The integral kernel in the Kuznetsov sum formula for \(\text{SU}(n+1,1)\). II: The case of one dimensional \(K\)-types
type: Item
instance of: scholarly article
title: The integral kernel in the Kuznetsov sum formula for \(\text{SU}(n+1,1)\). II: The case of one dimensional \(K\)-types
zbMATH Open document ID: 1066.22010
author: N. P. Kisbye
published in: Pacific Journal of Mathematics
publication date: 2004-11-01T00:00:00Z
review text: In the context of connected noncompact semisimple Lie groups of rank one, a Kuznetsov sum formula was determined by \textit{R. J. Miatello} and \textit{N. R. Wallach} [J. Funct. Anal. 93, 171--206 (1990; Zbl 0711.11023)]. It relates spectral data concerning automorphic forms to geometric data concerning the intersection of a discrete subgroup of finite covolume with the big cell in the Bruhat decomposition. Test functions on the spectral and on the geometric sides are linked by an integral transformation. The \(\tau\)-function appears in the kernel of this transformation. In the case of trivial \(K\)-types, explicit formulas for the \(\tau\)-function were determined by Miatello and Wallach [loc. cit.] for \(SO(n+1,1)\) and \(SU(2,1)\), and by the author in Part I of this paper [Indag. Math. 12, 83--101 (2001; Zbl 1039.11032)] for \(SU(n+1,1)\). The paper under review determines an explicit formula for the \(\tau\)-function in the case of the group \(SU(n+1,1)\) and one-dimensional \(K\)-types. The computation relies on the solution of complicated recurrence relations for the coefficients appearing in a series expansion of the \(\tau\)-function.
reviewed by: Angela Pasquale
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 2113418
zbMATH Keywords: Kuznetsov sum formula
zbMATH Keywords: \(\tau\)-function
zbMATH Keywords: Whittaker vectors
MaRDI profile type: Publication
full work available at URL: pjm.2004.213.295
OpenAlex ID: W2085881515
DOI: 10.2140/PJM.2004.213.295
description: scientific article; zbMATH DE number 2113418