$$P$$ P -alcoves, parabolic subalgebras and cocenters of affine Hecke algebras

He, Xuhua; Nie, Sian

doi:10.1007/s00029-014-0170-x

Cited by 9 publications

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“…By Proposition 2.8 there exists a standard Levi subgroup M ⊂ G and an Mbasic element x M ∈ Ω M such that x and x M areW G -conjugate. By[15, Prop. 4.5] any two such elements are even W -conjugate and thus correspond to the same element in X * (T ) dom .…”

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confidence: 99%

Irreducible components of minuscule affine Deligne–Lusztig varieties

Hamacher

Viehmann

2018

Alg. Number Th.

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We examine the set of J b (F )-orbits in the set of irreducible components of affine Deligne-Lusztig varieties for a hyperspecial subgroup and minuscule coweight µ. Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with µ of the dual group.The authors were partially supported by ERC starting grant 277889 "Moduli spaces of local G-shtukas".The following basic assertion seems to be well-known, but we could not find a reference in the literature.Lemma 1.1. The scheme X µ (b) is locally of finite type in the equal characteristic case and locally of perfectly finite type in the case of unequal characteristic.Proof. The proof of this is the same as the corresponding part of the analogous statement for moduli spaces of local G-shtukas, compare the proof of Theorem 6.3 in [12] (where only the first half of p. 113 of loc. cit. is needed). In that proof, the case of equal characteristic and split G is considered. However, the general statement follows from the same proof.Notice that in general X µ (b) is not quasi-compact since it may have infinitely many irreducible components. It is conjectured to be equidimensional, but this has not been proven in full generality yet. In Section 3 we give an overview about the cases where equidimensionality has been proven. In the case of µ minuscule, which we are primarily interested in here, there are only a few exceptional cases where this is not yet known.Definition 1.2. For a finite-dimensional k-scheme X we denote by Σ(X) the set of irreducible components of X and by Σ top (X) ⊂ Σ(X) the subset of those irreducible components which are top-dimensional.The affine Deligne-Lusztig varieties X µ (b) and X µ (b) carry a natural action (by left multiplication) by the groupThis action induces an action of J b (F ) on the set of irreducible components.A complete description of the set of orbits was previously only known for the groups GL n and GSp 2n and minuscule µ where the action is transitive ([24],[25]), and for some other particular cases, see for example [28] for a particular family of unitary groups and minuscule µ.To describe the (conjectured) number of orbits, denote byĜ the dual group of G in the sense of Deligne and Lusztig. That is,Ĝ is the reductive group scheme over O F that contains a Borel subgroupB with maximal torusT and maximal split torusŜ such that there exists an Galois equivariant isomorphism X * (T ) ∼ = X * (T ) identifying simple coroots ofT with simple roots of T . For any µ ∈ X * (T ) dom = X * (T ) dom we denote by V µ the associated Weyl module ofĜ OL .In the following we use an element λ G (b) ∈ X * (T Γ ) that we define in Section 2. Its restriction λ toŜ can be seen as a 'best integral approximation' of the Newton point ν b of [b], while its precise value in X * (T Γ ) will depend on the Kottwitz point κ G (b). We choose a liftλ ∈ X * (T ). Conjecture 1.3 (Chen, Zhu). There exists a canonical bijection between J b (F )\Σ(X µ (b)) and the basis of V µ (λ G ...

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confidence: 99%

Irreducible components of minuscule affine Deligne–Lusztig varieties

Hamacher

Viehmann

2018

Alg. Number Th.

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“…In this section, we recall the explicit description of the cocenter of H obtained in [HN1] and [HN2]. 5.1.…”

Section: Spanning Set Of the Cocentermentioning

confidence: 99%

“…However, the expression for a minimal length element T w in O is usually very complicated. To study the induction and restriction functors on the cocenter, we also need the Bernstein-Lusztig presentation ofH ′ established in [HN2]. 5.4.…”

Section: Spanning Set Of the Cocentermentioning

confidence: 99%

“…By [HN2,Lemma 5.8], the map from A to the set of conjugacy classes of W ′ sending (J, C) to the unique conjugacy class O of W with C ⊂ O gives a bijection from A/ ∼ to the set of conjugacy classes of W . We denote this map by τ .…”

Section: Spanning Set Of the Cocentermentioning

confidence: 99%

“…In section 4, we recall in our setting the main definitions and properties of the induction and restriction functors, as introduced for p-adic groups in [BDK] and studied further in [Da]. In section 5, we record the main cocenter results from [HN1,HN2] that we will use in the rest of the paper. In sections 6 and 7, we define the rigid and elliptic cocenters and the rigid and elliptic quotients of R(H), and prove the main results about the duality between the rigid cocenter and rigid quotient, in particular, Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

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Cocenters and representations of affine Hecke algebras

Ciubotaru¹,

He²

2017

J. Eur. Math. Soc.

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In this paper, we study the relation between the cocenter and the representation theory of affine Hecke algebras. The approach is based on the interaction between the rigid cocenter, an important subspace of the cocenter, and the dual object in representation theory, the rigid quotient of the Grothendieck group of finite dimensional representations.2010 Mathematics Subject Classification. 20C08, 22E50.

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A Gallery Model for Affine Flag Varieties via Chimney Retractions

Milićević

Naqvi

Schwer

et al. 2022

Transformation Groups

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This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat–Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat–Tits buildings in the function field case, we relate these retractions and their effect on minimal galleries to double coset intersections in the corresponding affine flag variety.

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$$P$$ P -alcoves, parabolic subalgebras and cocenters of affine Hecke algebras

Cited by 9 publications

References 15 publications

Irreducible components of minuscule affine Deligne–Lusztig varieties

Irreducible components of minuscule affine Deligne–Lusztig varieties

Cocenters and representations of affine Hecke algebras

A Gallery Model for Affine Flag Varieties via Chimney Retractions

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