Whittaker functions for the large discrete series representations of \(\mathrm{SU}(2,1)\) and related zeta integrals
type: Item
instance of: scholarly article
title: Whittaker functions for the large discrete series representations of \(\mathrm{SU}(2,1)\) and related zeta integrals
zbMATH Open document ID: 0867.11039
author: Harutaka Koseki
author: Takayuki Oda
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1996-08-21T00:00:00Z
review text: The authors study the smooth Whittaker functions for certain discrete series representations of the real semisimple group \(\mathrm{SU}(2,1)\). This group has three types of discrete series: holomorphic, anti-holomorphic, and neither. It is the third type which is considered here. These discrete series representations are those of Gelfand-Kirillov dimension 3, the dimension of the maximal unipotent subgroup; hence they are large in the sense of Vogan. By a theorem of Kostant, they possess a unique smooth Whittaker model. In this paper the authors give explicit formulas for these Whittaker functions and for their Mellin transforms. The authors first describe the Harish-Chandra parameters associated to the large discrete series representations, and compute their \(K\)-types. They compute the Whittaker functions by first identifying the relevant intertwining space with the solution space to a certain partial differential operator. In the case of minimal \(K\)-type, they further reduce this to a system of differential-difference equations, which they solve. This gives an explicit formula for (the radial part) of the minimal \(K\)-type Whittaker function in terms of classical Bessel functions. Next the authors give recurrence relations for the weight vectors in all \(K\)-type subspaces of the Whittaker model. This step uses the observation that the restriction of the Whittaker model of a given large discrete series representation of \(\mathrm{SU}(2,1)\) to \(K\) is multiplicity free. Using these formulas, the authors compute the Mellin transforms of the \(K\)-finite Whittaker functions in terms of polynomials and Gamma factors. The authors apply their results above to carry out the evaluation of certain local zeta integrals in the archimedean case. Namely, \textit{S. Gelbart} and \textit{I. Piatetski-Shapiro} [in: Lie group representations II, Lect. Notes Math. 1041, Springer-Verlag, 141--184 (1984; Zbl 0531.10034)] attached certain zeta-integrals to a generic representa…
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F60
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 906878
zbMATH Keywords: archimedean local \(L\)-factor
zbMATH Keywords: large discrete series representations
zbMATH Keywords: Mellin transforms
zbMATH Keywords: Harish-Chandra parameters
zbMATH Keywords: local zeta integrals
zbMATH Keywords: partial automorphic \(L\)-functions
zbMATH Keywords: unitary group
zbMATH Keywords: smooth Whittaker functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195163592
OpenAlex ID: W2064132246
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DOI: 10.2977/PRIMS/1195163592
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description: scientific article; zbMATH DE number 906878