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Let $(G,H)$ be a symmetric pair over a $p$-adic field and $\pi$ a discrete series representation of $G$. In this paper, for some type of symmetric pairs $(G,H)$, we show that local periods in $Hom_H(\pi,\mathbb{C})$ can be constructed by integrating the matrix coefficients of $\pi$ over $H$., Comment: contain the result in arXiv:1410.8274. comments welcome

It is known that the problem of classifying the irreducible unitary representations of a linear connected semisimple Lie group G comes down to deciding which Langlands quotients J ( MAN , σ , v ) are infinitesimally unitary. Here MAN is any cuspidal parabolic subgroup, σ is any discrete series or nondegenerate limit of discrete series representation of M , and v is any complex-valued linear functional on the Lie algebra of A satisfying certain positivity and symmetry properties. The authors determine which Langlands quotients are infinitesimally unitary under the conditions that G is simple, that dim A = 1, and that G is neither split F 4 nor split G 2 .

To each discrete series representation of a connected semisimple Lie group G with finite center, a G-equivariant embedding into a generalized principal series representation is given. This representation is induced from specified parameters on a maximal parabolic subgroup of G and the mapping is defined by an integral formula, analogous to the Szego integral introduced by Knapp and Wallach for a minimal parabolic subgroup. In a limiting case, embeddings of limits of discrete series representations are obtained and used to exhibit a reducibility theorem.

Osborne and Warner have given a formula for the multiplicity of an integrable discrete series representation in L2(GΓ) when G is real rank one semi-simple Lie group and Γ is a discrete subgroup of co-finite volume. We simplify and evaluate this formula to show that for most G (as above) the multiplicity is the formal degree of the representation times the volume of GΓ. When it is not we give a simple interpretation of the difference.

Blattner's conjecture gives a formula for the multiplicity with which a unitary irreducible representation of the maximal compact subgroup K appears in any discrete series representation of a semisimple Lie group G. We give an elementary derivation of this formula from Harish Chandra's character formula for G ∼- SOe(2n, 1) (n ⩾ 2). The idea is to regularize the character (on the Cartan subgroup), to show that the regularization is unique, and to derive the multiplicities by expanding the resulting distribution in a Fourier series. To prove uniqueness of the regularization one uses a priori constraints on the multiplicities (=Fourier coefficients) that follow from the subquotient theorem.

Let G be a noncompact connected simple Lie group of split-rank 1. Assume that G possesses a compact Cartan subgroup so that the discrete series for G is not empty. Let Γ be a nonuniform lattice in G. In this paper, we give an explicit formula for the multiplicity with which an integrable discrete series representation of G occurs in the space of cusp forms in L2(GΓ).

LetH⊂Gbe real reductive Lie groups. A discrete series representation for a homogeneous spaceG/His an irreducible representation ofGrealized as a closedG-invariant subspace ofL2(G/H). The condition for the existence of discrete series representations forG/Hwas not known in general except for reductive symmetric spaces. This paper offers a sufficient condition for the existence of discrete series representations forG/Hin the setting thatG/His a homogeneous submanifold of a symmetric space G/HwhereG⊂G⊃H. We prove that discrete series representations are non-empty for a number of non-symmetric homogeneous spaces such asSp(2n,R)/Sp(n0,C)×GL(n1,C)×…×GL(nk,C) (∑nj=n) andO(4m,n)/U(2m,j) (0⩽2j⩽n).

Let G be a semisimple Lie group which has a compact Cartan subgroup H , let K be a maximal compact subgroup of G containing H , and let n be the sum of the negative root spaces of G corresponding to an arbitrary choice of a positive root system of ( G , H ). We compute the n -cohomology of the K -finite vectors in a limit of a discrete series representation π of G . In the special case when π is a discrete series representation our result reduces to the result of W. Schmid. In the special case when π is a holomorphic limit we interpret its multiplicity in L 2 ( ΓβG ) cohomologically, where Γ is a discrete subgroup of G . In the general case we present a conjecture for this multiplicity.

We obtain the explicit asymptotic expansion of the matrix coefficients of the middle discrete series representation of SU(2,2) by means of degenerate Gaussian hypergeometric series.