Holomorphic Discrete Series Representations Research Articles

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups

For every simple Hermitian Lie group G, we consider a certain maximal parabolic subgroup whose unipotent radical N is either abelian (if G is of tube type) or two-step nilpotent (if G is of non-tube type). By the generalized Whittaker Plancherel formula we mean the Plancherel decomposition of L2(G/N,ω), the space of square-integrable sections of the homogeneous vector bundle over G/N associated with an irreducible unitary representation ω of N. Assuming that the central character of ω is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of G into L2(G/N,ω) and show that the multiplicities are equal to the dimensions of the lowest K-types. The construction is in terms of a kernel function which can be explicitly defined using certain projections inside a complexification of G. This kernel function carries all information about the holomorphic discrete series embedding, the lowest K-type as functions on G/N, as well as the associated Whittaker vectors.

Multivariate Meixner polynomials related to holomorphic discrete series representations of [formula omitted]

We show that Griffiths’ multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group SU(1,d). Using this interpretation we derive several fundamental properties of the multivariate Meixner polynomials, such as orthogonality relations and difference equations. Furthermore, we also show that matrix coefficients for specific group elements lead to degenerate versions of the multivariate Meixner polynomials and their properties.

Ornstein-Uhlenbeck operators and holomorphic representations for a class of bounded symmetric domains

Let G be the group of holomorphic automorphisms of a bounded symmetric domain D. We discuss the correspondence between unitarizing measures for scalar-valued holomorphic discrete series representations of G and Ornstein-Uhlenbeck operators on D. We complement the results obtained for the domains in [1] and [6] by showing that for Cartan domain II of n×n symmetric complex matrices and domain III of skew-symmetric complex matrices Z such that ZZ⁎<I, the vector fields in the infinitesimal holomorphic representation are the entries of Hua matrices. Similar results stay valid for some Kähler submanifolds M of D, assuming that the entries of the matrices in M depend linearly on fixed independent entries and M is isomorphic to a product of Cartan domains.

A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups

This paper is devoted to branching problems for holomorphic discrete series representations of a conformal group [Formula: see text] of a tube domain [Formula: see text] over a symmetric cone [Formula: see text] restricted to the conformal group [Formula: see text] of a tube domain [Formula: see text] holomorphically embedded in [Formula: see text]. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. We answer this program with the introduction of another functional model for holomorphic discrete series representations. This model leads to a geometrical interpretation and a close relation to the theory of orthogonal polynomials for such branching problems. This program is illustrated by three cases. First, we consider the [Formula: see text]-fold tensor product of holomorphic discrete series of the universal covering of [Formula: see text]. Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group [Formula: see text] to the subgroup [Formula: see text], and finally to the subgroup [Formula: see text].

Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Let pi _{alpha } be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space A^2_{alpha } (Omega ) on a bounded symmetric domain Omega , of formal dimension d_{pi _{alpha }} > 0. It is shown that if the Bergman kernel k^{(alpha )}_z is a cyclic vector for the restriction pi _{alpha } |_{Gamma } to a lattice Gamma le G (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma } is a frame for A^2_{alpha }(Omega )), then {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} le |Gamma _z|^{-1}. The estimate {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} ge |Gamma _z|^{-1} holds for k^{(alpha )}_z being a p_z-separating vector (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma / Gamma _z} being a Riesz sequence in A^2_{alpha } (Omega )). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for G ={mathrm {PSU}}(1, 1).

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ on $D$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the $K_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement $\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of this article is to compute explicitly the inner product $\big\langle f(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\big\rangle_\lambda$ for $f(x_2)\in\mathcal{P}(\mathfrak{p}^+_2)$, $x=(x_1,x_2)$, $z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$. For example, when $\mathfrak{p}^+$, $\mathfrak{p}^+_2$ are of tube type and $f(x_2)=\det(x_2)^k$, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ${}_2F_1$. Also, as an application, we construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $\lambda$.

On counting cuspidal automorphic representations for GSp(4)

Abstract We find the number s k ⁢ ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ⁢ ( 4 , A Q ) \mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ⁢ ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

New Atomic Decompositions of Bergman Spaces on Bounded Symmetric Domains

We provide a large family of atoms for Bergman spaces on irreducible bounded symmetric domains. The atomic decompositions are derived using the holomorphic discrete series representations for the domain, and the approach is inspired by recent advances in wavelet and coorbit theory. Our results vastly generalize previous work by Coifman and Rochberg. Their atoms correspond to translates of a constant function at a discrete subset of the automorphism group of the domain. In this paper we show that atoms can be obtained as translates of any holomorphic function with rapidly decreasing coefficients (including polynomials). This approach also settles the relation between atomic decompositions for the bounded and unbounded realizations of the domain.

Construction of discrete series representations of via algebraic Dirac induction

The notion of algebraic Dirac induction was introduced by P. Pandžić and D. Renard. This is a construction which gives representations with prescribed Dirac cohomology. They proved that all holomorphic discrete series representations can be constructed via Dirac induction. All discrete series representations of the group are nonholomorphic. In this paper we prove that they can also be constructed using algebraic Dirac induction.Communicated by Silvana Bazzoni

Some analytic aspects of automorphic forms on GL(2) of minimal type

Let $\pi$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $\phi$ be an $L^2$-normalized automorphic form in the space of $\pi$. The sup-norm problem asks for bounds on $\| \phi \|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|\phi|^2 (g) \, d g$ of $\phi$. All previous work on these problems in the conductor-aspect has focused on the case that $\phi$ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor $p$ of $C$, the local component $\pi_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $\phi$ for which the local components $\phi_p \in \pi_p$ are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of $PGL_2(\mathbb{R})$. For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if $\pi_\infty$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8 -\epsilon} k^{1/4 - \epsilon} \ll_{\epsilon} |\phi|_\infty \ll_{\epsilon} C^{1/8 + \epsilon} k^{1/4+\epsilon}$. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence for newforms of powerful level.

A construction by deformation of unitary irreducible representations of SU(1,n) and SU(n + 1)

We recover the holomorphic discrete series representations of [Formula: see text] as well as some unitary irreducible representations of [Formula: see text] by deformation of a minimal realization of [Formula: see text].

Construction of Intertwining Operators between Holomorphic Discrete Series Representations

In this paper we explicitly construct $G_1$-intertwining operators between holomorphic discrete series representations $\mathcal{H}$ of a Lie group $G$ and those $\mathcal{H}_1$ of a subgroup $G_1\subset G$ when $(G,G_1)$ is a symmetric pair of holomorphic type. More precisely, we construct $G_1$-intertwining projection operators from $\mathcal{H}$ onto $\mathcal{H}_1$ as differential operators, in the case $(G,G_1)=(G_0\times G_0,\Delta G_0)$ and both $\mathcal{H}$, $\mathcal{H}_1$ are of scalar type, and also construct $G_1$-intertwining embedding operators from $\mathcal{H}_1$ into $\mathcal{H}$ as infinite-order differential operators, in the case $G$ is simple, $\mathcal{H}$ is of scalar type,and $\mathcal{H}_1$ is multiplicity-free under a maximal compact subgroup $K_1\subset K$. In the actual computation we make use of series expansions of integral kernels and the result of Faraut-Kor&#225;nyi (1990) or the author's previous result (2016) on norm computation. As an application, we observe the behavior of residues of the intertwining operators, which define the maps from some subquotient modules, when the parameters are at poles.

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation $\pi$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $\nu$ occurring in the restriction $\pi|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $\pi$ and $\nu$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits $O^G$ of a semisimple Lie group $G$ corresponding to highest weight modules of scalar type. We prove that the Corwin--Greenleaf number $\sharp(O^G \cap pr^{-1}(O^H))/H$ is either zero or one for any $H$-coadjoint orbit $O^H$, whenever $(G,H)$ is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits $O^H$ with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as "classical limits" of the multiplicity-free branching laws of holomorphic discrete series representations (T.Kobayashi [Progr.Math.2007]).

Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs

Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of $$SL(2,\mathbb {R})$$ . We address a general problem to find explicit formulae for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose, we use a new method (F-method) developed in Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9 ) and based on the algebraic Fourier transform for generalized Verma modules. The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulae of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin–Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.

Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series

In this paper we show that the multiplicities of holomorphic discrete series representations relatively to reductive subgroups satisfy the credo Quantization commutes with reduction.

STABILITY OF BRANCHING LAWS FOR HIGHEST WEIGHT MODULES

In this paper, we study the irreducible decomposition of a (ℂ[X];G)-module M for a quasi-affine spherical variety X of a connected reductive algebraic group G over ℂ. We show that for sufficiently large parameters, the decomposition of M with respect to G is reduced to the decomposition of the ‘fiber’ M/\( \mathfrak{m} \)(x 0)M with respect to some reductive subgroup L of G. In particular, we obtain a method to compute the maximum value of multiplicities in M. Our main result is a generalization of earlier work by F. Satō in [17]. We apply this result to branching laws of holomorphic discrete series representations with respect to symmetric pairs of holomorphic type. We give a necessary and sufficient condition for multiplicity-freeness of the branching laws.

The quantum Rabi model and Lie algebra representations of

The aim of the present paper is to understand the spectral problem of the quantum Rabi model in terms of Lie algebra representations of . We define a second order element of the universal enveloping algebra of , which, through the image of a principal series representation of , provides a picture equivalent to the quantum Rabi model drawn by confluent Heun differential equations. By this description, in particular, we give a representation theoretic interpretation of the degenerate part of the spectrum (i.e., Juddʼs eigenstates) of the Rabi Hamiltonian due to Kuś in 1985, which is a part of the exceptional spectrum parameterized by integers. We also discuss the non-degenerate part of the exceptional spectrum of the model, in addition to the Judd eigenstates, from a viewpoint of infinite dimensional irreducible submodules (or subquotients) of the non-unitary principal series such as holomorphic discrete series representations of .

Classical limit of the tensor product of holomorphic discrete series representations

For three coadjoint orbits \(\mathcal {O}_1, \mathcal {O}_2\) and \(\mathcal {O}_3\) in \(\mathfrak {g}^*\), the Corwin–Greenleaf function\(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) is given by the number of \(G\)-orbits in \(\{(\lambda , \mu ) \in \mathcal {O}_1 \times \mathcal {O}_2 \, : \, \lambda + \mu \in \mathcal {O}_3 \}\) under the diagonal action. In the case where \(G\) is a simple Lie group of Hermitian type, we give an explicit formula of \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) for coadjoint orbits \(\mathcal {O}_1\) and \(\mathcal {O}_2\) that meet \(\left( [\mathfrak {k}, \mathfrak {k}] + \mathfrak {p}\right) ^{\perp }\), and show that the formula is regarded as the ‘classical limit’ of a special case of Kobayashi’s multiplicity-free theorem (Progr. Math. 2007) in the branching law to symmetric pairs.

Stability of branching laws for spherical varieties and highest weight modules

If a locally finite rational representation $V$ of a connected reductive algebraic group $G$ has uniformly bounded multiplicities, the multiplicities may have good properties such as stability. Let $X$ be a quasi-affine spherical $G$-variety, and $M$ be a $(\mathbf{C}[X],G)$-module. In this paper, we show that the decomposition of $M$ as a $G$-representation can be controlled by the decomposition of the fiber $M/\mathfrak{m}(x_{0})M$ with respect to some reductive subgroup $L \subset G$ for sufficiently large parameters. As an application, we apply this result to branching laws for simple real Lie groups of Hermitian type. We show that the sufficient condition on multiplicity-freeness given by the theory of visible actions is also a necessary condition for holomorphic discrete series representations and symmetric pairs of holomorphic type. We also show that two branching laws of a holomorphic discrete series representation with respect to two symmetric pairs of holomorphic type coincide for sufficiently large parameters if two subgroups are in the same $\epsilon$-family.

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].