Degenerate principal series representations for \(U(n,n)\)
type: Item
instance of: scholarly article
title: Degenerate principal series representations for \(U(n,n)\)
zbMATH Open document ID: 0896.22007
DOI: 10.1007/BF02937337
author: Stephen S. Kudla
author: W. Jay jun. Sweet
published in: Israel Journal of Mathematics
publication date: 1998-08-27T00:00:00Z
review text: Degenerate principal series representations of split classical groups over local fields which are induced from a character of the maximal parabolic subgroup with abelian unipotent radical play a central role in Howe correspondences. To understand this role it is essential to know points of reducibility, complete composition series, constituents arising from local theta-lifts, the behavior of the constituents under normalized intertwining operators and certain other information. In this paper the authors provide all this information for \(\text{U}(N,N)\) and \(\text{GL}_{2N}\) over a non-Archimedean local field. The methods used are similar to those used by the first author and \textit{S. Rallis} to study the case of \(\text{Sp}_{2N}\) [Isr. J. Math. 78, 209-256 (1992; Zbl 0787.22019)]. The results obtained have been applied to the study of local theta correspondences for unitary groups by the authors and \textit{M. Harris} [J. Am. Math. Soc. 9, 941-1004 (1996; Zbl 0870.11026)]. They are also of relevance for the extension of the Siegel-Weil formula to unitary groups. Questions of the reducibility of induced representations for classical groups have been studied by numerous authors; see, for instance, \textit{F. Shahidi} [Duke Math. J. 66, 1-41 (1992; Zbl 0785.22022), Invent. Math. 119, 1-36 (1995; Zbl 0852.22016)], \textit{D. Goldberg} [J. Reine Angew. Math. 448, 65-95 (1994; Zbl 0815.11029), Am. J. Math. 116, 1101-1151 (1994; Zbl 0851.22021)], \textit{Ch. Jantzen} [Compos. Math. 104, No. 1, 55-63 (1996; Zbl 0866.22017), Mem. Am. Math. Soc. 590, 100 p. (1996; Zbl 0866.22016)], \textit{M. Tadić} [Pitman Res. Notes Math. Ser. 311, 129-204 (1994; Zbl 0859.22011), J. Algebra 177, 1-33 (1995; Zbl 0874.22014)]. We will now summarize the main results of the paper under review. Let \(F\) be a non-Archimedean local field, and let \(E/F\) be a quadratic extension. Let \(G\) be the isometry group of the skew Hermitian form defined on the space of row vectors \(W=E^{2N}\) via \…
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 1033788
zbMATH Keywords: points of reducibility and composition series of principal series
zbMATH Keywords: Weil representation
zbMATH Keywords: split classical groups
zbMATH Keywords: Siegel-Weil formula
zbMATH Keywords: unitary group
zbMATH Keywords: Howe correspondences
zbMATH Keywords: intertwining operators
zbMATH Keywords: dual reductive pair
zbMATH Keywords: symplectic group
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02937337
OpenAlex ID: W1973198081
description: scientific article; zbMATH DE number 1033788