Legendre sums, Soto–Andrade sums and Kloosterman sums

Anthony C. Kable

Vol. 206 (2002), No. 1, 139–157

DOI: 10.2140/pjm.2002.206.139

Abstract

The three sums named in the title are all known to appear in connection with the complex representation theory of GL(2,q). The first two are incarnations of certain spherical vectors, whereas the third is a matrix coefficient for a parabolic basis. In this work, Legendre and Soto-Andrade sums are shown to occur in a second way, as parabolic Clebsch-Gordan coefficients for the tensor product of two Steinberg representations. This realization connects them with Kloosterman sums, and from it we derive a number of identities.

Milestones

Received: 28 November 2000

Revised: 21 November 2001

Published: 1 September 2002