Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for ${\mathop {\protect \mathit{C}}\limits ^{\sim }}_2$

Abstract

We prove Lusztig's conjectures P1-P15 for the affine Weyl group of typeC2 for all choices of positive weight function. Our approach to computing Lusztig's a-function is based on the notion of a "balanced system of cell representations". Once this system is established roughly half of the conjectures P1-P15 follow. Next we establish an "asymptotic Plancherel Theorem" for typeC2, from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig's conjectures for all rank 1 and 2 affine Weyl groups for all choices of parameters.1 Kazhdan-Lusztig theory and balanced cell representations qs = q s ′ whenever s and s ′ are conjugate in W . For a given weight function L, we denote the above specialisation by ΘL : Hg → H.While Kazhdan-Lusztig theory is setup in terms of the specialised algebra H = HL, we will also need the generic algebra Hg at times in this paper (particularly in Section 6). We sometimes write Qs = qs − q −1 s , or Qs = q L(s) − q −L(s) depending on context (particularly in matrices for typesetting purposes). If S = {s0, . . . , sn} we will also often write, for example, 0121 as shorthand for s0s1s2s1, and thus in the Hecke algebra T0121 = Ts 0 s 1 s 2 s 1 . In particular, note that 1 is shorthand for s1, and therefore to avoid confusion we denote the identity of W by e. The Kazhdan-Lusztig basisLet L be a positive weight function and let H = HL. The involution¯on R which sends q to q −1 can be extended to an involution on H by settingIn [13], Kazhdan and Lusztig proved that there exists a unique basis {Cw | w ∈ W } of H such that, for all w ∈ W ,This basis is called the Kazhdan-Lusztig basis (KL basis for short) of H. The polynomials Py,w are called the Kazhdan-Lusztig polynomials, and to complete the definition we set Pw,w = 1 and Py,w = 0 whenever y < w (here ≤ denotes Bruhat order on W ) and Pw,w = 1 for all w ∈ W . We note that the Kazhdan-Lusztig polynomials, and hence the elements Cw, depend on the the weight function L (see the following example).Example 1.1. Let (W, S, L) be a Coxeter group and let J ⊆ S be such that the group WJ generated by J is finite. Let wJ be the longest element of WJ . The Kazhdan-Lusztig element Cw J is equal to w∈W J q L(w)−L(w J ) Tw. Indeed, this element has the required triangularity with respect to the standard basis and it is stable under the bar involution. Further, if we set Cw J := w∈W qwq −1 w J Tw ∈ Hg then we have ΘL(Cw J ) = Cw J for all positive weight functions L on W . Now assume that S contains two elements s1, s2 such that (s1s2) 4 = e. If we set a = L(s1) and b = L(s2) then we haveIndeed, the expressions on the right-hand side are stable under the bar involution and since they have the required triangularity property, they have to be the Kazhdan-Lusztig element associated to 212. Unlike the case where w = wJ , there is no generic element in Hg that specialises to C212 ∈ H(W, S, L) for all positive weight functions L. We also note that when b > a we have P2,212 = q −b−a − q −b+a , showing t...

A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters

“…Our approach extends naturally to all rank 2 affine Weyl groups. The analysis for the three‐parameter case becomes rather involved due to the large number of distinct regimes of cell decompositions, and the details are provided in .…”

“…Furthermore, our methods provide a theoretical framework that one may hope to apply to other types of affine Weyl groups. For instance, the approach outlined in this paper can be applied to the case, however the analysis is rather involved in this three‐parameter setting and so we provide the details in .…”

Conjectures P1-P15 for Coxeter groups with complete graph

“…In [Gui08b,Gui10], Guilhot explicitly determined the left and two-sided cells of affine Weyl groups of types B2 (or C2 ) and G2 . Based on the cell partitions, Guilhot and Parkinson gave a proof of P1-P15 for affine Weyl groups of type C2 and G2 , see [GP19b,GP19a]. They introduced a notion, called a balanced system of cell representations, which was inspired by the work [Gec11] of Geck for the finite case.…”