Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

In this paper, we prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results from [KO13,KS16] on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. In order to deduce this application we prove relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group $G$ defined over $\mathbb{R}$, the space of $G(\mathbb{R})$-equivariant distributions on the manifold of real points of any algebraic $G$-manifold $X$ is finite-dimensional if $G$ has finitely many orbits on $X$.

1.1. The relative character. In this paper, we prove that a relative character (a.k.a. spherical character) of a smooth admissible Fréchet representation of moderate growth of a real reductive group is holonomic. The relative character is a basic notion of relative representation theory that generalizes the notion of a character of a representation. By a real reductive group we mean a connected algebraic reductive group defined over R. Unless confusion is possible, we will not distinguish between such a group and the group of its real points. Let us now recall the notions of spherical pair, relative character and holonomic distribution. For the notion of smooth admissible Fréchet representation of moderate growth we refer the reader to [Cas89] or [Wal88,Chapter 11].
Definition 1.1.1. Let G be a real reductive group and let H ⊂ G be its (algebraic) subgroup. Let P denote a minimal parabolic subgroup of G and B denote a Borel subgroup of the complexification G C . The subgroup H is called real spherical if it has finitely many orbits on G/P and spherical if its complexification has finitely many orbits on G C /B.
It seems that this theorem is not found in the literature in this formulation, however it has two proofs, one due to Kashiwara (see [Kas74;KK76] for similar statements) and another due to Bernstein (unpublished).
In order to make our applications in representation theory more precise, we need an effective version of this theorem. We prove such a version (see Theorem 3.2.2 below) following Bernstein's approach, as it is more appropriate for effective bounds. We use this effective version to derive a relative version. Namely, we show that if the system depends on a parameter in an algebraic way, then the dimension of the space of solutions is bounded (see §3.3 below).
This relative version allows us to deduce the following theorem.
Theorem D (See §3.3). Let a real algebraic group G act on a real algebraic manifold X with finitely many orbits. Let g be the Lie algebra of G. Let E be an algebraic G-equivariant bundle on X. Then, for any natural number n ∈ N, there exists C n ∈ N such that for every n-dimensional representation τ of g we have dim Hom g (τ, S * (X, E)) ≤ C n , where Hom g denotes the space of all continuous g-equivariant maps. dim Hom h (π, τ ) < ∞.
(ii) If the diagonal ∆H is a spherical subgroup in G×H then the multiplicities are universally bounded, i.e., there exists C ∈ N such that for every π ∈ Irr(G), τ ∈ Irr(H) we have dim Hom h (π, τ ) ≤ C.
This corollary follows from Theorem E since Hom h (π, τ ) lies in the space of ∆h-invariant functionals on the completed tensor product π ⊗τ ∈ Irr(G × H) (see [AG09a, Corollary A.0.7 and Lemma A.0.8]). All symmetric pairs satisfying the conditions of the corollary were classified in [KM14].
The inverse implications for Theorem E and Corollary F are proven in [KO13]. An advantage of Theorem E(i) over [KO13;KS] is that C n does not depend on τ . On the other hand, the results on multiplicities in [KO13;KS] are slightly stronger than Theorem E since they allow H to be any closed Lie subgroup and consider maps from the Harish-Chandra space of π to τ . In addition, [KO13,Theorem B] implies that if H ⊂ G is an algebraic spherical subgroup there exists C ∈ N such that dim Hom h (π, τ ) ≤ C dim τ, for every π ∈ Irr(G) and every finite-dimensional continuous representation τ of H. It is easy to modify our proof of Theorem E(ii) to show the boundedness of multiplicities for any π ∈ Irr(G) and any τ of a fixed dimension, but the proof that the bound depends linearly on this dimension would require more work.
Our methods are different from the methods of [KO13], which in turn differ from the ones of [KS], and the bounds given in the three works are probably very different. . This gives us hope that it can be extended to the non-Archimedean case. The main difficulty is the fact that our proof heavily relies on the theory of modules over the ring of differential operators, which does not act on distributions in the non-Archimedean case. However, in view of Theorem 1.4.1 we believe that this difficulty can be overcome. Namely, one can deduce an analog of Theorem E(ii) for many spherical pairs from the following conjecture .
Conjecture 1.6.1. Let G be a reductive group defined over a non-Archimedean field F of characteristic 0 and let H 1 , H 2 ⊂ G be its (algebraic) spherical subgroups. Let χ i be characters of H i . Fix a character λ of the Bernstein center z(G).
Note that Theorem B and Theorem 1.4.1(i) imply that the dimension of (the Zariski closure of) the wave front set of a distribution that satisfies (1-3) does not exceed dim G. In many ways the wave front set replaces the singular support, in absence of the theory of differential operators (see, e.g., [Aiz13; AD; AGS; AGK]). Thus, in order to prove Conjecture 1.6.1, it is left to prove analogs of Theorems 1.2.1 and 3.2.2 for the integral system of equations (1-3).
1.7. Structure of the paper. In §2, we prove Theorem B using a theorem of Steinberg [Ste76] concerning the Springer resolution.
In §3, we prove an effective version of Theorem 1.2.1, and then adapt it to algebraic families. We also derive Theorem D.
In §4, we derive Theorem E from Theorem B and §3. We do that by embedding the multiplicity space into a certain space of relative characters.
In Appendix A, we prove Lemma 3.1.1 which computes the pullback of the D-module of distributions with respect to a closed embedding. We use this lemma in §3.
1.8. Acknowledgements. We thank Eitan Sayag and Bernhard Kroetz for fruitful discussions. We thank Joseph Bernstein for telling us the sketch of his proof of Theorem 1.2.1. The three authors were partially supported by the Minerva foundation with funding from the Federal German Ministry for Education and Research; A.A. was also partially supported by ISF grant 687/13, and D.G. and A.M. by ISF grant 756/12; D.G. was also supported in part by the ERC grant 291612.

PROOF OF THEOREM B
It is enough to prove the theorem for a reductive group G defined over an algebraically closed field of characteristic 0. Since S includes the zero section of T * G ∼ =G × g * , we have dim S ≥ dim G. Thus, it is enough to prove that dim S ≤ dim G. Let B denote the flag variety of G and N ⊂ g * denote the nilpotent cone. Since G is reductive, we can identify Recall the Springer resolution µ : T * B → N defined by µ(B, X) = X and consider the following diagram. (1) Here, α is defined by α(g, X) = (X, Ad * (g −1 )X), and res is the restriction. Passing to the fiber of 0 ∈ h * 1 × h * 2 , we obtain the following diagram. (2) We need to estimate dim S. We do it using the following lemma.
. By this lemma, applied to φ 1 = α ′ and φ 2 = µ ′ , it is enough to estimate the dimensions of L i and of the fibers of µ ′ and α ′ .
Proof. Since H i has finitely many orbits in B, it is enough to show that L i is the union of the conormal bundles to the orbits of H i in B. Let B ∈ B, and b = LieB, and identify is isomorphic to the stabilizer G η , and the dimension of the fiber (µ ′ ) −1 (η, Ad * (g)η) is twice the dimension of the Springer fiber µ −1 (η). Recall the following theorem of Steinberg (conjectured by Grothendieck): Using Lemma 2.0.1, we obtain for some (η, ad * (g)η): an open dense subset such that ϕ −1 2 (V ) intersects those and only those irreducible components C 1 , . . . , C j of ϕ −1 2 (W ) that map dominantly to W . Note that j > 0 since ϕ 2 is surjective. Moreover, without loss of generality, we may assume that, for every 1 ≤ i ≤ j, all fibers over V of the restriction of ϕ 2 to C i are of the same dimension. Since one of these dimensions has to be equal to dim ϕ −1 . Thanks to dim V = dim W , taking any y ∈ V , formulas (3) and (4) imply the statement.

DIMENSION OF THE SPACE OF SOLUTIONS OF A HOLONOMIC SYSTEM
In this section, we prove an effective version of Theorem 1.2.1, and then adapt it to algebraic families. We also derive Theorem D.

Preliminaries.
3.1.1. D-modules. In this section, we will use the theory of D-modules on algebraic varieties over an arbitrary field k of characteristic zero. We will now recall some facts and notions that we will use. For a good introduction to the algebraic theory of D-modules, we refer the reader to [Ber] and [Bor87]. For a short overview, see [AG09a,Appendix B].
By a D-module on a smooth algebraic variety X we mean a coherent sheaf of right modules over the sheaf D X of algebras of algebraic differential operators. Denote the category of D Xmodules by M(D X ).
For a smooth affine variety V , we denote D(V ) := D V (V ). Note that the category M(D V ) of D-modules V is equivalent to the category of D(V )-modules. We will thus identify these categories.
The algebra D(V ) is equipped with a filtration which is called the geometric filtration and defined by the degree of differential operators. The associated graded algebra with respect to this fitration is the algebra O(T * V ) of regular functions on the total space of the cotangent bundle of V . This allows us to define the singular support of a D-module M on V in the following way. Choose a good filtration on M, i.e. a filtration such that the associated graded module is a finitely-generated module over O(T * V ), and define the singular support SS(M) to be the support of this module. One can show that the singular support does not depend on the choice of a good filtration on M.
This definition easily extends to the non-affine case. A D-module M on X is called smooth if SS(M) is the zero section of T * X. This is equivalent to being coherent over O X and to be coherent and locally free over O X . The Bernstein inequality states that, for any non-zero M, If V is an affine space then the algebra D(V ) has an additional filtration, called the Bernstein filtration. It is defined by deg(∂/∂x i ) = deg(x i ) = 1, where x i are the coordinates in V . This gives rise to the notion of Bernstein's singular support, that we will denote We will also use the theory of analytic D-modules. By an analytic D-module on a smooth complex analytic manifold X we mean a coherent sheaf of right modules over the sheaf D An X of algebras of differential operators with analytic coefficients. All of the above notions and statements, except those concerning the Bernstein filtration, have analytic counterparts. In addition, all smooth analytic D-modules of the same rank are isomorphic.
3.1.2. Distributions. We will use the theory of distributions on differentiable manifolds and the theory of tempered distributions on real algebraic manifolds, see e.g. [Hör90;AG08]. For a real algebraic manifold X, we denote the space of distributions on X by D ′ (X) := (C ∞ c (X)) * and the space of tempered distributions (a.k.a. Schwartz distributions) by S * (X) := (S(X)) * . Similarly, for an algebraic bundle E over X we denote D ′ (X, E) := (C ∞ c (X, E)) * and S * (X, E) := (S(X, E)) * . The spaces D ′ (X) and S * (X) form (right) D-modules over X. The space D ′ (X) is also an analytic D-module. We define the singular support of a distribution to be the singular support of the D-module it generates. It is well-known that this definition is equivalent to Definition 1.1.3. We say that a distribution is holonomic if it generates a holonomic D-module.
Lemma 3.1.1 (See Appendix A). Let i : X → Y be a closed embedding of smooth affine real algebraic varieties. Then Since M An is also smooth, M An ∼ = An(C n ) r . Thus it is left to prove that and the latter space is one-dimensional. This follows from the fact that a distribution with vanishing partial derivatives is a multiple of the Lebesgue measure.

Lie algebra actions.
Definition 3.1.4. Let X be an algebraic manifold over a field k and g be a Lie algebra over k.
(i) An action of g on X is a Lie algebra map from g to the algebra of algebraic vector fields on X. (ii) Assume that X is affine, fix an action of g on X and let E be an algebraic vector bundle on X. Let M be the space of global regular (algebraic) sections of E. An action of g on E is a linear map T : g → End k (M) such that, for any α ∈ g, f ∈ O(X), v ∈ M, we have (iii) The definition above extends to non-affine X in a straightforward way.

Weil representation.
Definition 3.1.5. Let V be a finite-dimensional real vector space. Let ω be the standard symplectic form on V ⊕ V * . Denote by p V : V ⊕ V * → V and p V * : V ⊕ V * → V * the natural projections. Define an action of the symplectic group Sp(V ⊕ V * , ω) on the algebra D(V ) by where v ∈ V, w ∈ V * , ∂ v denotes the derivative in the direction of v, and elements of V * are viewed as linear polynomials and thus differential operators of order zero. For a D(V )-module M and an element g ∈ Sp(V ⊕ V * ), we will denote by M g the D(V )-module obtained by twisting the action of D(V ) by π(g).
In fact, this corollary can be derived directly from the Stone-von-Neumann theorem.
is locally constant.  In this subsection, we prove We will need the following geometric lemmas Lemma 3.2.3. Let V be a vector space, L ⊂ V be a subspace and C ⊂ V be a closed conic algebraic subvariety such that L ∩ C = {0}. Then the projection p : C → V /L is a finite map.
Proof. By induction, it is enough to prove the case dim L = 1. Choose coordinates x 1 , . . . , x n on V such that the coordinates x 1 , . . . , x n−1 vanish on L. Let p be a homogeneous polynomial that vanishes on C but not on L. Proof. Let L denote the variety of all Lagrangian subspaces of W . Note that dim L = n(n + 1)/2. Let P (C) ⊂ P(W ) be the projectivizations of C and W . Consider the configuration space We have to show that p(X) = L where p : X → L is the projection. Let q : X → P (C) be the other projection. Note that dim q −1 (x) = n(n − 1)/2 for any x ∈ P (C). Thus dim X = n(n − 1)/2 + n − 1 < n(n + 1)/2 = dim L, and thus p : X → L cannot be onto.
. Thus it is enough to show that for any holonomic D-module N on an affine space A n we have dim Hom(N, S * (R n )) ≤ deg b (N).
Let C ⊂ A 2n be the singular support of N with respect to the Bernstein filtration. By Corollary 3.2.5, there exists g ∈ Sp 2n such that p| gC is a finite map, where p : A 2n → A n is the projection on the first n coordinates. By Corollary 3.1.8 we have dim Hom(N, S * (R n )) = dim Hom(N g , S * (R n ) g ) = dim Hom(N g , S * (R n )).
By Lemma 3.1.6 we have SS b (N g ) = gC. Let F be a good filtration on N g (with respect to the Bernstein filtration on D(A n )). We see that Gr N g is finitely generated over O(A n ), and thus so is N g . Thus N g is a smooth D-module. Note that rk O(A n ) N g ≤ deg b N g = deg b N. By Lemma 3.1.2 dim Hom(N g , S * (R n )) ≤ rk O(A n ) N g .

Families of D-modules.
In this section we discuss families of D-modules on algebraic varieties over an arbitrary field k of zero characteristic. Notation 3.3.1. Let φ : X → Y be a map of algebraic varieties and M be a quasi-coherent sheaf of O X -modules. For any y ∈ Y , denote by M y the pullback of M to φ −1 (y).
Definition 3.3.2. Let X, Y be smooth algebraic varieties.
• If X and Y are affine we define the algebra D(X, Y ) to be D(X) ⊗ k O(Y ).
• Extending this definition we obtain a sheaf of algebras D X,Y on X × Y .
• By a family of D X -modules parameterized by Y , we mean a sheaf of right modules over the sheaf of algebras D X,Y on X × Y which is quasicoherent as a sheaf of O X×Ymodules. • We call a family of D X -modules parameterized by Y coherent if it is locally finitely generated as a D X,Y -module. • For a family M of D X -modules parameterized by Y and a point y ∈ Y , we call M y the specialization of M at y and consider it with the natural structure of a D X -module. • We say that a coherent family M is holonomic if every specialization is holonomic. Proof. Without loss of generality, we can assume that X = A n and Y is an affine variety, and prove that deg b (M y ) is bounded. We will prove this by induction on dim Y . The Bernstein filtration on D(A n ) gives rise to a filtration on D(A n , Y ). Choose a filtration F on M which is good with respect to this filtration and let N := Gr M, considered as a graded O(A 2n × Y )-module. Associate to N a coherent sheaf N on P 2n−1 × Y . Let N y be the pullback of N under the embedding of P 2n−1 into P 2n−1 × Y given by x → (x, y). By definition, the Hilbert polynomial of M y with respect to the filtration induced by F is the Hilbert polynomial of N y . By Corollary 3.1.11, there exists an open dense subset U ⊂ Y such that the Hilbert polynomial of N y does not depend on y as long as y ∈ U. By the induction hypothesis, deg b (M y ) is bounded on Y \ U, and thus bounded on Y .
For an application of this theorem we will need the following lemma. Lemma 3.3.4. Let a real Lie algebra g act on a real algebraic manifold X and on an algebraic vector bundle E on X. Fix a natural number n and let Y be the variety of all representations of g on C n . Then there exists a coherent family M of D X -modules parameterized by Y such that, for any τ ∈ Y , we have (1) Hom g (τ, S * (X, E)) = Hom D X (M τ , S * (X)).
(2) The singular support of M τ (with respect to the geometric filtration) is included in Proof. It is enough to prove the lemma for affine X. Let N be the coherent sheaf of the regular (algebraic) sections of E (considered as a sheaf of O X -modules). Let N be the pullback of N to X × Y . Let N ′ := N ⊗ O X×Y D X,Y ⊗ C C n , and N ′′ ⊂ N ′ be the D X,Y -submodule generated by elements of the form where α ∈ g, ξ α is the vector field on X corresponding to α, and f α (v) ∈ D X,Y ⊗ C C n is the C n -valued regular function on X × Y given by f α (v)(x, τ ) = τ (α)v. Then M := N ′ /N ′′ satisfies the requirements.
Theorems 3.2.2 and 3.3.3 and Lemma 3.3.4 imply Theorem D.

PROOF OF THEOREMS A AND E
In this section, we derive Theorems A and E from Theorem B and §3. We do that by embedding the multiplicity space into a certain space of relative characters. 4.1. Preliminaries. For a real reductive group G, we denote by Irr(G) the collection of irreducible admissible smooth Fréchet representation of G of moderate growth. We refer to [Cas89;Wal88] for the background on these representations.  Proof of Proposition 1.1.4. Let ξ be a relative character of a smooth admissible Fréchet representation π of G of moderate growth with respect to a pair of subgroups (H 1 , H 2 ) and their characters χ 1 , χ 2 . By Theorem 4.1.1, there exists an ideal I ⊂ z(U(g)) of finite codimension that annihilates π and thus annihilates ξ. For any element z ∈ z(U(g)), there exists a polynomial p such that p(z) ∈ I and thus p(z)ξ = 0. This implies that the symbol of any z ∈ z(U(g)) of positive degree vanishes on the singular support of ξ. It is well-known that the joint zero-set of these symbols over each point g ∈ G is the nilpotent cone N (g * ). Since ξ is (h 1 × h 2 , χ 1 × χ 2 )-equivariant, this implies that the singular support of ξ lies in S.

4.3.
Proof of Theorem E. Part (i) follows immediately from Theorem D and the Casselman embedding theorem. If G is quasi-split then so does part (ii). For the proof of part (ii) in the general case, we will need the following lemma. Lemma 4.3.1. Let G be a real reductive group and H 1 , H 2 be spherical subgroups. Let Y = Spec(z(U(g)))×Y 1 × Y 2 , where Y i is the variety of characters of h i = Lie H i . For any λ ∈ Y (C), define U λ,χ 1 ,χ 2 := S * (G) h 1 ×h 2 ,(χ 1 ,χ 2 ),(z(U (g)),λ) to be the space of tempered distributions on G that are left χ 1 -equivariant with respect to h 1 , right χ 2 -equivariant with respect to h 2 and are eigendistributions with respect to the action of z(U(g)) with eigencharacter λ. Then dim U λ,χ 1 ,χ 2 is bounded over Y (C).
Proof. Let us construct a family of D(G)-modules M parameterized by Y . For any α ∈ g, let r α and l α be the corresponding right and left invariant vector fields on G considered as elements in D(G, Y ). For any β ∈ z(U(g)), α i ∈ h i , let f β , g i α i be the functions on Y that send (µ, γ 1 , γ 2 ) ∈ Y to µ(β), γ i (α i ) respectively. Let also d β be the differential operator on G corresponding to β, such that d β ξ = βξ for any distribution ξ on G. We consider d β , r α 1 , l α 2 , f β , g i α i as elements of D(G, Y ). Let I ⊂ D(G, Y ) be the ideal generated by r α 1 −g 1 α 1 , l α 2 −g 2 α 2 and f β −d β where α i ∈ h i and β ∈ z(U(g)). Define M := D(G, Y )/I.

Proof of Theorem E(ii).
We choose an involution θ as in Lemma 4.1.2, let H 1 := H, H 2 := θ(H), and define the spaces U λ as in Lemma 4.3.1. Now let π ∈ Irr(G) and let χ be a character of h such that (π * ) h,χ = 0. Let λ stand for the infinitesimal character of π. By Lemma 4.1.2, (π * ) dθ(h),dθ(χ) = 0. Fix a non-zero φ ∈ (π * ) dθ(h),dθ(χ) . Then φ defines an embedding (π * ) h,χ ֒→ U λ,χ,dθ(χ) by ψ → ξ ψ,φ , where ξ ψ,φ is the relative character, which is defined by ξ ψ,φ (f ) := ψ, π(f )φ . Thus, dim(π * ) h,χ ≤ dim U λ,χ,dθ(χ) , which is bounded by Lemma 4.3.1. APPENDIX A. PROOF OF LEMMA 3.1.1 For the proof, we will need the following standard lemmas. Let M be a smooth manifold and N ⊂ M be a closed smooth submanifold. Proof. Using partition of unity, it is enough to show that, for any f ∈ I N and x ∈ M, there exists f ′ ∈ J such that f coincides with f ′ in a neighborhood of x. For x / ∈ N this is obvious, so we assume that x ∈ N. We prove the statement by induction on the codimension d of N in M. The base case d = 1 follows, using the implicit function theorem, from the case N = R n−1 ⊂ R n = M, which is obvious.
For the induction step, take an element g ∈ J such that d x g = 0. Let Z := {y ∈ M | g(y) = 0} and U := {y ∈ M | d y g = 0}.
By the induction hypothesis,f ∈J. Thus, there exists f ′′ ∈ J such that f − f ′′ vanishes in a neighborhood of x in Z. Now, the case d = 1 implies that there exists α ∈ C ∞ c (M) such that f − f ′′ coincides with αg in a neighborhood of x. Let Y be a real algebraic manifold and X be a closed algebraic submanifold. Let i : X → Y denote the embedding.
Lemma A.0.3. Let ξ be a distribution on X such that i * ξ is a tempered distribution. Then ξ is a tempered distribution.
Proof. The map i * is dual to the pullback map C ∞ c (Y ) → C ∞ c (X). This can be extended to a continuous map i * : S(Y ) → S(X) which is onto by [AG08, Theorem 4.6.1]. The Banach open map theorem implies that i * is an open map. It is easy to see that i * ξ : S(Y ) → C vanishes on Ker(i * ), and thus it gives rise to a continuous map S(X) → C, which extends ξ.
Lemma A.0.4. Let ξ be a distribution on Y such that pξ = 0 for any polynomial p on Y that vanishes on X. Then ξ is a pushforward of a distribution on X.
Proof. Let J(X) be the ideal of all polynomials on Y that vanish on X. Let J := J(X)C ∞ c (Y ). By Lemma A.0.1 we have J = I X . Thus, ξ vanishes on I X and thus, by Lemma A.0.2, ξ is a pushforward of a distribution on X.