On the decomposition of a representation of \(GL(3)\) restricted to \(GL(2)\) over a \(p\)-adic field
type: Item
instance of: scholarly article
title: On the decomposition of a representation of \(GL(3)\) restricted to \(GL(2)\) over a \(p\)-adic field
zbMATH Open document ID: 0827.22006
DOI: 10.1215/S0012-7094-93-06908-6
author: Dipendra Prasad
published in: Duke Mathematical Journal
publication date: 1993-08-08T00:00:00Z
review text: Let \(F\) be a nonarchimedean local field. Consider irreducible admissible representations \(V\) of \(\text{GL}_N(F)\) and \(W\) of \(\text{GL}_{n- 1} (F)\). By restriction \(V\) is also a \(\text{GL}_{n - 1} (F)\)-module. A theorem of Bernstein asserts that the dimension of the space \(\text{Hom}_{\text{GL}_{n-1} (F)} (V,W)\) is at most one. The present paper is concerned with the question which pairs \((V,W)\) do have nonzero homomorphisms. A complete answer in terms of induced representations and, via Langlands-correspondence, in terms of Weil group representations is given for the case \(W = \mathbb{C}\) and \(n =3\). A conjecture for \(n > 3\) is stated. It further is proven that there are always homomorphisms if \(V\) and \(W\) are generic, i.e. possess unique Whittaker models.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 179234
zbMATH Keywords: nonarchimedean local field
zbMATH Keywords: irreducible admissible representations
zbMATH Keywords: restriction
zbMATH Keywords: induced representations
zbMATH Keywords: Weil group representations
zbMATH Keywords: Whittaker models
reviewed by: Anton Deitmar
MaRDI profile type: Publication
cites work: Q4110092
cites work: Induced representations of reductive ${\germ p}$-adic groups. I
cites work: Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup
cites work: Q4118032
cites work: Automorphic forms on \(GL(3)\). I, II
cites work: Rankin-Selberg Convolutions
cites work: Q3682555
full work available at URL: s0012-7094-93-06908-6
OpenAlex ID: W1511916494
description: scientific article; zbMATH DE number 179234