On extensions of principal series representations

Linear representations of Lie groups are useful in both physics and mathematics. The most important applicable aspects of representations theory are matrix elements and Clebsch-Gordan (CG) coefficients of group representations. New results of representation theory allow us to give a new approach to matrix elements and CG coefficients. At first, matrix elements and CG coefficients of unitary irreducible representations were studied separately for compact and non-compact semisimple Lie groups and for each series of representations (the principal unitary series, the supplementary series, the discrete series and so on). These studies can now be linked together. This link is realized by the principal nonunitary series representations (“analytic continuation” in the continuous parameters of the principal unitary series representations) of a semisimple noncompact Lie group. The point is that every completely irreducible representation of such a group is contained in some representation of the principal nonunitary series. On the other hand, matrix elements of the principal nonunitary series representations at fixed group element are entire analytic functions of continuous parameters defining representations. Therefore, if matrix elements of representations of one of the continuous series (for example, of the principal unitary series) are known then those of representations of other series can be obtained by an appropriate analytic Continuation. However, most of the unitary representations which are contained in the principal nonunitary series representations are nonunitary. Therefore, it is necessary to evaluate the matrices of unitarization of unitarizable representations. These matrices bear a simple relationship with intertwining operators for the principal nonunitary series representations. For this reason an evaluation of matrices of intertwining operators is an important task. Different intertwining operators are linked by CG coefficients of the tensor product of the principal nonunitary series representations with finite dimensional representations of the group.

The decomposition of the tensor product of two principal series representations is determined for the simply connected double covering, G = Spin(4, 1), of the DeSitter group. The main result is that this decomposition consists of two pieces, T, and Td, where T, is a continuous direct sum with respect to Plancherel measure on G of representations from the principal series only and Td is a discrete sum of representations from the discrete series of G. The multiplicities of representations occurring in T, and Td are all finite. Introduction. Let G = Spin(4, 1) be the simply connected double covering of the DeSitter group, G = KAN an Iwasawa decomposition of G, M the centralizer of A in K, and P = MAN the associated minimal parabolic subgroup of G. For a E M and T E A, a x T is a representation of P via a X T(man) = a(m)T(a) and a representation of the form 7(a, T) = Indp a x T is called a principal series representation of G. The main goal of this paper is to determine the decomposition of the tensor product of two principal series representations of G into irreducibles. It was shown in [7], by using Mackey's tensor product theorem and the Mackey-Anh reciprocity theorem, that this problem reduces to knowing how to decompose the restriction to MA of almost every principal series representation of G and each discrete series representation of G. For a representation v7 belonging to the principal series of G, the restriction of v7 to MA, (7)MAA was determined by using Mackey's subgroup theorem. However, in that paper, we were not able to determine explicitly (7)MA for a representation 'r belonging to the discrete series of G. This we do in ?3 of this paper by using Lie algebraic methods and the realizations of these representations given by Dixmier in [2]. This paper is organized as follows. In ?? 1 and 2 we summarize the main results concerning the structure and representation theory of G that we shall use. In ?3 we determine (7)MA when v7 is a discrete series representation of G. We also include the results of [7] concerning the decomposition of (@)MA when v7 is a principal series representation of G. In ?4 we show how to decompose the tensor product of two principal series representations of G. The main results are contained in Theorem 4. The basic methodology used in this paper to decompose principal series tensor products originates in the works of G. Mackey [6], N. Anh [1], and F. Williams [11]. Received by the editors April 15, 1980. 1980 Mathematics Subject Classification Primary 22E43, 81C40. 'This research was partially supported by a grant from the National Science Foundation. ? 1981 American Mathematical Society 0002-9947/81 /0000-0207/$04.75 121 This content downloaded from 157.55.39.118 on Fri, 22 Apr 2016 04:32:41 UTC All use subject to http://about.jstor.org/terms

Lecture 5 - Principal series representations

In this paper we study p p -adic principal series representation of a p p -adic group G G as a module over the maximal compact subgroup G 0 G_0 . We show that there are no non-trivial G 0 G_0 -intertwining maps between principal series representations attached to characters whose restrictions to the torus of G 0 G_0 are distinct, and there are no non-scalar endomorphisms of a fixed principal series representation. This is surprising when compared with another result which we prove: that a principal series representation may contain infinitely many closed G 0 G_0 -invariant subspaces. As for the proof, we work mainly in the setting of Iwasawa modules, and deduce results about G 0 G_0 -representations by duality.

Schur Orthogonality Relations for Non Square Integrable Representations of Real Semisimple Linear Group and Its Application

There are some mistakes in the paper cited in the title. By correcting these mistakes, we find that parameters of the spherical function are rational with respect to parameters of the (generalized principal series) representation. As an additional remark, we see that the shift operator is the Dunkl operator of A_1 -type.

Introduction. In this paper we study the question of reducibility of principal series representations of p-adic Chevalley groups. Let G be a p-adic Chevalley group, and let B be a Borel subgroup of G. If 'y is an irreducible finite-dimensional unitary representation of B, let lY = Ind y. Then flY is a BIG unitary (principal series) representation of G. It is well known that if w is an element of the restricted Weyl group of G,

In the next few chapters we will need an understanding of elements of the representation theory of noncompact semisimple Lie groups – esp. those representations which occur in the Plancherei theory. These representations fall into two large classes – the discrete series and the principal series. We will study the principal series in this chapter.

The tensor product of a positive and a negative discrete series representation of the quantum algebra U q (su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q−1, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of 2φ1-series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.

The decomposition of the tensor product of a positive and a negative discrete\nseries representation of the Lie algebra su(1,1) is a direct integral over the\nprincipal unitary series representations. In the decomposition discrete terms\ncan occur, and the discrete terms are a finite number of discrete series\nrepresentations or one complementary series representation. The interpretation\nof Meixner functions and polynomials as overlap coefficients in the four\nclasses of representations and the Clebsch-Gordan decomposition, lead to a\ngeneral bilinear generating function for the Meixner polynomials. Finally,\nrealizing the positive and negative discrete series representations as\noperators on the spaces of holomorphic and anti-holomorphic functions\nrespectively, a non-symmetric type Poisson kernel is found for the Meixner\nfunctions.\n

The quantum group analogue of the normalizer of SU (1, 1) in SL(2, C) is an important and non-trivial example of a non-compact quantum group.The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group.The first main goal of the paper is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra U q (su(1, 1)).It turns out that U q (su(1, 1)) does not suffice to generate the dual quantum group.The dual quantum group is graded with respect to commutation and anticommutation with a suitable analogue of the Casimir operator characterized by an affiliation relation to a von Neumann algebra.This is used to obtain an explicit set of generators.Having the dual quantum group the left regular corepresentation of the quantum group analogue of the normalizer of SU (1, 1) in SL(2, C) is decomposed into irreducible corepresentations.Upon restricting the irreducible corepresentations to U q (su(1, 1))-representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations.The detailed analysis of this example involves analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately.The paper is split into two parts; the first part gives almost all of the statements and the results, and the statements in the first part are independent of the second part.The second part contains the proofs of all the statements.

In this article explicit expressions are obtained for the action of the infinitesimal operators of the principal nonunitary series representations of the groups U(p,q) in a U(p) ×U(q) basis. It is moreover shown how the finite dimensional irreducible representations of the group U(p,q) and the group U(p+q) with respect to a U(p) ×U(q) basis are obtained from the principal nonunitary series representations of the group U(p,q).

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a special melodic condition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions.

We combine the geometric realization of principal series representations on partially holomorphic cohomology spaces, with the Bott–Borel–Weil theorem for direct limits of compact Lie groups, obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm-preserving representations on a Banach space or unitary representations on a Hilbert space. We specialize the results to diagonal embedding direct limit groups. Finally we discuss the possibilities of extending the results to limits of tempered series other than the principal series.