Supercuspidal representations and the theta correspondence
type: Item
instance of: scholarly article
title: Supercuspidal representations and the theta correspondence
zbMATH Open document ID: 0780.22003
DOI: 10.1016/0021-8693(92)90120-B
author: David Manderscheid
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: Howe's local theta correspondence is carefully re-examined for the dual pairs \(((SL_ 2(F), O(F))\), where \(F\) is a nonarchimedean local field of odd residual characteristic, and \(O(F)\) is the orthogonal group of a quadratic form in one or two variables; in particular, the representations of \(O(F)\) which occur in this correspondence are described, and the corresponding representations of \(SL_ 2(F)\) (if super-cuspidal) are determined. As the author himself emphasizes, the correspondences studied in this paper have been studied already elsewhere [cf. the reviewer's survey paper in Proc. Symp. Pure Math. 33, 1, 287-296 (1979; Zbl 0425.22024)]; therefore, the thrust of this paper is not the results obtained, but rather the techniques used. Indeed, the two main methods used are the lattice model for the oscillator (or ``Weil'') representation, and the parametrization of supercuspidal representations via induction from compact open subgroups.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 165843
zbMATH Keywords: Howe's local theta correspondence
zbMATH Keywords: dual pairs
zbMATH Keywords: nonarchimedean local field
zbMATH Keywords: orthogonal group
zbMATH Keywords: quadratic form
zbMATH Keywords: representations
zbMATH Keywords: supercuspidal representations
reviewed by: Stephen Gelbart
MaRDI profile type: Publication
cites work: Hereditary orders, Gauss sums and supercuspidal representations of GLN.
cites work: Non-Abelian Congruence Gauss Sums and p -Adic Simple Algebras
cites work: On the Representations of SL 2 (k) Related to Binary Quadratic Forms
cites work: Q3848237
cites work: Weil's representation and the spectrum of the metaplectic group
cites work: Q3860034
cites work: Weil representations associated to finite fields
cites work: Q3856974
cites work: Tamely ramified supercuspidal representations of \(Gl_n\)
cites work: Q3939864
cites work: On the supercuspidal representations of ${\rm GL}_N,\;N$ the product of two primes
cites work: On the supercuspidal representations of \(SL_ 2\) and its twofold cover. I, II
cites work: Supercuspidal Duality for the Two-Fold Cover of SL 2 and the Split O 3
cites work: Howe correspondences on a \(p\)-adic field
cites work: Some Tamely Ramified Supercuspidal Representations of Symplectic Groups
cites work: On an Explicit Construction of a Certain Class of Automorphic Forms
cites work: Q5679031
cites work: On irreducible unitary representations of some special linear groups of the second order. I, II
cites work: Construction and classification of irreducible representations of special linear group of the second order over a finite field
cites work: Q3209342
cites work: Q4090435
cites work: Sur certains groupes d'opérateurs unitaires
full work available at URL: 0021-8693(92)90120-b
OpenAlex ID: W2074144506
description: scientific article; zbMATH DE number 165843