Allan J. Silberger and Ernst-Wilhelm Zink
The Shintani descent of a cuspidal representation of GL_n(k_d)
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

20C33 Representations of finite groups of Lie type

22E50 Representations of Lie and linear algebraic groups over local fields

Abstract: Let k be a finite field, k_d | k the degree d extension of k and G(k_d) the general linear group GL_n with entries in k_d. Shintani (T. Shintani, Two Remarks on Irreducible Characters of Finite General Linear Groups, J. Math. Society of Japan 28 (1976), 396-414) showed how to associate to any generator of the Galoisgroup Gal(k_d | k) a bijective mapping j from the set of Galois invariant characters of G(k_d) to the set of all irreducible characters of G(k) which in a natural way generalizes the bijection X ° Norm \mapsto X for G = G₁.
Let Theta be a Galois invariant cuspidal character of G(k_d). Then (d,n) = 1, and the Green's parameter of Theta is of the form [X ° Norm] where X is a character of k^x_n and the norm refers to k_dn | k_n. In this paper we show that j(Theta) is cuspidal and has [X] as its Green's parameter, independently of the choice which has been made to define j. These facts follow from more general assertions proved by global methods for GL_n over local fields (J. G. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Study 120, Princeton U. Press, 1989). The arguments of the present paper make use of only the classical theory of representations of general linear groups over finite fields.

Keywords: general linear group, finite field, characters, base change, Shintani descent