An explicit integral representation of Whittaker functions on \(\text{Sp}(2;\mathbb{R})\) for the large discrete series representations
type: Item
instance of: scholarly article
title: An explicit integral representation of Whittaker functions on \(\text{Sp}(2;\mathbb{R})\) for the large discrete series representations
zbMATH Open document ID: 0812.11033
author: Takayuki Oda
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1994-10-16T00:00:00Z
review text: The author remarks that it is striking that the Whittaker functions for the discrete series representations of \(\text{SL} (2,\mathbb{R})\) are much simpler functions in comparison to the general case. He sets out to determine whether a similar phenomenon takes place for \(\text{Sp} (2,\mathbb{R})\) (of rank 2). He develops a theory of such functions and determines which differential equations they satisfy. He shows also that for generic discrete series representations the space of solutions is of dimension 4; otherwise, and in general, it is of dimension 0 or 8. The functions of moderate growth in this space form a subspace of dimension 0 or 1. He is then able to determine this function, when it exists, as a simple integral involving a confluent hypergeometric function. This is done by verifying that it satisfies the differential equations.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 650900
zbMATH Keywords: Whittaker functions
zbMATH Keywords: discrete series representations
zbMATH Keywords: \(\text{Sp} (2,\mathbb{R})\) (of rank 2)
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225761
OpenAlex ID: W1733933781
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DOI: 10.2748/TMJ/1178225761
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description: scientific article; zbMATH DE number 650900