Branching rules obtained from explicit correspondences of automorphic representations
type: Item
instance of: scholarly article
title: Branching rules obtained from explicit correspondences of automorphic representations
zbMATH Open document ID: 1194.22018
author: David Ginzburg
published in: Israel Journal of Mathematics
publication date: 2009-10-02T00:00:00Z
review text: The paper under review is concerned with applications of the classical Casselman--Shalika formula [\textit{W. Casselman} and \textit{J. Shalika}, Compos. Math. 41, 207--231 (1980; Zbl 0472.22005), Theorem 5.4] to ``branching rules'' for finite-dimensional representations of complex groups. More specifically, note that, via the Weyl character formula, the cited formula expresses the unramified Whittaker function (of a generic, unramified representation) on a \(p\)-adic group \(G(F_v)\) in terms of characters of its complex Langlands dual group \({^LG}(\mathbb C)\). Historically, one has combined this formula with an understanding of the finite-dimensional, irreducible representations of the complex \(L\)-groups to deduce instances of Langlands functoriality, i.e., lifting of automorphic representations between adèlic groups \(H(\mathbb A)\) and \(G(\mathbb A)\). In the paper under review, Ginzburg demonstrates that the current state of understanding of explicit liftings is sufficiently advanced to allow information to flow the other way; that is, that one may use known examples of Langlands functoriality to deduce information about the decomposition into irreducible representations of \({^LH}(\mathbb C)\) of the restriction of an irreducible representation of \({^LG}(\mathbb C)\). The paper starts by recovering the classical branching rule for the restriction to \(\text{SO}_{2n - 1}(\mathbb C) = {^L\text{Sp}_{2(n - 1)}}(\mathbb C)\) of a representation of \(\text{SO}_{2n}(\mathbb C) = {^L\text{SO}_{2n}}(\mathbb C)\). We describe this in detail, since it is the model for what follows. The groups \(\text{SO}_{2n}\) and \(\text{Sp}_{2(n - 1)}\) (almost) form a dual pair in a large symplectic group, and one may use the Weil representation on the corresponding metaplectic group to obtain a lifting. (As observed on p.~31, \textit{R. Howe, E.-C. Tan} and \textit{J. F. Willenbring} [Trans. Am. Math. Soc. 357, No. 4, 1601--1626 (2005; Zbl 1069.22006)] have previously consider…
reviewed by: Loren Spice
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 5610595
zbMATH Keywords: branching rules
zbMATH Keywords: Langlands functoriality
zbMATH Keywords: Casselman-Shalika
MaRDI profile type: Publication
full work available at URL: s11856-009-0061-6
OpenAlex ID: W2092726533
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DOI: 10.1007/S11856-009-0061-6
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description: scientific article; zbMATH DE number 5610595