Gelfand's trick for the spherical derived Hecke algebra

TL;DR

This paper extends Gelfand's classical method to demonstrate that the spherical derived Hecke algebra is graded-commutative, broadening understanding of algebraic structures in p-adic groups.

Contribution

It adapts Gelfand's trick to prove graded-commutativity of the spherical derived Hecke algebra under mild conditions.

Findings

Spherical derived Hecke algebra is graded-commutative.

Gelfand's trick can be adapted to derived settings.

Provides conditions for graded-commutativity.

Abstract

Gelfand's trick shows that the spherical Hecke algebra of a $p$-adic split reductive group is commutative. We adapt this strategy in order to show that the spherical derived Hecke algebra is graded-commutative under mild assumptions on the coefficient ring.