Infinitely generated Hecke algebras with infinite presentation

TL;DR

This paper investigates the structure of Hecke algebras associated with certain totally disconnected groups, revealing that for some groups they are infinitely generated and presented, contrasting with the finitely generated case when the group is 2-transitive.

Contribution

It proves that Hecke algebras for universal groups U(F)^+ with primitive but not 2-transitive F are infinitely generated and presented, expanding understanding of their algebraic complexity.

Findings

Hecke algebras are infinitely generated for primitive but not 2-transitive F.

Hecke algebras are finitely generated and commutative when F is 2-transitive.

Results may impact the construction of irreducible unitary representations and the classification of U(F)^+.

Abstract

For a locally compact group G and a compact subgroup K, the corresponding Hecke algebra consists of all continuous compactly supported complex functions on G that are K-bi-invariant. There are many examples of totally disconnected locally compact groups whose Hecke algebras with respect to the maximal compact subgroups are not commutative. One of those is the universal group U(F)^+, when F is primitive but not 2-transitive. For this class of groups we prove that the Hecke algebra with respect to the maximal compact subgroup K is infinitely generated and infinitely presented. This may be relevant for constructing irreducible unitary representations of U(F)^+ whose subspace of K-fixed vectors has dimension at least two, or answering the question whether U(F)^+ is a type I group or not. On the contrary, when F is 2-transitive that Hecke algebra of U(F)^+ is commutative, finitely generated admitting only one generator.