Induced representations for finite groups of Lie type as Hecke algebra modules

Abstract

Let G be a split, connected reductive algebraic group taken over some p-adic field F having ring of integers O. Given a character 𝜒T/T(O)→Cx on the torus T, extended trivially to the Borel subgroup B, an extremely important class of functions in representation theory are linear functionals of the form: L : iG B(χ) → C. Consider a related type of linear functional that arises from doing a similar induction on the universal character χuniv : T/T(O) → C[P∨] mapping πµ → πµ. We are interested in the special functionals induced from this universal character: L : iG B(χ−1 univ) → C[P∨]. Some important examples of these special functionals include the Spherical functional, the Whittaker functional, and the Bessel model. These linear functionals also arise from our group G within a different context. Let I be the Iwahori subgroup of G and let ℌ denote the Hecke algebra, and ℌₒ the finite Hecke algebra of G. Take 𝜖 to be a character on ℌₒ and let Vϵ = IndH H0 (ϵ), then we can consider the special functions FVϵ : H → Vϵ as done in the work of Brubaker, Bump and Friedberg [1]. The purpose of this work is to provide an algebraic framework for explaining this coincidence and to use it as a dictionary to find new examples of interesting functionals. In particular we explore this topic from the finite field perspective as motivated by certain conjectures in the work of Kawanaka [2] concerning the Gelfand-Graev representation, which can be viewed as the Whittaker functional for the finite field case. Kawanaka provides a construction of generalized Gelfand-Graev representations (gGGrs) with the goal of classifying the irreducible representations of a finite reductive group G by using multiplicities of irreducible representations of G in gGGrs. We are interested in better understanding these types of linear functionals as they are inherently important in representation theory. The functionals resulting from these constructions provide information about the local components needed to construct integral representations of L-functions, around which many conjectures of the Langlands program are centered.