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

Guilhot, Jeremie; Parkinson, James

doi:10.1112/plms.12211

Cited by 6 publications

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“…Finally we note that we have slightly changed the numbering from [12], where B5 was denoted B4 ′ , and B6 was denoted B5.…”

Section: )mentioning

confidence: 99%

“…In [12] we showed that the existence of a balanced system of cell representations is sufficient to compute Lusztig's a-function. In particular, we have: Note that the first part of this theorem implies that the bounds aπ Γ in Definition 1.5 are in fact unique.…”

Section: )mentioning

confidence: 99%

“…We begin this section with some basic facts about root systems and Weyl groups. We then recall the combinatorial language of alcove paths from [21], and the concept of alcove paths confined to strips from [12]. We also discuss the combinatorics of the affine Hecke algebra (and extended affine Hecke algebra) of typeC2.…”

Section: Affine Weyl Groups Affine Hecke Algebras and Alcove Pathsmentioning

confidence: 99%

“…One of the main challenges in proving Lusztig's conjectures is to compute Lusztig's a-function since, in principle, it requires us to have information on all the structure constants with respect to the Kazhdan-Lusztig basis. In [12] we showed that the existence of a balanced system of cell representations is sufficient to compute the a-function. Such a system is a family (πΓ) Γ∈Λ of representations of H, each equipped with a distinguished basis, satisfying various axioms including (1) πΓ(Cw) = 0 for all w ∈ Γ ′ with Γ ′ ≥LR Γ, (2) the maximal degree of the coefficients that appear in the matrix πΓ(Cw) is bounded by a constant aπ Γ , (3) this bound is attained if and only if w ∈ Γ.…”

Section: Introductionmentioning

confidence: 99%

“…We conclude this introduction with an outline of the structure of the paper. In Section 1 we recall the basics of Kazhdan-Lusztig theory, and we recall the axioms of a balanced system of cell representations from [12]. Section 2 provides background on affine Weyl groups, root systems, the affine Hecke algebra, and the combinatorics of alcove paths.…”

Section: Introductionmentioning

confidence: 99%

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

Guilhot¹,

Parkinson²

2019

Algebraic Combinatorics

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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...

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“…Finally we note that we have slightly changed the numbering from [12], where B5 was denoted B4 ′ , and B6 was denoted B5.…”

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%