The generalized Whittaker functions for several admissible representations of \(Sp(2,\mathbb{R})\)
type: Item
instance of: scholarly article
title: The generalized Whittaker functions for several admissible representations of \(Sp(2,\mathbb{R})\)
zbMATH Open document ID: 0905.11025
author: Takuya Miyazaki
published in: Proceedings of the Japan Academy. Series A
publication date: 1999-01-18T00:00:00Z
review text: This note studies the generalized Whittaker functions attached to various types of admissible representations \(\pi\) of the rank two symplectic group \(G=Sp(2,{\mathbb R})\), including the large discrete series. The Whittaker functions in question transform under a character \(\eta\) of the unipotent radical of the Siegel parabolic \(P\) and also by a character of the identity component of the stabilizer subgroup \(SO(\eta)\) in the Levi of \(P\). For given \(K\)-type (\(K\) a fixed maximal parabolic subgroup), these functions are determined by restriction to the split maximal torus \(A\) of \(G\). Under certain assumptions on both the character \(\eta\) and the representation \(\pi\), the author obtains an explicit system of differential equations for the restriction of such a generalized Whittaker function to \(A\) by using the \(K\)-equivariant shift operators defined by Schmid. He moreover observes that this system of differential equations is completely integrable, and in certain cases he expresses the solution in terms of integrals of products of \(J\)-Bessel functions of the first kind and hypergeometric functions. In all cases treated, he computes the Mellin transform of this integral, which arises as the archimedean part of the Rankin-Selberg integral for the spinor \(L\)-function obtained by Andrianov, and he shows that this Mellin transform is a product of Gamma functions.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 1160952
zbMATH Keywords: generalized Whittaker function
zbMATH Keywords: shift operator
zbMATH Keywords: spin \(L\)-function
zbMATH Keywords: admissible representations of the rank two symplectic group
zbMATH Keywords: Mellin transform
reviewed by: Solomon Friedberg
MaRDI profile type: Publication
full work available at URL: pjaa.73.130
OpenAlex ID: W2019726849
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DOI: 10.3792/PJAA.73.130
description: scientific article; zbMATH DE number 1160952