models. Adin, Postnikov, and Roichman [@APR] constructed a GIM for $G(r,n)$ extending Inglis, Richardson, and Saxl’s original model for $S_n$, which provides the starting point of [@Ma1; @Ma].

As mentioned at the outset, these sorts of classifications are interesting because they lead to interesting representations. We close this introduction with some recent evidence of this phenomenon. The model in Example \[exa2\] with $r=1$ gives rise via induction to a representation of $S_n$ on the vector space spanned by its involutions. This representation turns out to have a simple combinatorial definition [@APR2007 §1.1], which surprisingly makes sense *mutatis mutandis* for any Coxeter group. The generic Coxeter group representation we get in this way corresponds to an involution model (in the finite cases) in precisely types $A_n$, $H_3$, and $I_2(2m+1)$. What’s more, recent work of Lusztig and Vogan [@LV1; @LV2; @LV3] indicates that this representation is the specialization of a Hecke algebra representation which for Weyl groups is expected to have deep connections to the unitary representations of real reductive groups.

Preliminaries {#prelim-sect}
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Throughout we let $ [n] {\stackrel{\rm def}{=}}\{ i \in {\mathbb{Z}}: 1 \leq i \leq n \}$ denote the set of the first $n$ positive integers. Fix positive integers 