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.