Infinitely Generated Hecke Algebras with Infinite Presentation

TY - JOUR

T1 - Infinitely Generated Hecke Algebras with Infinite Presentation

AU - Ciobotaru, Corina

N1 - Funding Information: We would like to thank Anders Karlsson for his encouragement to continue working on this article and his useful comments and Laurent Bartholdi, Ofir David, Iain Gordon, Alex Lubotzky and Agata Smoktunowicz for further discussions regarding finitely and infinitely generated algebras. This article was written when the author was a post-doc at the University of Geneva and University of M?nster. We would like to thank those institutions for their hospitality and good conditions of working. We also thank the referee for carefully reading the article and providing useful comments to improve the exposition. Publisher Copyright: © 2019, Springer Nature B.V.

PY - 2020/12/1

Y1 - 2020/12/1

N2 - 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 a 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 the Hecke algebra with respect to a 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. On the contrary, when F is 2–transitive that Hecke algebra of U(F)+ is commutative, finitely generated admitting a single generator.

AB - 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 a 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 the Hecke algebra with respect to a 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. On the contrary, when F is 2–transitive that Hecke algebra of U(F)+ is commutative, finitely generated admitting a single generator.

KW - Hecke algebras

KW - Locally compact groups

UR - http://www.scopus.com/inward/record.url?scp=85076576654&partnerID=8YFLogxK

U2 - 10.1007/s10468-019-09939-8

DO - 10.1007/s10468-019-09939-8

M3 - Journal article

AN - SCOPUS:85076576654

SN - 1386-923X

VL - 23

SP - 2275

EP - 2293

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 6

ER -