Hypergeometric functions over finite fields and representations of \(\mathrm{SL}(2,q)\)
From the author's abstract and introduction: ``It is well known that the matrix coefficients of the representations of the discrete series of \(\mathrm{SL}(2, \mathbb{R})\) with respect to an appropriate basis are essentially hypergeometric functions. A parallel theory is presented here for the finite group \(\mathrm{SL} (2,\mathbb{F}_q)\).'' ``The organization of the paper is as follows: A construction of the principal series is given in section 2 (after a short introduction containing some historical remarks and motivation in section 1); matrix elements of these representations are given with respect to a canonical basis in section 3; these matrix elements are described in terms of hypergeometric functions over finite fields [as defined by the author in Trans. Am. Math. Soc. 301, 77--101 (1987; Zbl 0629.12017)] in section 4; this section contains also a description of how properties of hypergeometric functions over finite fields can be derived from this framework. Finally comments, questions and related presentations are given in section 5''.
scientific article; zbMATH DE number 515189
scientific article