Principal series and generalized principal series Whittaker functions with peripheral \(K\)-types on the real symplectic group of rank 2
type: Item
instance of: scholarly article
title: Principal series and generalized principal series Whittaker functions with peripheral \(K\)-types on the real symplectic group of rank 2
zbMATH Open document ID: 1229.11069
author: Yasuko Hasegawa
published in: Manuscripta Mathematica
publication date: 2011-01-14T00:00:00Z
review text: The paper under review contains a variety of results pertaining to Whittaker functions belonging to the standard representations of the group \(\mathrm{Sp}(2,\mathbb{R})\). These results generalize those of \textit{T. Ishii} [J. Funct. Anal. 225, No. 1, 1--32 (2005; Zbl 1078.11031)] which performed similar computations for principal series representations having minimal \(K\)-types; those of \textit{T. Miyazaki} and \textit{T. Oda} [Tohoku Math. J., II. Ser. 50, No. 2, 243--260 (1998; Zbl 1003.11020)] with computations for representations induced from the Jacobi maximal parabolic subgroup; and \textit{T. Oda} [Tohoku Math. J., II. Ser. 46, No. 2, 261--279 (1994; Zbl 0812.11033)] with computations for the discrete series representations. The paper is divided into three chapters each containing a different main result. The first main result of the paper under review computes the explicit formulas of principal series Whittaker functions on \(\mathrm{Sp}(2,\mathbb{R})\) with arbitrary one-dimensional \(K\)-types. In fact, the author deduces these results from those of Ishii by applying various Maass shift operators to the functions whose explicit formulas were computed by Ishii. These shift operators move the \(K\)-type from the Whittaker functions with minimal scalar \(K\)-types to ones having arbitrary scalar \(K\)-types. Additionally, the effect that these shift operators have on the explicit formula for the Whittaker functionals from representations with minimal scalar \(K\)-type are tractable enough for the author to use to derive the explicit formulas for the Whittaker functions from representations with with arbitrary scalar \(K\)-types. The second main result of the paper under review pertains to vector-valued Whittaker functions for representations \(\Pi\) which are induced from a certain maximal parabolic subgroup and have peripheral \(K\)-types. The maximal parabolic subgroup in question is the Siegel parabolic subgroup (denoted \(P_S\)). This parabolic subgr…
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 5836518
zbMATH Keywords: Whittaker functions
zbMATH Keywords: peripheral K-types
zbMATH Keywords: real symplectic group
zbMATH Keywords: Maass shift operator
zbMATH Keywords: holonomic systems
zbMATH Keywords: Siegel parabolic
reviewed by: Christian A. Zorn
MaRDI profile type: Publication
full work available at URL: s00229-010-0385-x
OpenAlex ID: W2002976062
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DOI: 10.1007/S00229-010-0385-X
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description: scientific article; zbMATH DE number 5836518