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Let π α be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space A α 2 (Ω) on a bounded symmetric domain Ω , of formal dimension d π α > 0 . It is shown that if the Bergman kernel k z (α) is a cyclic vector for the restriction π α | Γ to a lattice Γ ≤ G (resp. (π α (γ) k z (α)) γ ∈ Γ is a frame for A α 2 (Ω) ), then vol (G / Γ) d π α ≤ | Γ z | - 1 . The estimate vol (G / Γ) d π α ≥ | Γ z | - 1 holds for k z (α) being a p z -separating vector (resp. (π α (γ) k z (α)) γ ∈ Γ / Γ z being a Riesz sequence in A α 2 (Ω) ). 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 = PSU (1 , 1) . [ABSTRACT FROM AUTHOR]

Let G / H {G/H} be a reductive symmetric space of split rank one and let K be a maximal compact subgroup of G. In a previous article the first two authors introduced a notion of cusp forms for G / H {G/H}. We show that the space of cusp forms coincides with the closure of the space of K-finite generalized matrix coefficients of discrete series representations if and only if there exist no K-spherical discrete series representations. Moreover, we prove that every K-spherical discrete series representation occurs with multiplicity one in the Plancherel decomposition of G / H {G/H}. [ABSTRACT FROM AUTHOR]

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spin c -Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete series representations of semisimple groups, where we assume the Riemannian metric to have a certain product form. The other is an index defined in terms of sections invariant under a group action. We obtain a relation with the analytic assembly map, quantisation commutes with reduction results, and Atiyah–Hirzebruch type vanishing theorems. The arguments are based on an explicit decomposition of Spin c -Dirac operators with respect to a global slice for the action. [ABSTRACT FROM AUTHOR]

We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Γ \ G / H , beyond the classical cases where H is compact (automorphic forms) or Γ is trivial (analysis on symmetric spaces). For any non-Riemannian reductive symmetric space X = G / H on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Γ whose action on X is “sufficiently proper”, we construct L 2 -eigenfunctions of the Laplacian on X Γ : = Γ \ X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to X Γ of sums, over the Γ-orbits, of eigenfunctions of the Laplacian on X . We prove that the Poincaré series we construct still converge, and define nonzero L 2 -functions, after any small deformation of Γ inside G , for a large class of groups Γ. Thus the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface. We actually construct joint L 2 -eigenfunctions for the whole commutative algebra of invariant differential operators on X Γ . [ABSTRACT FROM AUTHOR]

Abstract: Let F be a non-Archimedean local field, and let A be a central simple F-algebra. In the present paper, based on the Bushnell–Kutzko theory of types for , which has recently been developed by Sécherre and Stevens, we derive an explicit formula for the formal degree of an irreducible discrete series representation of the multiplicative group . [Copyright &y& Elsevier]

Abstract: Let denote the split symplectic group of rank 2 over . Fix a prime p. Let be a parahoric subgroup of . An arithmetic subgroup Γ is defined by , where . In this paper, we calculate Arthurʼs -Lefschetz trace formula for in order to obtain an explicit formula for multiplicities of discrete series and some non-tempered unitary representations in the discrete spectrum of for each such Γ. From them we derive explicit multiplicity formulas for large discrete series, which are our main results. The multiplicity formulas are applied to a study on numbers of cuspidal automorphic representations of . [Copyright &y& Elsevier]

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, ). This generalizes the results already obtained in the case when ρ is a unitary character of K, see Cahen (Beiträge Algebra Geom 51:301-311, ). [ABSTRACT FROM AUTHOR]

Abstract: We obtain the explicit asymptotic expansion of the matrix coefficients of the middle discrete series representation of by means of degenerate Gaussian hypergeometric series. [Copyright &y& Elsevier]

Considers the deformation of irreducible unitary representations of discrete series of Hermitian symmetric simple Lie groups in the class of pure pseudorepresentations. Proof of the theorem; Hilbert space; Isomorphism.