Infinitely Generated Hecke Algebras with Infinite Presentation
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Infinitely Generated Hecke Algebras with Infinite Presentation Corina Ciobotaru1 Received: 29 April 2017 / Accepted: 20 November 2019 / © Springer Nature B.V. 2019
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 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. Keywords Hecke algebras · Locally compact groups Mathematics Subject Classification (2010) 22D15
1 Introduction The Hecke algebras are very useful tools to study the representation theory of locally compact groups. For example, in the particular case of a semi-simple algebraic group G over a non-Archimedean local field there are two important Hecke algebras that can be associated with: the Hecke algebra of G with respect to a good maximal compact subgroup, called the spherical Hecke algebra of G, and the Hecke algebra of G with respect to a Iwahori subgroup, which is a smaller compact subgroup. The latter algebra is called the Iwahori–Hecke algebra of G and plays a very important role in the representation theory of algebraic groups, especially in the Kazhdan–Lusztig theory, being an intense and rich field of research. The former one is used to study the spherical unitary dual of semi-simple and analogous groups. That Hecke algebra is moreover commutative and finitely generated with respect to the Presented by: Iain Gordon Corina Ciobotaru
[email protected] 1
Universite de Fribourg, Fribourg, Switzerland
C. Ciobotaru
convolution product. The representation theory of both algebras is intimately related to the representation theory of G. In this article we restrict our attention to Hecke algebras associated with specific totally disconnected locally compact groups and their maximal compact subgroups. First, let us recall the general definition of a Hecke algebra and some more specific motivation. Definition 1.1 Let G be a locally compact group and K ≤ G be a compact subgroup. We denote by Cc (G, K) the space of continuous, compactly supported complex-valued functions φ : G → C that are K-bi-invariant, i.e., functions that satisfy the equality φ(kgk ) = φ(g) for every g ∈ G and all k, k ∈ K. We view the C-vector space Cc (G, K) as an algebra whose multiplication is given by the convolution product φ(xg)ψ(g −1 )dμ(g) φ ∗ ψ : x → G
where μ is the left Haar measure on G. Moreover, Cc (G,
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