Hecke Algebras of Classical Groups over p-adic Fields II

Abstract

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p -adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair ( J Σ , ρ Σ ) of an open compact subgroup J Σ and its irreducible representation ρ Σ which is constructed from given data Σ = (Γ, P ′ 0 , ϱ). Here, Γ is a semisimple element in the Lie algebra of G, P ′ 0 is a parahoric subgroup in the centralizer of Γ in G, and ϱ is a cuspidal representation on the finite reductive quotient of P ′ 0 . In this paper, we explicitly describe those Hecke algebras when P ′ 0 is a minimal parahoric subgroup, ϱ is trivial and ρ Σ is a character.