Alcove walks and nearby cycles on affine flag manifolds

Görtz, Ulrich

doi:10.1007/s10801-007-0063-6

Cited by 16 publications

(19 citation statements)

References 14 publications

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“…Consider the associated straight alcove walk

(v_{0}, v_{1} \dots, v_{ℓ})

, where

v_{0} = 1

and

v_{k} = s_{i_{1}} \dots s_{i_{k}}

. Let

ε_{1}, \dots, ε_{ℓ}

be defined using the periodic orientation on hyperplanes as follows:

ε_{k} = \{\begin{matrix} left + 1 & left if v_{k - 1}^{-} ∣^{+} v_{k} (that is, a positive crossing) \\ left - 1 & left if v_{k}^{-} ∣^{+} v_{k - 1} (that is, a negative crossing). \end{matrix}

It turns out that the element

X_{v} = T_{s_{i_{1}}}^{ε_{1}} \dots T_{s_{i_{ℓ}}}^{ε_{ℓ}}

scriptH

does not depend on the particular expression

v = s_{i_{1}} \dots s_{i_{ℓ}}

we have chosen (see ). If

λ \in Q

we write

X^{λ} = X_{t_{λ}}

…”

Section: Affine Weyl Groups Affine Hecke Algebras and Alcove Walksmentioning

confidence: 99%

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

Guilhot

Parkinson

2018

Proc. London Math. Soc.

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We prove Lusztig's conjectures P1–P15 for the affine Weyl group of type G∼2 for all choices of parameters. Our approach to compute Lusztig's bolda‐function is based on the notion of a ‘balanced system of cell representations’ for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the bolda‐function and we explicitly construct such a system in type G∼2 for arbitrary parameters. We then investigate the connection between Kazhdan–Lusztig cells and the Plancherel theorem in type G∼2, allowing us to prove P1 and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types G2 and A1, along with some explicit computations for the finite cells.

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“…Consider the associated straight alcove walk

(v_{0}, v_{1} \dots, v_{ℓ})

, where

v_{0} = 1

and

v_{k} = s_{i_{1}} \dots s_{i_{k}}

. Let

ε_{1}, \dots, ε_{ℓ}

be defined using the periodic orientation on hyperplanes as follows:

ε_{k} = \{\begin{matrix} left + 1 & left if v_{k - 1}^{-} ∣^{+} v_{k} (that is, a positive crossing) \\ left - 1 & left if v_{k}^{-} ∣^{+} v_{k - 1} (that is, a negative crossing). \end{matrix}

It turns out that the element

X_{v} = T_{s_{i_{1}}}^{ε_{1}} \dots T_{s_{i_{ℓ}}}^{ε_{ℓ}}

scriptH

does not depend on the particular expression

v = s_{i_{1}} \dots s_{i_{ℓ}}

we have chosen (see ). If

λ \in Q

we write

X^{λ} = X_{t_{λ}}

…”

Section: Affine Weyl Groups Affine Hecke Algebras and Alcove Walksmentioning

confidence: 99%

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

Guilhot

Parkinson

2018

Proc. London Math. Soc.

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“…The regimes (r1, r2) ∈ Rj with j = 1, 2, 3 are "generic", and admit cell factorisations where 1,10,12) if j = 2 (e, 0, 01, 010) if j = 3…”

Section: The Cell γmentioning

confidence: 99%

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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“…As for the zν ,J , the main problem is to compute them explicitly when J = I is an Iwahori subgroup of G(F ), in terms of the Iwahori-Matsumoto generators T w (w ∈ W τ ) for the Iwahori-Hecke algebra H(G(F ), I). This can be done using the theory of alcove-walks, see [Gor07] and [HR12,Appendix]. Thus, in principle, the geometric basis elements Cλ ,J can be computed explicitly, in every case.…”

Section: 2mentioning

confidence: 99%

Dualities for root systems with automorphisms and applications to non-split groups

Haines

2018

Represent. Theory

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This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over nonarchimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the {µ}-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Titséchelonnage root system Σ0, the Knop root system Σ0, and the Macdonald root system Σ1, in terms of Galois actions on the absolute roots Φ; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis. The latter gives an explicit form of the test function conjecture for general Shimura varieties with parahoric level structure. ) is an Iwahori subgroup, and if M µ is the associated Rapoport-Zink local model [RZ], then Theorem C ensures that we have a good understanding of the set of irreducible components in the special fiber of M µ . See §4 for more discussion.We now assume again that G/F is quasi-split. With Theorems B and C in hand, we can construct and study a geometric basis for the center Z(G(F ), J) of the parahoric Hecke algebra associated to G(F ) and an parahoric subgroup J ⊂ G(F ). Recall that W F acts on the dual group G, preserving a splitting ( B, T , X). A dual-group consequence of Theorem C is that there is an isomorphism of G I -modulesfor aλ ,µ ∈ Z ≥0 . Hereλ ∈ X * ( T I ) is the image of λ ∈ X * ( T ), V µ and Vλ are irreducible highest weight representations of the reductive groups G and G I , and Wt(μ) is the set of T I -weights in Vμ. These extend to representations V I µ and Vλ ,1 of G I ⋊ τ (see §5.2 and §7.1). The following theorem summarizes Lemma 7.4 and Theorem 7.5.Theorem D. There is a basis {Cλ ,J }λ for Z(G(F ), J) indexed byλ ∈ X * ( T I ) +,τ , characterized by:Cλ ,J acts on π J by the scalar tr(s(π) ⋊ τ | Vλ ,1 ) whenever π is an irreducible smooth representation of G(F ) with π J = 0 and Satake parameter s(π) (see [H15]). Furthermore, in terms of Bernstein functions zν ,J ∈ Z(G(F ), J) and certain Kazhdan-Lusztig polynomials P wν ,wλ (q 1/2 ) associated to the affine Hecke algebra with parameters H( W τ , S τ aff , L) there is an equalityWe call the Cλ ,J the geometric basis elements for the following reason: when F = F q ((t)) and) is a very special maximal parahoric subgroup of G(F q ((t))), Cλ ,J is the element of H(G(F q ((t))), J) arising, via the function-sheaf dictionary, from the equivariant perverse sheaf on the affine flag variety LG/L + P f which corresponds to Vλ ∈ Rep( G I ) under the geometric Satake isomorphism (see [Ric2], [Zhu1]). Moreover, in the

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Alcove walks and nearby cycles on affine flag manifolds

Abstract: Using Ram's theory of alcove walks we give a proof of the Bernstein presentation of the affine Hecke algebra. The method works also in the case of unequal parameters. We also discuss how these results help in studying sheaves of nearby cycles on affine flag manifolds.

Cited by 16 publications

References 14 publications

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

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

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

Dualities for root systems with automorphisms and applications to non-split groups

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