˜ into C_G/K⁰ .

Proof : Let ˜ξ ∈ C_G/⁰_˜ _K_˜ and ξ =π_g,˜_g(˜ξ). Then hξ˜+ ˜η,gzi ≥˜ 0, ∀˜g ∈ G˜ if ˜η ∈ ˜g^∗ is small enough. It follows that hξ+π_g,˜_g(˜η), gzi = hξ˜+ ˜η, gzi ≥ 0, ∀g ∈G if ˜η is small enough.

Sinceπg,˜g is an open map, we can conclude that ξ∈ C_G/K⁰ . ✷

Let ˜T be a maximal torus of ˜K, with Lie algebra ˜t. The ˜G-orbits in the interior of CG/˜ K˜ are parametrized by the holomorphic chamber ˜Chol ⊂˜t^∗. The previous Lemma says that the projectionπ_g,˜_g( ˜O) of any ˜G-orbit ˜O ⊂ C_G/⁰_˜ _K_˜ is the union ofG-orbits O ⊂ C_G/K⁰ .

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So it is natural to study the following object (8) Π_hol( ˜G, G) :=n

(˜ξ, ξ)∈C˜hol× Chol; Gξ ⊂ π_g,˜_g G˜ξ˜o .

Here is a first result.

Proposition 2.5 Π_hol( ˜G, G) is a closed cone of C˜_hol× C_hol.

Proof : Suppose that a sequence (˜ξn, ξn)∈Π_hol( ˜G, G) converge to (˜ξ∞, ξ∞)∈C˜_hol×C_hol. By definition, there exists a sequence (˜g_n, g_n)∈ G˜ ×G such that g_nξ_n =π_g,˜_g(˜g_nξ˜_n). Let

˜h_n := g_n⁻¹g˜_n, so that ξ_n = πg,˜g(˜h_nξ˜_n) and h˜h_nξ˜_n, zi = hξ_n, zi. We use now that the sequencehξ_n, zi is bounded, and that the sequence ˜ξ_n belongs to a compact subset of ˜C_hol. Thanks to the second point of Proposition 2.2, these facts imply thatk˜h⁻¹_n zkis a bounded sequence. Hence ˜h_nadmits a subsequence converging to ˜h_∞. So we getξ_∞=π_g,˜_g(˜h_∞ξ˜_∞), and that proves that (˜ξ_∞, ξ_∞)∈Π_hol( ˜G, G). ✷

2.3 Rational and weakly regular points

Let (M,Ω) be a symplectic manifold. We suppose that there exists a line bundle L with connection ∇ that prequantizes the 2-form Ω: in other words, ∇² = −iΩ. Let K be a compact connected Lie group acting onL →M, and leaving the connection invariant. Let Φ_K :M →k^∗ be the moment map defined by Kostant’s relations

(9) LX− ∇X =ihΦK, Xi, ∀X∈k.

HereL_X is the Lie derivative acting on the sections of L →M.

Remark that relations (9) imply, via the equivariant Bianchi formula, the relations (10) ι(X_M)Ω =−dhΦ_K, Xi, ∀X∈k,

whereX_M(m) := _dt^d|t=0e^−tXm is the vector field on M generated byX ∈k.

Definition 2.6 Let dim_K(M) := min_m∈Mdimk_m. An element ξ∈k^∗ is a weakly-regular value ofΦ_K if for all m∈Φ⁻¹_K (ξ) we have dimk_m = dim_K(M).

When ξ∈k^∗ is a weakly-regular value of Φ_K, the constant rank theorem tells us that Φ⁻¹_K (ξ) is a submanifold of M stable under the action of the stabilizer subgroupK_ξ. We see then that the reduced space

(11) M_ξ:= Φ⁻¹_K (ξ)/K_ξ

is a symplectic orbifold.

Let T ⊂ K be a maximal torus with Lie algebra t. We consider the lattice ∧ :=

1

2πker(exp :t →T) and the dual lattice ∧^∗ ⊂t^∗ defined by ∧^∗ = hom(∧,Z). We remark

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that iη is a differential of a character of T if and only if η ∈ ∧^∗. The Q-vector space generated by the lattice ∧^∗ is denoted by t^∗_Q: the vectors belonging to t^∗_Q are designed as rational. An affine subspace V ⊂ t^∗ is called rational if it is affinely generated by its rational points.

We also fix a closed positive Weyl chambert^∗_≥0. For each relatively open faceσ⊂t^∗_≥0, the stabilizer K_ξ of points ξ ∈ σ under the coadjoint action, does not depend on ξ, and will be denotedK_σ. The Lie algebra g_σ decomposes into its semi-simple and central parts k_σ = [k_σ,k_σ]⊕z_σ. The subspacez^∗_σ is defined to be the annihilator of [k_σ,k_σ], or equivalently the fixed point set of the coadjointK_σ action. Notice thatz^∗_σ is a rational subspace oft^∗, and that the faceσ is an open cone ofz^∗_σ,

We suppose that the moment map Φ_K is proper. The Convexity Theorem [1, 10, 16, 34, 22] tells us that ∆_K(M) := Image(Φ_K)T

t^∗_≥0 is a closed, convex, locally polyhedral set.

Definition 2.7 We denote by ∆K(M)⁰ the subset of ∆K(M) formed by the weakly- regular valuesof the moment map Φ_K contained in ∆_K(M).

We will use the following remark in the next sections.

Lemma 2.8 The subset ∆_K(M)⁰∩t^∗_Q is dense in ∆_K(M).

Proof : Let us first explain why ∆_K(M)⁰ is a dense open subset of ∆_K(M). There exists a unique open faceτ of the Weyl chambert^∗_≥0 such as ∆_K(M)∩τ is dense in ∆_K(M) : τ is called the principal face in [22]. The Principal-cross-section Theorem [22] tells us that Y_τ := Φ⁻¹(τ) is a symplectic K_τ-manifold, with a trivial action of [K_τ, K_τ]. The line bundle L_τ := L|_Y_τ prequantizes the symplectic structure on Y_τ, and relations (10) show that [K_τ, K_τ] acts trivially on Lτ. Moreover the restriction of Φ_K on Y_τ is the moment map Φ_τ :Y_τ →z^∗_τ associated to the action of the torusZ_τ = exp(z_τ) onLτ.

LetI ⊂z^∗_τ be the smallest affine subspace containing ∆K(M). LetzI ⊂zτ be the annihilator of the subspace parallel toI : relations (10) show thatz_I is the generic infinitesimal stabilizer of the z_τ-action onY_τ. Hence z_I is the Lie algebra of the torusZ_I := exp(z_I).

We see then that any regular value of Φτ :Yτ → I, viewed as a map with codomain I, is a weakly-regular value of the moment map Φ_K. It explains why ∆_K(M)⁰ is a dense open subset of ∆_K(M).

The convex set ∆K(M)∩τ is equal to ∆Zτ(Yτ) := Image(Φτ), it is sufficient to check that ∆_Z_τ(Y_τ)⁰∩t^∗_Q is dense in ∆_Z_τ(Y_τ). The subtorus Z_I ⊂Z_τ acts trivially onY_τ, and it acts on the line bundleLτ through a characterχ. Letη∈ ∧^∗∩t^∗_τ such thatdχ=iη|zI. The affine subspaceI which is equal to η+ (zI)^⊥ is rational. Since the open subset ∆Zτ(Yτ)⁰ generates the rational affine subspace I, we can conclude that ∆_Z_τ(Y_τ)⁰∩t^∗_Q is dense in

∆_Z_τ(Y_τ). ✷

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2.4 Weinstein’s theorem

Let ˜a∈C˜hol. Consider the Hamiltonian action of the group Gon the coadjoint orbit ˜G˜a.

The moment map Φ^˜â_G : ˜Gã→ g^∗ corresponds to the restriction of the projection πg,˜g to G˜ã. In this setting, the following conditions holds :

1. The action ofGon ˜G˜ais proper.

2. The moment map Φ^˜^a_Gis a proper map since the maphΦ^˜^a_G, ziis proper (see the second point of Proposition 2.2).

Conditions 1. and 2. imposes that the image of Φ^˜^a_G is contained in the open subset g^∗_se of strongly elliptic elements [30]. Thus, theG-orbits contained in the image of Φ^˜^a_G are parametrized by the following subset of the holomorphic chamberChol :

∆_G( ˜G˜a) := Image(Φ^˜^a_G) \ t^∗_≥0. We notice that Π_hol( ˜G, G) =S

˜

a∈C˜_hol{˜a} ×∆_G( ˜G˜a).

Like in Definition 2.6, an element ξ ∈ g^∗ is a weakly-regular value of Φ^˜â_G if for all m ∈ (Φâ_G^˜)⁻¹(ξ) we have dimg_m = min_x∈_G˜_˜_adim(g_x). We denote by ∆_G( ˜Gã)⁰ the set of elements ξ∈∆_G( ˜Gã) that are weakly-regular for Φ^˜â_G.

Theorem 2.9 (Weinstein) For any ˜a ∈ C˜_hol, ∆G( ˜G˜a) is a closed convex subset con- tained in Chol.

Proof : We recall briefly the arguments of the proof (see [37] or [30][§2]). Under Conditions 1. and 2., one checks easily that Y_˜_a := (Φ^˜^a_G)⁻¹(k^∗) is a smooth K-invariant symplectic submanifold of ˜G˜asuch that

(12) G˜˜a≃G×_KY_˜_a.

The moment map Φâ_K^˜ :Y_˜_a→k^∗, which corresponds to the restriction of the map Φ^˜â_GtoY_a_˜, is a proper map. Hence the Convexity Theorem tells us that ∆_K(Y_˜_a) := Image(Φ^˜â_K)T

t^∗_≥0 is a closed, convex, locally polyhedral set. Thanks to the isomorphism (12) we see that

∆_G( ˜G˜a) coincides with the closed convex subset ∆_K(Y_˜_a). The proof is completed. ✷ The next Lemma is used in §4.1.

Lemma 2.10 Letã∈C˜_hol be a rational element. Then∆_G( ˜Gã)⁰∩t^∗_Qis dense in∆_G( ˜Gã).

Proof : Thanks to (12), we know that ∆_G( ˜G˜a) = ∆_K(Y_˜_a). Relation (12) shows also that

∆_G( ˜G˜a)⁰ = ∆_K(Y_a_˜)⁰. Let N ≥1 such that ˜µ=Na˜∈ ∧^∗∩ Chol. The stabilizer subgroup G˜_µ_˜ is compact, equal to ˜K_µ_˜. Let us denote byC_µ_˜ the one-dimensional representation of K˜µ˜ associated to ˜µ. The convex set ∆G( ˜G˜a) is equal to _N¹∆G( ˜G˜µ), so it is sufficient to check that ∆_G( ˜G˜µ)⁰∩t^∗_Q = ∆_K(Y_µ_˜)⁰∩t^∗_Q is dense in ∆_G( ˜Gµ) = ∆˜ _K(Y_µ_˜). The coadjoint orbit ˜G˜µ is prequantized by the line bundle ˜G×Kµ˜ C_µ_˜ and the symplectic slice Y_µ_˜ is prequantized by the line bundleLµ˜:= ˜G×Kµ˜C_µ_˜|Yµ˜. Thanks to Lemma 2.8, we know that

∆_K(Y_µ_˜)⁰∩t^∗_Q is dense in ∆_K(Y_µ_˜) : the proof is complete. ✷

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3 Holomorphic discrete series

3.1 Definition

We return to the framework of §2.1. We recall the notion of holomorphic discrete series representations associated to a Hermitian symmetric spacesG/K. Let us introduce

C^ρ_hol:=

ξ ∈t^∗_≥0|(ξ, β)≥(2ρ_n, β), ∀β ∈R^+,z_n where 2ρ_n =P

β∈R^+,zn β isW-invariant.

Lemma 3.1 1. We have C_hol^ρ ⊂ Chol.

2. For any ξ ∈ Chol there exists N ≥1 such that N ξ ∈ C_hol^ρ .

Proof : The first point is due to the fact that (β₀, β₁) ≥ 0 for any β₀, β₁ ∈ R^+,z_n . The second point is obvious. ✷

We will be interested in the following subset of dominants weights : Gb_hol:=∧^∗₊\

C_hol^ρ .

Let Sym(p) be the symmetric algebra of the complexK-module (p,ad(z)).

Theorem 3.2 (Harish-Chandra) For anyλ∈Gb_hol, there exists an irreducible unitary representation ofG, denoted V_λ^G, such that the vector space ofK-finite vectors isV_λ^G|_K :=

V_λ^K⊗Sym(p).

The set V_λ^G, λ ∈ Gb_hol corresponds to the holomorphic discrete series representations associated to the complex structure ad(z).

3.2 Restriction

We come back to the framework of§2.2. We consider two compatible Hermitian symmetric spaces G/K ⊂ G/˜ K˜, and we look after the restriction of holomorphic discrete series representations of ˜Gto the subgroupG.

Let ˜λ∈Gb˜_hol. Since the representationV_˜^G^˜

λ is discretely admissible relatively the circle group exp(Rz), it is also discretely admissible relatively to G. We can be more precise [15, 24, 21] :

Proposition 3.3 We have an Hilbertian direct sum V_˜^G^˜

λ |G= M

λ∈Gbhol

m^λ_˜

λ V_λ^G, where the multiplicitym^λ_˜

λ := [V_λ^G:V_˜^G^˜

λ ]is finite for any λ.

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The Hermitian ˜K-vector space ˜p, when restricted to theK-action, admits an orthogonal decomposition ˜p=p⊕q. Notice that the symmetric algebra Sym(q) is an admissible K-module.

In [15], H.P. Jakobsen and M. Vergne obtained the following nice characterization of the multiplicities [V_λ^G :V_˜^G^˜

λ ]. Another proof is given in [30],§4.4.

Theorem 3.4 (Jakobsen-Vergne) Let (˜λ, λ)∈Gb˜_hol×Gb_hol. The multiplicity [V_λ^G:V_˜^G^˜

λ ] is equal to the multiplicity of the representation V_λ^K in Sym(q)⊗V_˜^K^˜

λ |_K. 3.3 Discrete analogues of Πhol( ˜G, G)

We define the following discrete analogues of the cone Π_hol( ˜G, G):

(13) Π^Z_hol( ˜G, G) :=n

(˜λ, λ)∈Gb˜_hol×Gb_hol ; [V_λ^G:V_λ_˜^G^˜]6= 0o . and

(14) Π^Q_hol( ˜G, G) :=n

(˜ξ, ξ)∈C˜hol× Chol ; ∃N ≥1, (N ξ, Nξ)˜ ∈Π^Z_hol( ˜G, G)o . We have the following key fact.

Proposition 3.5

• Π^Z_hol( ˜G, G) is a subset of ∧˜^∗× ∧^∗ stable under the addition.

• Π^Q_hol( ˜G, G) is a Q-convex cone of theQ-vector space˜t^∗_Q×t^∗_Q.

Proof : Suppose that a1 := (˜λ1, λ1) and a2 := (˜λ2, λ2) belongs to Π^Z_hol( ˜G, G). Thanks to Theorem 3.4, we know that theK-modules Sym(q)⊗(V_λ^K

j)^∗ ⊗V_˜^K^˜

λj|K possess a non-zero invariant vectorφ_j, forj= 1,2.

Let X := K/T ×K/˜ T˜ be the product of flag manifolds. The complex structure is normalized so that T_([e],[˜_e])X≃ n₋⊕˜n₊, where n₋ =P

α<0(k_C)_α and ˜n₊ =P

˜

α>0(˜k_C)_α_˜. We associate to each dataa_j, the holomorphic line bundleLj :=K×T C_−λ_j⊠K˜×T˜C₋_λ_˜

j

on X. Let H⁰(X,Lj) be the space of holomorphic sections of the line bundle Lj. The Borel-Weil Theorem tell us thatH⁰(X,L_j)≃(V_λ^K_j)^∗⊗V_˜^K^˜

λj|_K,∀j ∈ {1,2}.

We have φ_j ∈

Sym(q)⊗H⁰(X,Lj)K

,∀j, and then φ₁φ₂∈Sym(q)⊗H⁰(X,L1⊗ L2) is a non-zero invariant vector. Hence [Sym(q)⊗(V_λ^K₁_+λ₂)^∗⊗V_˜^K^˜

λ1+˜λ2|K]^K 6= 0. Thanks to Theorem 3.4, we can conclude that a₁ +a₂ = (˜λ₁+ ˜λ₂, λ₁ +λ₂) belongs to Π^Z_hol( ˜G, G).

The first point is proved. From the first point, one checks easily that - Π^Q_hol( ˜G, G) is stable under addition,

- Π^Q_hol( ˜G, G) is stable by expansion by a non-negative rational number.

The second point is settled. ✷

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3.4 Riemann-Roch numbers We come back to the framework of§2.3.

We associate to a dominant weight µ∈ ∧^∗₊ the (possibly singular) symplectic reduced spaceM_µ:= Φ⁻¹_K (µ)/K_µ, and the (possibly singular) line bundle over M_µ :

L_µ:=

L|_Φ⁻¹

K(µ)⊗C_−µ /K_µ.

Suppose first thatµ is a weakly-regular value of ΦK. ThenMµis an orbifold equipped with a symplectic structure Ω_µ, and Lµ is a line orbi-bundle over M_µ that prequantizes the symplectic structure. By choosing an almost complex structure on M_µ compatible with Ωµ, we get a decomposition∧T^∗Mµ⊗C=⊕i,j∧^i,jT^∗Mµof the bundle of differential forms. Using Hermitian structure in the tangent bundleTM_µ of M_µ, and in the fibers of Lµ, we define a Dolbeaut-Dirac operator

D⁺_µ :A^0,+(M_µ,Lµ)−→ A^0,−(M_µ,Lµ) whereA^i,j(Mµ,Lµ) = Γ(Mµ,∧^i,jT^∗Mµ⊗ Lµ).

Definition 3.6 Let µ ∈ ∧^∗₊ be a weakly-regular value of the moment map Φ_K. The Riemann-Roch number RR(M_µ,Lµ) ∈ Z is defined as the index of the elliptic operator D_µ⁺: RR(M_µ,Lµ) = dim(ker(D⁺_µ))−dim(coker(D_µ⁺)).

Suppose that µ /∈∆_K(M). ThenM_µ=∅and we take RR(M_µ,Lµ) = 0.

Suppose now that µ∈∆_K(M) is not (necessarily) a weakly-regular value of Φ_K. Take a small elementǫ∈t^∗such thatµ+ǫis a weakly-regular value of Φ_K belonging to ∆_K(M).

We consider the symplectic orbifoldM_µ+ǫ: if ǫis small enough, L_µ,ǫ:=

L|_Φ⁻¹

K (µ+ǫ)⊗C_−µ

/K_µ+ǫ. is a line orbi-bundle overMµ+ǫ .

We have the following important result (see §3.4.3 in [33]).

Proposition 3.7 Let µ∈∆_K(M)∩ ∧^∗. The Riemann-Roch number RR(M_µ+ǫ,Lµ,ǫ) do not depend on the choice of ǫ small enough so that µ+ǫ ∈ ∆_K(M) is a weakly-regular value ofΦ_K.

We can now introduce the following definition.

Definition 3.8 Let µ∈ ∧^∗₊. We define Q(M_µ,Ω_µ) =

(0 if µ /∈∆_K(M), RR(M_µ+ǫ,Lµ,ǫ) if µ∈∆_K(M).

Above,ǫ is chosen small enough so thatµ+ǫ∈∆_K(M) is a weakly-regular value of Φ_K.

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Let n ≥ 1. The manifold M, equipped with the symplectic structure nΩ, is prequantized by the line bundle L^⊗n: the corresponding moment map is nΦ_K. For any dominant weightµ∈ ∧^∗₊, the symplectic reduction of (M, nΩ) relatively to the weight nµ is (M_µ, nΩ_µ). Like in definition 3.8, we consider the following Riemann-Roch numbers

Q(Mµ, nΩ_µ) =

(0 if µ /∈∆K(M),

RR(M_µ+ǫ,(Lµ,ǫ)^⊗n) if µ∈∆_K(M) andkǫk<<1.

Kawasaki-Riemann-Roch formula shows thatn≥17→ Q(Mµ, nΩ_µ) is a quasi-polynomial map [36, 23]. When µ is a weakly-regular value of Φ_K, we denote by vol(M_µ) :=

1 dµ

R

Mµ(^Ω_2π^µ)^dim²^Mµ the symplectic volume of the symplectic orbifold (M_µ,Ω_µ). Here d_µ is the generic value of the mapm∈Φ⁻¹_K (µ)7→cardinal(K_m/K_m⁰).

The following proposition is a direct consequence of Kawasaki-Riemann-Roch formula (see [23] or§1.3.4 in [29]).

Proposition 3.9 Let µ ∈ ∆_K(M) ∩ ∧^∗₊ be a weakly-regular value of Φ_K. Then we have Q(Mµ, nΩ_µ) ∼ vol(M_µ)n^dim²^Mµ, when n → ∞. In particular the map n ≥ 1 7→

Q(Mµ, nΩ_µ) is non-zero.

3.5 Quantization commutes with reduction

Let us explain the “Quantization commutes with Reduction” Theorem proved in [30].

We fix ˜λ∈Gb˜_hol. The coadjoint orbit ˜Gλ˜is prequantized by the line bundle ˜G×Kλ˜C_˜_λ, and the moment map Φ^˜^λ_G: ˜G˜λ→g^∗ corresponding to the G-action on ˜G×K_λ_˜ C_˜_λ is equal to the restriction of the mapπ_g,˜_g to ˜Gλ.˜

The symplectic sliceY_λ_˜ = (Φ^λ_G^˜)⁻¹(k^∗) is prequantized by the line bundleLλ˜ := ˜G×Kλ˜

C_λ_˜|Y_λ_˜. The moment map Φ^λ_K^˜ : Y_λ_˜ → k^∗ corresponding to the K-action is equal to the restriction of Φ^˜^λ_G to Y_λ_˜.

For anyλ∈Gb_hol, we consider the (possibly singular) symplectic reduced space X_λ,λ_˜ := (Φ^˜^λ_K)⁻¹(λ)/K_λ,

equipped with the reduced symplectic form Ω_λ,λ_˜ , and the (possibly singular) line bundle L_˜_λ,λ:=

Lλ˜|_(Φ^˜λ

K)⁻¹(λ)⊗C_−λ /K_λ.

Thanks to Definition 3.8, the geometric quantization Q(X˜λ,λ,Ω_˜_λ,λ)∈Zof those compact symplectic spaces (Xλ,λ˜ ,Ω˜λ,λ) are well-defined even if they are singular. In particular Q(Xλ,λ˜ ,Ω_λ,λ_˜ ) = 0 when X_˜_λ,λ:=∅.

The following Theorem is proved in [30].

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Theorem 3.10 Let λ˜∈Gb˜_hol. We have an Hilbertian direct sum V_˜_λ^G^˜|G= M

λ∈Gbhol

Q(X˜λ,λ,Ω_λ,λ_˜ ) V_λ^G,

It means that for anyλ∈Gb_hol, the multiplicity of the representation V_λ^G in the restriction V_˜^G^˜

λ |G is equal to the geometric quantization Q(Xλ,λ˜ ,Ω_λ,λ_˜ ) of the (compact) reduced space X_λ,λ_˜ .

Remark 3.11 Let (˜λ, λ)∈Gb˜_hol×Gb_hol. Theorem 3.10. shows that hV_nλ^G :V_n^G^˜_λ_˜i

=Q(Xλ,λ˜ , nΩ˜λ,λ) for any n≥1.

4 Proofs of the main results

We come back to the setting of §2.2: G/K is a complex submanifold of a Hermitian symmetric space ˜G/K˜. It means that there exits a ˜K-invariant element z ∈k such that ad(z) defines complex structures on ˜p and p. We consider the orthogonal decomposition

˜p = p⊕q, and we denote by Sym(q) the symmetric algebra of the complex K-module (q,ad(z)).

4.1 Proof of Theorem A The set Π_hol( ˜G, G) is equal to S

˜

a∈C˜hol{˜a} ×∆_G( ˜G˜a). We define Π_hol( ˜G, G)⁰ := [

˜ a∈C˜_hol

{˜a} ×∆_G( ˜G˜a)⁰. We start with the following result.

Lemma 4.1 The setΠ_hol( ˜G, G)⁰T˜t^∗_Q×t^∗_Q is dense in Π_hol( ˜G, G).

Proof : Let (˜ξ, ξ)∈Π_hol( ˜G, G): take ˜g∈G˜ such thatξ =π_g,˜_g(˜gξ). We consider a sequence˜ ξ˜_n ∈ C˜hol ∩˜t^∗_Q converging to ˜ξ. Then ξ_n := π_g,˜_g(˜gξ˜_n) is a sequence of C_G/K⁰ converging to ξ ∈ Chol. Since the map p :C_G/K⁰ → Chol is continuous (se Lemma 2.3), the sequence η_n:= p(ξ_n) converges to p(ξ) =ξ. By definition, we have η_n ∈∆_G( ˜Gξ˜_n) for any n∈N.

Since ˜ξn are rational, each subset ∆G( ˜Gξ˜n)⁰∩t^∗_Q is dense in ∆G( ˜Gξ˜n) (see Lemma 2.10).

Hence,∀n∈N, there exists ζ_n ∈∆_G( ˜Gξ˜_n)⁰∩t^∗_Q such that kζn−η_nk ≤2⁻ⁿ. Finally, we see that (˜ξ_n, ζ_n) is a sequence of rational elements of Π_hol( ˜G, G)⁰ converging to (ξ,ξ).˜ ✷

The main purpose of this section is the proof of the following theorem.

Horn problem for quasi-hermitian Lie groups

HAL Id: hal-02867106

https://hal.archives-ouvertes.fr/hal-02867106

Horn problem for quasi-hermitian Lie groups

Paul-Emile Paradan

To cite this version:

Horn problem for quasi-hermitian Lie groups

Contents

1 Introduction

2 The cone Π

( ˜ G, G ) : first properties

3 Holomorphic discrete series

4 Proofs of the main results