Whittaker supports for representations of reductive groups

Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $\pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\mathbb{R}$). For each nilpotent orbit $\mathcal{O}$ we consider a certain Whittaker quotient $\pi_{\mathcal{O}}$ of $\pi$. We define the Whittaker support WS$(\pi)$ to be the set of maximal $\mathcal{O}$ among those for which $\pi_{\mathcal{O}}\neq 0$. In this paper we prove that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $\pi$ is quasi-cuspidal then we show that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17]

Analytic continuation of equivariant distributions

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun

Raising and Lowering Operators for Askey-Wilson Polynomials

In this paper we describe two pairs of raising/lowering operators for Askey-Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the ''classical'' properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra

Bandes de « microbes » et insécurité à Abidjan

Le phénomène des gangs de « microbes » a tendance à s’étendre sur l’ensemble du territoire du district d’Abidjan, mais surtout à accroître le nombre des victimes. L’objectif de cette étude est de déterminer les actes criminels perpétrés par les « microbes » et de montrer comment ce phénomène est vécu au sein des populations. La méthodologie est basée sur une recherche documentaire et de terrain auprès des populations. L’analyse qualitative a permis de mettre en évidence la criminalité et le sentiment d’insécurité au sein de la population. Les résultats ont mis en exergue le fait que les multiples agressions des « microbes », à la fois violentes et récurrentes, ont conduit à de nombreux blessés voire décès des victimes. Aussi, ont-ils concouru à l’instauration d’un sentiment d’insécurité au sein des populations abidjanaise

Stein--Sahi complementary series and their degenerations

The aim of the paper is an introduction to Stein--Sahi complementary series, holomorphic series, and 'unipotent representations'. We also discuss some open problems related to these objects. For the sake of simplicity, we consider only the groups U(n,n).Comment: 40pp, 7fig, revised versio

Biorthogonal Expansion of Non-Symmetric Jack Functions

We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function

Isolation and expression analysis of salt stress-associated ESTs from contrasting rice cultivars using a PCR-based subtraction method

Salt stress adversely affects the growth of rice plants. To understand the molecular basis of salt-stress response, four subtracted cDNA libraries were constructed employing specific NaCl-stressed tissues from salt-tolerant (CSR 27 and Pokkali) and salt-sensitive (Pusa basmati 1) rice cultivars. An efficient PCR-based cDNA subtraction method was employed for the isolation of the salt-stress responsive cDNA clones. In all, 1,266 cDNA clones were isolated in the course of this study, out of which 85 clones were end-sequenced. Database search of the sequenced clones showed that 22 clones were homologous to genes that have earlier been implicated in stress response, 34 clones were novel with respect to their function and six clones showed no homology to sequences in any of the public database. Northern analysis showed that the transcript expression pattern of selected clones was variable amongst the cultivars tested with respect to stress-regulation

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic