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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

$ {\rm GL}(4,{\bf R})$-Whittaker functions and $ {}\sb 4F\sb 3(1)$ hypergeometric series


Author: Eric Stade
Journal: Trans. Amer. Math. Soc. 336 (1993), 253-264
MSC: Primary 22E30; Secondary 11F55, 33C15, 33C20
MathSciNet review: 1102226
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Abstract: In this paper we consider spaces of $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions, which are special functions that arise in the study of $ {\text{GL}}(4,\mathbb{R})$ automorphic forms. Our main result is to determine explicitly the series expansion for a $ {\text{GL}}(4,\mathbb{R})$-Whittaker function that is "fundamental," in that it may be used to generate a basis for the space of all $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions of fixed eigenvalues.

The series that we find in the case of $ {\text{GL}}(4,\mathbb{R})$ is particularly interesting in that its coefficients are not merely ratios of Gamma functions, as they are in the lower-rank cases. Rather, these coefficients are themselves certain series-- namely, they are finite hypergeometric series of unit argument. We suspect that this is a fair indication of what will happen in the general case of $ {\text{GL}}(n,\mathbb{R})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1102226-1
PII: S 0002-9947(1993)1102226-1
Keywords: Whittaker functions, automorphic forms, hypergeometric series
Article copyright: © Copyright 1993 American Mathematical Society