Hecke Algebras of Group Extensions

Abstract

We describe the Hecke algebra ℋ(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ⊂ Γ/N satisfies S ⁻¹ S = Γ/N, we show that ℋ (Γ,Γ₀) is the twisted crossed product of ℋ (N,Γ₀) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.

Keywords:

• Hecke algebras
• Representation
• Twisted crossed product by semigroups

Mathematics Subject Classification:

• Primary 20C08
• Secondary 20C25, 46L55, 22D25, 43A62, 43A65

ACKNOWLEDGMENT

This research was supported by the Australian Research Council and the Higher Education Equity Program.

Notes

Communicated by D. Easdown.