Search

Let $F$ be a $p$-adic field of characteristic zero and odd residual characteristic. Let $\mathbf{Sp}_{2n}(F)$ denote the symplectic group defined over $F$, where $n\geq 2$. We prove that the Speh representations $\mathcal{U}(\delta,2)$, where $\delta$ is a discrete series representation of $\mathbf{GL}_n(F)$, lie in the discrete spectrum of the $p$-adic symmetric space $\mathbf{Sp}_{2n}(F) \backslash \mathbf{GL}_{2n}(F)$., Comment: 30 pages, to appear in Represent. Theory; v3: Section 4.1 rewritten and Proposition 4.5 added, several minor errors corrected, bibliography updated

Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation pi such that O occurs in the wave front cycle of pi. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space., Comment: The class of nilpotent orbits studied in this paper is different from the class of noticed nilpotent orbits studied by Noel. A previous version of this paper erroneously stated that these two classes are the same. Representation Theory, Volume 16, 2012

Let E / F E/F be a quadratic extension of p p -adic fields. If π \pi is an admissible representation of G L n ( E ) GL_n(E) that is parabolically induced from discrete series representations, then we prove that the space of P n ( F ) P_n(F) -invariant linear functionals on π \pi has dimension one, where P n ( F ) P_n(F) is the mirabolic subgroup. As a corollary, it is deduced that if π \pi is distinguished by G L n ( F ) GL_n(F) , then the twisted tensor L L -function associated to π \pi has a pole at s = 0 s=0 . It follows that if π \pi is a discrete series representation, then at most one of the representations π \pi and π ⊗ χ \pi \otimes \chi is distinguished, where χ \chi is an extension of the local class field theory character associated to E / F E/F . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

Let G G be a reductive p \mathfrak {p} -adic group, K K a good maximal compact subgroup, K 1 ⊂ K {K_1} \subset K any open subgroup, and π \pi an admissible representation of G G of finite type. In A submersion principle and its applications, Harish-Chandra proves the theorem that ∫ K π ( k g k − 1 ) d k \int _K {\pi (kg{k^{ - 1}})\,dk} is a finite-rank operator for g g in the regular set G ′ G’ in order to show that the character Θ π ( g ) {\Theta _\pi }(g) is a locally constant class function on G ′ G’ . From this, the authors derive the formula θ ( 1 ) Θ π ( g ) = d ( π ) ∫ G / Z ∫ K 1 θ ( x k g k − 1 x − 1 ) d k d x ˙ ( g ∈ G ′ ) \theta (1){\Theta _\pi }(g) = d(\pi )\int _{G/Z} {\int _{{K_1}} {\theta (xkg{k^{ - 1}}{x^{ - 1}})\,dk\,d\dot x} \quad (g \in G’)} for any K K -finite matrix coefficient θ \theta of a discrete series representation π \pi with formal degree d ( π ) d(\pi ) . They use another technical result of the paper to prove that invariant integrals of Schwartz space functions converge absolutely. None of these results depends upon a characteristic zero assumption.

Let G G be a connected reductive linear Lie group with compact center and real rank l l . For each integer k ( 1 ⩽ k ⩽ l ) k(1 \leqslant k \leqslant l) and each discrete series representation π \pi of G G , an explicit embedding of π \pi into a generalized principal series representation induced from a parabolic subgroup of rank k k is given. The existence of such embeddings was proved by W. Schmid. In this paper an explicit integral formula with Szegö kernel is given which provides these mappings.

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