Holomorphic Discrete Series Research Articles

Deformations of Irreducible Unitary Representations of the Discrete Series of Hermitian-Symmetric Simple Lie Groups in the Class of Pure Pseudo-Representations

We prove that irreducible unitary representations of the discrete series of simple Hermitian-symmetric Lie groups can be continuously deformed in the class of pure unitary pseudo-representations (note that these representations cannot be continuously deformed in the class of unitary representations). We also briefly recall the main definitions and facts related to the notions of quasi-symmetry and pseudo-symmetry and to the realization of representations of the holomorphic discrete series of simple hermitian-symmetric Lie groups.

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.

Hilbert spaces of tensor-valued holomorphic functions on the unit ball of ℂn

We study expansion of reproducing kernels for Hilbert spaces of holomorphic functions on the unit ball in C with values in the antisymmetric tensor of the tangent and the cotangent spaces. As an application we find the composition series for the analytic continuation of certain families of holomorphic discrete series.

Transfer of Unitary Representations

Introduction. This paper has as its primary purpose a clear exposition of the idea in [21] which we describe as transfer between real forms of a semi-simple Lie group over C. The basic point is that there are representations of one real form that can be more easily understood in the context of another real from. An interesting example is a minimal representation (that is annihilated by the Joseph ideal) of a split group over R. In the case when the complexification admits a Hermitian symmetric real form, minimal representations of that real form are part of the “analytic continuation of the holomorphic discrete series” (cf. [5]). For example, in the case of SO(4, 4) or split E7 one can “transfer” the holomorphic minimal representations of SO(6, 2) and Hermitian symmetric E7 respectively. If there is no Hermitian symmetric real form then since there is always a quaternionic real form one can do a similar transfer. We feel that this exposition is necssary since the original discussion had many misprints which could be confusing and also lacked, in some cases, proper reference to earlier related work. The secondary purpose is to give some new examples of its applicability. These examples give more evidence of ap ossible deep connection between the notion of transfer and Howe’s theory of dual pairs. We will now give a description of the paper. Let GC be a connected, simply connected semi-simple Lie group over C .L etG and

The Meromorphic continuation of the resolvent of the Laplacian on line bundles over ℂH(n)

Let G = SU(n,1), K = S(U(n) × U(1)), and for l ∈ Z, let {τ l } l ∈ Z be a one-dimensional K-type and let E l the line bundle over G/K associated to τ l . In this work we prove that the resolvent of the Laplacian, acting on C∞c-sections of E l is given by convolution with a kernel which has a meromorphic continuation to C. We prove that this extension has only simple poles and we identify the images of the corresponding residues with (g, K)-submodules of the principal series representations. We show that for certain values of the parameters these modules are holomorphic (or antiholomorphic) discrete series.

Branching Coefficients of Holomorphic Representations and Segal–Bargmann Transform

Let D = G/ K be a complex bounded symmetric domain of tube type in a Jordan algebra V C , and let D= H/ L= D ∩ V be its real form in a Jordan algebra V⊂ V C . The analytic continuation of the holomorphic discrete series on Dopf; forms a family of interesting representations of G. We consider the restriction on D and the branching rule under H of the scalar holomorphic representations. 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 find a canonical basis of L-invariant polynomials in the components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal–Bargmann transform of those L-invariant polynomials are, under the spherical transform on D, multi-variable Wilson-type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on D, when extended to a holomorphic function in a neighborhood of 0∈ D , in terms of the L-spherical holomorphic polynomials on D , the coefficients being the Wilson polynomials.

Nearly holomorphic functions and relative discrete series of weighted $L^2$-spaces on bounded symmetric domains

Let $\Omega = G/K$ be a bounded symmetric domain in a complex vector space $V$ with the Lebesgue measure $dm(z)$ and the Bergman reproducing kernel $h(z,w)^{-p}$. Let $d\mu _{\alpha}(z) = h(z, \bar{z})^{\alpha}dm(z)$, $\alpha > -1$, be the weighted measure on $\Omega$. The group $G$ acts unitarily on the space $L^{2}(\Omega , \mu_\alpha )$ via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irreducible decomposition of the $L^{2}$-space. Let $\bar{D} = B(z, \bar{z})\partial$ be the invariant Cauchy-Riemann operator. We realize the relative discrete series as the kernels of the power $\bar{D}^{m+1}$ of the invariant Cauchy-Riemann operator $\bar{D}$ and thus as nearly holomorphic functions in the sense of Shimura. We prove that, roughly speaking, the operators $\bar{D}^{m}$ are intertwining operators from the relative discrete series into the standard modules of holomorphic discrete series (as Bergman spaces of vector-valued holomorphic functions on $\Omega$).

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 transform on real bounded symmetric domains

Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$ with a real form $V$ and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space $V$. We construct the Berezin kernel and consider the Berezin transform on the $L^2$-space on $D$. The corresponding representation of $G$ is then unitarily equivalent to the restriction to $G$ of a scalar holomorphic discrete series of holomorphic functions on $\mathbb D$ and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the $L^2$-space.

The Plancherel Theorem for invariant Hilbert spaces

We calculate explicitly the Plancherel measure for certain Hilbert spaces of holomorphic functions naturally associated to affine symmetric spaces with holomorphic discrete series.

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.

Formal Dimension for Semisimple Symmetric Spaces

If G is a semisimple Lie group and (π, ) belonging to the holomorphic discrete series of H\G, then one can define a formal dimension d(π) in an analogous manner. In this paper we compute d(π) for these classes of representations.

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.

Holomorphic Discrete Models of Semisimple Lie Groups and their Symplectic Constructions

Let G be a connected real semisimple Lie group which contains a compact Cartan subgroup such that it has non-empty discrete series. A holomorphic discrete model of G is a unitary G-representation consisting of all its holomorphic discrete series with multiplicity one. We perform geometric quantization to a class of G-invariant pseudo-Kähler manifolds and construct a holomorphic discrete model. The construction of discrete series which are not holomorphic is also discussed.

On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of 𝑆𝑙(2,ℝ)

Analogous to the holomorphic discrete series ofSl(2,R)Sl(2,\mathbb R)there is a continuous family{πr}\{\pi _r\},−1>r>∞-1>r>\infty, of irreducible unitary representations ofGG, the simply-connected covering group ofSl(2,R)Sl(2,\mathbb R). A construction of this series is given in this paper using classical function theory. For allrrthe Hilbert space isL2((0,∞))L_2((0,\infty )). First of all one exhibits a representation,DrD_r, ofg=LieG\mathfrak g=Lie\,Gby second order differential operators onC∞((0,∞))C^\infty ((0,\infty )). Forx∈(0,∞)x\in (0,\infty ),−1>r>∞-1>r>\inftyandn∈Z+n\in \mathbb Z_+letφn(r)(x)=e−xxr2Ln(r)(2x)\varphi _n^{(r)}(x)= e^{-x}x^{\frac {r}{2}}L_n^{(r)}(2x)whereLn(r)(x)L_n^{(r)}(x)is the Laguerre polynomial with parameters{n,r}\{n,r\}. LetHrHC\mathcal H_r^{HC}be the span ofφn(r)\varphi _n^{(r)}forn∈Z+n\in \mathbb Z_+. Next one shows, using a famous result of E. Nelson, thatDr|HrHCD_r|{\mathcal H}_r^{HC}exponentiates to the unitary representationπr\pi _rofGG. The power of Nelson’s theorem is exhibited here by the fact that if0>r>10>r>1, one hasDr=D−rD_r=D_{-r}, whereasπr\pi _ris inequivalent toπ−r\pi _{-r}. Forr=12r=\frac 12, the elements in the pair{π12,π−12}\{\pi _{\frac {1}{2}},\pi _{-\frac {1}{2}}\}are the two components of the metaplectic representation. Using a result of G.H. Hardy one shows that the Hankel transform is given byπr(a)\pi _r(a)wherea∈Ga\in Ginduces the non-trivial element of a Weyl group. As a consequence, continuity properties and enlarged domains of definition, of the Hankel transform follow from standard facts in representation theory. Also, ifJrJ_ris the classical Bessel function, then for anyy∈(0,∞)y\in (0,\infty ), the functionJr,y(x)=Jr(2xy)J_{r,y}(x)=J_r(2\sqrt {xy})is a Whittaker vector. Other weight vectors are given and the highest weight vector is given by a limiting behavior at00.

The expressions of the Harish-Chandra C-functions of semisimple Lie groups Spin(n,1), SU(n,1)

We give the recursion formula of the Harish-Chandra C-function with respect to the highest weight of the representations of K. Using this formula, we get the explicit expressions of the Harish-Chandra C-functions for Spin(n,1) and SU(n,1). As an application, by using these expressions, we get the realizations of discrete series representations of SU(n,1) as subquotients of nonunitary principal series representations. We shall also get the decompositions of holomorphic and antiholomorphic discrete series when restricted to U(n-1,1). By using the structures of K-spectra of discrete series representations, we can concretely construct the invariant subspaces of the representation spaces of holomorphic and antiholomorphic discrete series.

Weighted Laplace transforms and Bessel functions on Hermitian symmetric spaces

This paper defines $\pi$-weighted Laplace transforms on the spaces of $\pi$-covariant functions. By the inverse Laplace transform we define operator-valued Bessel functions. We also study the holomorphic discrete series of the automorphism group of a Siegel domain of type II.

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.

The Plancherel Theorem for Biinvariant Hilbert Spaces

We calculate explicitly the Plancherel measure for certain Hilbert spaces of holomorphic functions naturally associated to affine symmetric spaces with holomorphic discrete series.

Representations of the Infinite Unitary Group from Constrained Quantization

We attempt to reconstruct the irreducible unitary representations of the Banach Lie group U 0(ℋ) of all unitary operators U on a separable Hilbert space ℋ for which U − I is compact, originally found by Kirillov and Ol’shanskii, through constrained quantization of its coadjoint orbits. For this purpose the coadjoint orbits are realized as Marsden-Weinstein quotients. The unconstrained system, given as a Weinstein dual pair, is quantized by a corresponding Howe dual pair. Constrained quantization is then performed in replacing the classical procedure of symplectic reduction by the C ∗-algebraic method of Rieffel induction. Reduction and induction have to be performed with respect to either U(M), which is straightforward, or U(M, N). In the latter case one induces from holomorphic discrete series representations, and the desired result is obtained if one ignores half-forms, and induces from a representation, ‘half’ of whose highest weight is shifted relative to the naive orbit correspondence. This is only possible when ℋ is finite-dimensional.