Holomorphic Discrete Series Representations Research Articles

BRANCHING RULES OF SINGULAR UNITARY REPRESENTATIONS WITH RESPECT TO SYMMETRIC PAIRS (A2n-1, Dn)

The irreducible decomposition of scalar holomorphic discrete series representations when restricted to semisimple symmetric pairs (G, H) is explicitly known by Schmid [Die Randwerte holomorphe funktionen auf hermetisch symmetrischen Raumen, Invent. Math.9 (1969–1970) 61–80] for H compact and by Kobayashi [Multiplicity-Free Theorems of the Restrictions of Unitary Highest Weight Modules with Respect to Reductive Symmetric Pairs, Progress in Mathematics, Vol. 255 (Birhäuser, 2007), pp. 45–109] for H non-compact. In this paper, we deal with the symmetric pair (U(n, n), SO* (2n)), and extend the Kobayashi–Schmid formula to certain non-tempered unitary representations which are realized in Dolbeault cohomology groups over open Grassmannian manifolds with indefinite metric. The resulting branching rule is multiplicity-free and discretely decomposable, which fits in the framework of the general theory of discrete decomposable restrictions by Kobayashi [Discrete decomposability of the restriction of A𝔮(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support, Ann. Math.147 (1998), 709–729].

DISCRETE DECOMPOSABLE BRANCHING LAWS AND PROPER MOMENTUM MAPS

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions. (1) The restriction π|H is discretely decomposable in the sense of Kobayashi. (2) The momentum map [Formula: see text] is proper. In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

Moritaʼs duality for split reductive groups

In this paper, we extend the work in [Z. Qi, C. Yang, Moritaʼs theory for the symplectic groups, Int. J. Number Theory 7 (2011) 2115–2137 [7] ] to split reductive groups. We construct and study the holomorphic discrete series representations and the principal series representations of a split reductive group G over a p -adic field F as well as a duality between certain sub-representations of these two representations.

MORITA'S THEORY FOR THE SYMPLECTIC GROUPS

We construct and study the holomorphic discrete series representations and the principal series representations of the symplectic group Sp (2n, F) over a p-adic field F as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for SL (2, F) defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.

Rankin-Cohen Brackets for Jacobi Forms and Tensor Products of Discrete Series Representations of the Jacobi Group

Explicit formulae for Rankin–Cohen brackets for Jacobi forms are obtained by pure Lie theoretic means. The Lie algebraic nature of these holomorphic bi-differential operators, which map pairs of Jacobi forms onto another such form, is unveiled, along with their relationship with the decompositions of tensor products of two (anti-)holomorphic discrete series representations of the Jacobi group into direct sums of similar representations. These decompositions, which turn out to be nonmultiplicity-free, are a problem of an independent interest, which is explicitly solved. Some of the obtained brackets are shown to be intertwiners, projecting the appropriate tensor product onto a direct summand. The first mentioned goal is reached by characterizing all the -equivariant holomorphic bi-differential operators, through establishing a one-to-one correspondence between them and the null vectors in the tensor products of two Verma -modules, for the complexified Lie algebra of . A large class of pairs of such modules is considered, the remaining situations being relegated to a future work. A thorough analysis of the structure of the tensor products, based on Borho’s isomorphism, which provides a factorization of the universal enveloping algebra of , yields explicit formulae for the sought after null vectors; they turn out to be completely determined by the null vectors in tensor products of three Verma -modules. The bi-differential operators associated with the obtained null vectors are then shown to be Rankin–Cohen brackets for Jacobi forms. The Lie theoretic approach developed here supplies a unifying framework within which results of previous investigations undertaken by different authors, using techniques pertaining to the realm of number theory, are recovered, provided with new interpretations and generalized. In particular, Bocherer’s generic dimensions are shown to be given by the above-mentioned multiplicities.

Erratum to “Mass equidistribution for automorphic forms of cohomological type on $GL_{2}$”

Corollary 3 of [4] is not known unconditionally, as cohomological automorphic forms on GL2 over an imaginary quadratic field are not known to satisfy the Ramanujan conjecture. We shall briefly describe the reason for this and discuss what information Theorem 1 of [4] does give in the case of imaginary quadratic fields. Let K be an imaginary quadratic field with nontrivial automorphism c, and let π be a cuspidal automorphic representation of GL2(AK) with unitary central character ω. Suppose that ω = ω and that π∞ has Langlands parameter WC = C× → GL2(C) given by z → diag(z1−k, z1−k) for some integer k ≥ 2 (i.e. so that π is any cohomological representation up to twist). It is then known (see Theorem 1.1 of [1]) that for any one may associate a continuous irreducible representation ρ : Gal(K/K) → GL2(Q ) to π such that the characteristic polynomial of ρ(Frobv) agrees with the Hecke polynomial of πv at all places v which do not divide and at which K/Q, π, and π are unramified. However, because ρ is constructed via an -adic limiting process, it is not known to arise from a motive and so is not known to be pure. To construct ρ, one first makes a theta lift from π to a holomorphic limit of discrete series representation Π on Sp4/Q as in [3]. Weissauer [6] has proven that if Π′ is a holomorphic discrete series representation of Sp4/Q which is not a CAP representation, then one may associate a Galois representation to it which is pure and locally compatible with Π′ at all unramified places, so that Π′ satisfies Ramanujan wherever it is unramified. These results are not known for the limit of discrete series representation Π, and to associate a Galois representation to it one must apply techniques of Taylor [5] which are similar to those used by Deligne and Serre to associate Galois representations to classical weight 1 modular forms. These involve multiplying a holomorphic form in Π by a well understood regular holomorphic form of large weight, applying the results of Weissauer and recovering ρ from these products by an -adic limiting process. Any Archimedean information about the Frobenius eigenvalues of ρ is lost during this, and neither does one know that ρ arises from a motive. As a result, the algebraic information we obtain about π is insufficient to deduce Ramanujan for it. We may still draw interesting conclusions from Theorem 1 of [4] in the imaginary quadratic case. Cohomological forms on GL2/K which are base changes from Q will satisfy Ramanujan, and so Theorem 1 establishes their equidistribution as their weight becomes large. Moreover, the experimental results of [2] suggest that all

Berezin quantization for holomorphic discrete series representations: the non-scalar case

Let M = G/K be a Hermitian symmetric space of the non-compact type and let π be a discrete series representation of G which is holomorphically induced from a unitary irreducible representation ρ of K. We define and we study a notion of Berezin symbol for operators acting on the space of π. In particular, we give explicit formulas for the Berezin symbols of the representation operators \({\pi(g) (g\in G)}\) and dπ(X) (X in the Lie algebra of G). Moreover, we construct an adapted Weyl correspondence for π in the sense of Cahen (Differ Geom Appl 25:177–190, 2007). This generalizes the results already obtained in the case when ρ is a unitary character of K, see Cahen (Beitrage Algebra Geom 51:301–311, 2010).

Boundary orbit strata and faces of invariant cones and complex Ol’shanskiĭ semigroups

Let D=G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification, and there exists a closed complex subsemigroup, the so-called minimal Olshanskii semigroup, of the complexification characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to it. Parallel to the classical theory of boundary strata for the symmetric domain D, due to Koranyi and Wolf, we give a detailed and complete description of the K-orbit type strata of the minimal Olshanskii semigroup, as K-equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra of G.

CORWIN–GREENLEAF MULTIPLICITY FUNCTIONS FOR HERMITIAN SYMMETRIC SPACES AND MULTIPLICITY-ONE THEOREM IN THE ORBIT METHOD

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.For this, we study the Corwin–Greenleaf multiplicity function [Formula: see text] for Hermitian symmetric spaces G/K. First, we prove that [Formula: see text] for any G-coadjoint orbit [Formula: see text] and any K-coadjoint orbit [Formula: see text] if [Formula: see text]. Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.Second, we find a necessary and sufficient condition for [Formula: see text] by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

Bessel Models for Lowest Weight Representations of

We prove uniqueness and give precise criteria for existence of split and nonsplit Bessel models for a class of lowest and highest weight representations of the groups and including all holomorphic and antiholomorphic discrete series representations. Explicit formulas for the resulting Bessel functions are obtained by solving systems of differential equations. The formulas are applied to derive an integral representation for a global L-function on GSp(4) × GL(2) involving a vector-valued Siegel modular form of degree 2.

Positive energy representations, holomorphic discrete series and finite-dimensional irreps

Let G be a semi-simple non-compact Lie group with unitary lowest/highest weight representations. We consider explicitly the relation among three types of representations of G: positive energy (unitary lowest weight) representations, (holomorphic) discrete series representations and non-unitary finite-dimensional irreps. We consider mainly the conformal groups SOo(n, 2) treating in full detail the cases n = 1, 3, 4.

Corwin–Greenleaf multiplicity functions for Hermitian symmetric spaces

Kobayashi’s multiplicity-free theorem asserts that irreducible unitary highest weight representations $\pi$ are multiplicity-free when restricted to any symmetric pairs if $\pi$ is of scalar type. The Hua–Kostant–Schmid–Kobayashi branching laws embody this abstract theorem with explicit irreducible decomposition formulas of holomorphic discrete series representations with respect to symmetric pairs. In this paper, we study the ‘classical limit’ (geometry of coadjoint orbits) of a special case of these representation theoretic theorems in the spirit of the Kirillov–Kostant–Duflo orbit method. \\ First, we consider the Corwin–Greenleaf multiplicity function $n (\mathcal{O}^{G},\,\mathcal{O}^{K})$ for Hermitian symmetric spaces $G/K$. The first main theorem is that $n(\mathcal{O}^{G},\,\mathcal{O}^{K}) \le 1$ for any $G$-coadjoint orbit $\mathcal{O}^{G}$ and any $K$-coadjoint orbit $\mathcal{O}^{K}$ if $\mathcal{O}^{G} \cap \sqrt{-1} ([\mathfrak{k}, \mathfrak{k}] + \mathfrak{p})^{\bot} \ne \emptyset$. Here, $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ is the Cartan decomposition of the Lie algebra $\mathfrak{g}$ of $G$. The second main theorem is a necessary and sufficient condition for $n (\mathcal{O}^{G},\,\mathcal{O}^{K}) \ne 0$ by means of strongly orthogonal roots.

Mayer’s transfer operator and representations of GL2

In this paper we relate Mayer’s transfer operator for the geodesic flow of the modular surface to the representation theory of the semigroup of invertible 2×2-matrices with non-negative entries. It turns out that similarly to the case of the Kac-Baker model (see Hilgert et al., Convex Cones, and Semigroups, Oxford University Press, London, 1989 and Hilgert and Mayer, Commun. Math. Phys. 232:19–58, 2002) from statistical mechanics which is related to Howe’s oscillator semigroup one has to introduce an additional multiplication operator to obtain a self-adjoint Hilbert space operator of trace class with the correct spectrum from the natural operators provided by the representation theory. In the present case the representations naturally live on weighted Bergman spaces, but can also be realized on weighted L 2-spaces. Using the representation theory of Ol’shanskiĭ semigroups the semigroup representations can be analytically extended to the simply connected covering of SL(2,ℝ) where they can be identified as holomorphic discrete series representations.

The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators

The Lie theoretic nature of the Rankin–Cohen brackets is here uncovered. These bilinear operations, which, among other purposes, were devised to produce a holomorphic automorphic form from any pair of such forms, are instances of SL ( 2 , R ) -equivariant holomorphic bi-differential operators on the upper half-plane. All of the latter are here characterized and explicitly obtained, by establishing their one-to-one correspondence with singular vectors in the tensor product of two sl ( 2 , C ) Verma modules. The Rankin–Cohen brackets arise in the generic situation where the linear span of the singular vectors of a given weight is one-dimensional. The picture is completed by the special brackets which appear for the finite number of pairs of initial lowest weights for which the above space is two-dimensional. Explicit formulæ for basis vectors in both situations are obtained and universal Lie algebraic objects subsuming all of them are exhibited. A few applications of these results and Lie theoretic approach are then considered. First, a generalization of the latter yields Rankin–Cohen type brackets for Hilbert modular forms. Then, some Rankin–Cohen brackets are shown to intertwine the tensor product of two holomorphic discrete series representations of SL ( 2 , R ) with another such representation occurring in the tensor product decomposition. Finally, the sought for precise relationship between the Rankin–Cohen brackets and Gordan's transvection processes of the nineteenth century invariant theory is unveiled.

Integral structures in the 𝑝-adic holomorphic discrete series

For a local non-Archimedean field K K we construct G L d + 1 ( K ) {\mathrm {GL}}_{d+1}(K) -equivariant coherent sheaves V O K {\mathcal V}_{{\mathcal O}_K} on the formal O K {\mathcal O}_K -scheme X {\mathfrak X} underlying the symmetric space X X over K K of dimension d d . These V O K {\mathcal V}_{{\mathcal O}_K} are O K {\mathcal O}_K -lattices in (the sheaf version of) the holomorphic discrete series representations (in K K -vector spaces) of G L d + 1 ( K ) {\mathrm {GL}}_{d+1}(K) as defined by P. Schneider. We prove that the cohomology H t ( X , V O K ) H^t({\mathfrak X},{\mathcal V}_{{\mathcal O}_K}) vanishes for t > 0 t>0 , for V O K {\mathcal V}_{{\mathcal O}_K} in a certain subclass. The proof is related to the other main topic of this paper: over a finite field k k , the study of the cohomology of vector bundles on the natural normal crossings compactification Y Y of the Deligne-Lusztig variety Y 0 Y^0 for G L d + 1 / k {\mathrm {GL}}_{d+1}/k (so Y 0 Y^0 is the open subscheme of P k d {\mathbb P}_k^d obtained by deleting all its k k -rational hyperplanes).

Fonctions holomorphes � croissance mod�r�e et vecteurs distributions

The space of distribution vectors of a holomorphic discrete series representation of scalar type can be identified to a space of holomorphic functions with moderate growth. This is the main result of the paper. We give an application of it to the Hankel transform and to multivariate Laguerre series expansions.

Composition series for analytic continuations of holomorphic discrete series representations of SU p, q

A family of holomorphic discrete series representations of SU p, q is studied, and analytic continuation in terms of the reproducing kernel parameter ν is discussed. The composition series of the module of all polynomials on the hermitian symmetric domain D z of SU p, q is determined for those values of ν for which the representations are reducible.

Berezin Kernels of Tube Domains

We study the Berezin kernel decomposition into positive-definite spherical functions on real forms of tube domains. In other words, we obtain, for all tube domains associated to causal pairs (G, H), the decomposition of holomorphic discrete series representations of G when restricted to the “fully restrictive” subgroup H.

Some Formulas for the Canonical Kernel Function

In this paper we present some expressions for the canonical kernel function of a bounded symmetric domain. We show how it is related both to the reproducing kernel of a holomorphic discrete series representation and to the contravariant form on highest weight modules over semisimple Lie algebras of Hermitian type.

Quaternionic discrete series

This work investigates the discrete series of linear connected semi- simple noncompact groups G G . These are irreducible unitary representations that occur as direct summands of L 2 ( G ) L^{2}(G) . Harish-Chandra produced discrete series representations, now called holomorphic discrete series representations, for groups G G with the property that, if K K is a maximal compact subgroup, then G / K G/K has a complex structure such that G G acts holomorphically. Holomorphic discrete series are extraordinarily explicit, it being possible to determine all the elements in the space and the action by the Lie algebra of G G . Later Harish-Chandra parametrized the discrete series in general. His argument did not give an actual realization of the representations, but later authors found realizations in spaces defined by homology or cohomology. These realizations have the property that it is not apparent what elements are in the space and what the action of the Lie algebra G G is. The point of this work is to find some intermediate ground between the holomorphic discrete series and the general discrete series, so that the intermediate cases may be used to get nontrivial insights into the internal structure of the discrete series in the general case. The author examines the Vogan-Zuckerman realization of discrete series by means of cohomological induction. An explicit complex for computing the homology on the level of a K K module was already known. Also, Duflo and Vergne had given information about how to compute the action of the Lie algebra of G G . The holomorphic discrete series are exactly those cases where the representations can be realized in homology of degree 0. The intermediate cases that are studied are those where the representation can be realized in homology of degree 1. Many of the intermediate cases correspond to the situation where G / K G/K has a quaternionic structure. The author obtains general results for A q ( λ ) {\text {A}_{\mathfrak {q}}(\lambda )} discrete series in the intermediate case.