Holomorphic H-spherical distribution vectors in principal series representations
Simon Gindikin, Bernhard Kroetz, Gestur Olafsson
备注：
Due to a very careful and professional referee's work we could strongly improve on both exposition and detail. To appear in Invent. math., 35 pages, 1 figure
Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L^2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown $Ξ$. In this article we construct a minimal G-invariant subdomain $Ξ_H$ of $Ξ$ with G/H as Shilov boundary. Let $π$ be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of $π$, which admit a holomorphic extension to $Ξ_H$, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L^2(G/H)_mc in a space of holomorphic functions.
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is described in a systematic way.
Let $X=H/L$ be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain $D=G/K$. The intersection $S$ of the Shilov boundary of $D$ with $X$ defines a distinguished subset of the topological boundary of $X$ and is invariant under $H$ and can also be realized as $S=H/P$ for certain parabolic subgroup $P$ of $H$. We study the spherical representations $Ind_P^H(\lam)$ of $H$ induced from $P$. We find formulas for the spherical functions in terms of the Macdonald ${}_2F_1$ hypergeometric function. This generalizes the earlier result of Faraut-Koranyi fo…
For infinite reductive groups with Frobenius maps, we show that certain subquotients of abstract representations of the groups induced from 1-dimensional representations of Borel subgroups are irreducible.
Let $\mathbb D=G/K$ be a complex bounded symmetric domain of tube type in a Jordan algebra $V_{\mathbb C}$, and let $D=H/L =\mathbb D\cap V$ be its real form in a Jordan algebra $V\subset V_{\mathbb C}$. The analytic continuation of the holomorphic discrete series on $\mathbb D$ forms a family of interesting representations of $G$. We consider the restriction on $D$ of the scalar holomorphic representations of $G$, as a representation of $H$. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group $L$ is a spherical subgroup of $K$ and we …
The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmertic domains.
For appropriate $N\ge 3$ and $d<0,$ the moduli space of principally polarized abelian surfaces with level $N$ structure and anti-holomorphic multiplication by $\mathcal O_d$ (the ring of integers in $\mathbb Q(\sqrt{d})$) is shown to consist of the real points of a quasi-projective algebraic variety defined over $\mathbb Q$, and to coincide with finitely many copies of the quotient of hyperbolic 3-space by the principal congruence subgroup of level $N$ in $\mathbf{SL}(2, \mathcal O_d).$
We compute generalized Bernstein-Reznikov integrals associated with standard complex symplectic forms by studying Knapp-Stein intertwining operators between spherical degenerate principal series of complex symplectic groups.
Let $(G,K)$ be one of the following classical irreducible Hermitian symmetric pairs of noncompact type: $(SU(p,q), S(U(p) \times U(q))),(Sp(n,R), U(n))$, or $(SO*(2n), U(n))$. Let $G_{\mathbb C}$ and $K_{\mathbb C}$ be complexifications of $G$ and $K$, respectively, and let $P$ be a maximal parabolic subgroup of $G_{\mathbb C}$ whose Levi subgroup is $K_{\mathbb C}$. Let $V$ be the holomorphic part of the complexifiaction of the tangent space at the origin of $G/K$. It is well known that the ring of $K_{\mathbb C}$-invariant differential operators on $V$ has a generating system $\{\varGamma_k …
For Iwahori-spherical representations of non-Archimedean general linear groups, Chan-Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori-Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan's formula for the Harish-Chandra--Howe character as a Grothendieck group expansion of the same module. Comparing the two approaches through the lens of Zelevinsky's PSH-algebras, we obtain an explicit unitriangular transition matrix between coefficients of the character ex…
By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of ${\rm GL}(n,\mathbb{R})$ an…
In this article we give a precise description of the Plancherel decomposition of the most continuous part of $L^{2}(Z)$ for a real spherical homogeneous space $Z$. Our starting point is the recent construction of Bernstein morphisms by Delorme, Knop, Krötz and Schlichtkrull. The most continuous part decomposes into a direct integral of unitary principal series representations. We give an explicit construction of the $H$-invariant functionals on these principal series. We show that for generic induction data the multiplicity space equals the full space of $H$-invariant functionals. Finally, we …
A holomorphic discrete series can be realized as the space of the holomorphic sections of a homogeneous vector bundle on a bounded symmetric domain. We can embed it into a degenerate principal series realized as the space of hyperfunction sections of a vector bundle on the Shilov boundary by taking the boundary value. We consider an algebraic version of higher cohomological analogue of such an embedding for complex reductive groups and classical groups of type A.
We find the complete branching law for the restriction of certain unitary representations of $O(1,n+1)$ to the subgroups $O(1,m+1)\times O(n-m)$, $0\leq m\leq n$. The unitary representations we consider belong either to the unitary spherical principal series, the spherical complementary series or are unitarizable subquotients of the spherical principal series. In the crucial case $0<m<n$ the decomposition consists of a continuous part and a discrete part. The continuous part is given by a direct integral of unitary principal series representations whereas the discrete part consists of finitely…
We say a representation V of a group G has stability if its multiplicities m^{G}_{V}(λ) is dependent only on some equivalence class of λfor a sufficiently large parameter λ. In this paper, we prove that the restriction of a holomorphic discrete series representation with respect to any holomorphic symmetric pairs has stability. As a corollary, we give a necessary and sufficient condition on multiplicity-freeness of the branching laws in this setting. This condition is same as the sufficient condition given by the theory of visible actions. We prove a general theorem before we show the stabilit…
Let $\mathcal{G}$ be a split reductive $p$-adic group with connected centre. We show that each Bernstein block in the principal series of $\mathcal{G}$ admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form $T//W$ where $T$ is a maximal torus in the Langlands dual group of $\mathcal{G}$ and $W$ is the Weyl group of $\mathcal{G}$.
We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p--adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p--adic type.
We~identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in~the sense of Shimura, on~bounded symmetric domains. This also yields a description of the analogous kernels for spaces of ``invariantly-polyanalytic'' functions -- a~generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In~both cases, the~kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to …
Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ the total space of a principal bundle $G\to M\to X$ so that $M$ is also a strongly pseudoconvex complex manifold. In this work, we show that if $G$ acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on $M$ is infinite $G$-dimensional.
Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by holomorphic automorphisms. Let G be the complexification of K and g the Lie algebra of G. Following the theory of algebraic transformation groups, we call the complex space X spherical if X is normal and its tangent space at some point is generated by the vector fields from a Borel subalgebra b or g. We give several characterizations of spherical Stein spaces. In particular, we prove that a connected Stein manifold X is spherical if and only if the algebra of K-invariant differential operators on X i…
We prove that there exists no irreducible representation of the identity component of the isometry group ${\rm PO}(1,n)$ of the real hyperbolic space of dimension $n$ into the group ${\rm O}(2,\infty)$, if $n\geq 3$. This is motivated by the existence of irreducible representations (arising from the spherical principal series) of ${\rm PO}(1,n)^{\circ}$ into the groups ${\rm O}(p,\infty)$ for other values of $p$.
We obtain a slow exponential growth estimate for the spherical principal series representation rho_s of Lie group Sp(n, 1) at the edge (Re(s)=1) of Cowling's strip (|Re(s)|<1) on the Sobolev space H^alpha(G/P) when alpha is the critical value Q/2=2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy rho_s (s in [0, 1]) of the spherical principal series which is required for the first author's program for proving the Baum--Connes conjecture with coefficients for Sp(n,1).
The group $Diff$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $U(p,q)$, $Sp(2n,R)$, $SO^*(2n)$; the space $Ξ$ of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of $Diff$ in the space of holomorphic functionals on $Ξ$, reproducing kernels on $Ξ$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.
Matrix elements of spinor and principal series representations of the Lorentz group are studied in the basis of complex angular momentum (helicity basis). It is shown that matrix elements are expressed via hyperspherical functions (relativistic spherical functions). In essence, the hyperspherical functions present itself a four-dimensional (with respect to a pseudo-euclidean metrics of Minkowski spacetime) generalization of the usual three-dimensional spherical functions. An explicit form of the hyperspherical functions is given. The hypespherical functions of the spinor representations are re…
