Representations of Hecke Algebras and Dilations of Semigroup Crossed Products

We consider a family of Hecke C∗-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group. Suppose that M is a subgroup of a group Γ such that |M ∩ γMγ| is finite for every γ ∈ Γ; we say that M is an almost normal subgroup or that (Γ,M) is a Hecke pair. The Hecke algebra H(Γ,M) is a convolution ∗-algebra of functions on the double coset space M\Γ/M , which can be represented in the commutant of the quasi-regular representation IndΓM 1 [8]; C -algebraic completions of Hecke algebras played a fundamental role in the analysis by Bost and Connes of phase transitions in number theory [2]. Several authors have since investigated classes of C-Hecke algebras which can be realised as semigroup crossed products [13, 1, 3, 11], and these realisations have provided valuable insight to the work of Bost and Connes [9, 18]. A recent theorem of Hall [5] asserts that, for a large class of Hecke pairs (Γ,M), the category of nondegenerate representations of the Hecke algebra H(Γ,M) is equivalent to the category C(Γ,M) of unitary representations of Γ which are generated by their M-fixed vectors. From an operator-algebraic point of view, C(Γ,M) seems an unusual category: it is not obvious, for example, whether it is the category of representations of some familiar C-algebra. Here we consider a class of Hecke algebras H(Γ,M) which can be realised as semigroup crossed products, and show that for these pairs (Γ,M), a representation of Γ is in C(Γ,M) precisely when it has been dilated from a representation of the corresponding semigroup crossed product. We then use the dilation-extension theory of Laca [10] to identify a single locally compact group Γ∞ whose category of continuous unitary representations is equivalent to C(Γ,M). We begin in §1 by describing a family (Γ,M) of Hecke pairs in which Γ is a semidirect product N ⋊ G, M is a normal subgroup of N , and G has the form SS for some subsemigroup S. It is shown in [11], generalising results in [3] and [1], that the associated Hecke algebra H(Γ,M) is isomorphic to a semigroup crossed product C(N/M)⋊αS. In Theorem 1.1, we show that for each covariant representation (π, V ) of (C(N/M), S, α) and each minimal unitary dilation U of V , there is a representation of N ⋊ G on HU which is generated by its M-fixed vectors; more importantly, we show that they all arise this way. Date: 8 June 2001. 1991 Mathematics Subject Classification. 46L55. The first author and her research were supported by the Danish Natural Science Research Council, and the research was also supported by the Australian Research Council.