Abstract
Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author have attached to any finite complex reflection group $W$ several notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular characters), completing the notion of Calogero-Moser families defined by Gordon. If moreover $W$ is a Coxeter group, they conjectured that these notions coincide with the analogous notions defined using the Hecke algebra by Kazhdan and Lusztig (or Lusztig in the unequal parameters case). In the present paper, we aim to investigate these conjectures whenever $W$ is a dihedral group.
Highlights
We do not prove all the conjectures but we get at least the following results:
If d is odd, Calogero–Moser cells coincide with the Kazhdan–Lusztig cells: this is a particular case of [6, Conjs
B ∈ R, the Kazhdan–Lusztig cellular characters for the dihedral groups are computable and a comparison with Table 4.1 shows that they coincide with Calogero–Moser cellular characters: this is [6, Conj
Summary
W is a dihedral group of order 2d, and (W, {s, t}) is a Coxeter system, where s2 = t2 = (st)d = 1. If d = 2e − 1 is odd, ξ = −ζ e and so τ = −se induces an inner automorphism of W (the conjugacy by se). Irreducible characters We denote by 1W the trivial character of W and let ε : W → C×, w → det(w). It is checked from (1.1) that ρk is a morphism of groups (that is, a representation of W). If R is any C-algebra, we still denote by ρk : RW → Mat2(R) the morphism of algebras induced by ρk. (2) If d is odd and k 0 mod d, χk is irreducible.
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