THE SHINTANI DESCENT OF A CUSPIDAL REPRESENTATION OF GLn(kd)

Let k be a nite eld, k d jk the degree d extension of k, and G(k d) := GL n (k d) the general linear group with entries in k d. showed how to associate to any generator of the Galois group Gal(k d jk) a bijective mapping j from the set of Gal(k d jk)-invariant characters of G(k d) to the set of all irreducible characters of G(k) := GL n (k) which in a natural way generalizes the bijection N k d jk 7 ! from the set of Gal(k d jk)-invariant characters of k d = GL 1 (k d) to the set of characters of k = GL 1 (k). Let be a Gal(k d jk)-invariant cuspidal character of G(k d). Then (d; n) = 1 and the Green's parameter dn ] of is of the form n N k dn jk n ], where n ] is a Gal(k n jk)-orbit of k n jk-regular characters of k n and N k dn jk n : k dn ! k n is the norm mapping. In this paper we show that j (() is cuspidal and that the Green's parameter of j (() is n ], independent of the choice of generator. These facts follow from more general assertions proved by global methods for GL n over local elds (J. The arguments of the present paper make use of only the classical theory of representations of general linear groups over nite elds. 0. Introduction x0.1 Some Notation. Let k = F q be a nite eld of cardinality q, let k be an algebraic closure of k, and let F : x 7 ! x q be the Frobenius morphism of kjk. Let k ` , k k ` k, denote the xed eld of the cyclic group hFì for any`1. For any positive integer m we write G m := GL m , considered as an algebraic group. We x a positive integer n and write G := G n. For any k `-subgroup H of G m we write H(k `) to denote the group of k `-points of H. For any nite group Y we write Y ^ to denote the set of irreducible unitary representations of Y and X(Y) for the group of one-dimensional characters of Y or Y=Y; Y]. We let F act on G(k d) by letting it …