Property (RD) for Hecke pairs

TL;DR

This paper investigates property (RD) for Hecke pairs, establishing conditions under which Hecke pairs inherit (RD) from groups and exploring implications for noncommutative geometry and K-theory of Hecke C*-algebras.

Contribution

It characterizes property (RD) for Hecke pairs with finite subgroups and adapts existing results to the setting of Hecke C*-algebras, linking (RD) to algebraic and K-theoretic properties.

Findings

Hecke pairs with finite subgroup H have (RD) iff G has (RD).

The algebra of rapidly decreasing functions is closed under holomorphic functional calculus.

Hecke pairs with (RD) share the same K_0-groups as their associated C*-algebras.

Abstract

As the first step towards developing noncommutative geometry over Hecke C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989 to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C*-algebra. Hence they have the same K_0-groups.