Notes on the norm map between the Hecke algebras of the Gelfand–Graev representations of GL(2,q2) and U(2,q)

Let be a connected reductive algebraic group defined over the field Fq and let F and F∗ be two Frobenius maps such that Fm=m(F∗) for some integer m. Let , and be the finite groups of fixed points. In this article we consider the case where , F is the usual Frobenius map so that and F∗ is the twisted Frobenius map such that . In this case, F2=2(F∗) and . This article provides connections between the complex representation theory of these groups using the norm maps (see [C. Curtis, T. Shoji, A norm map for endomorphism algebras of Gelfand–Graev representations, in: Progr. Math., vol. 141, 1997, pp. 185–194]) from the Gelfand–Graev Hecke algebra of GL(2,q2) to the Gelfand–Graev Hecke algebras of both GL(2,q) and U(2,q).