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# Minimal length elements in some double cosets of Coxeter groups

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*in*Advances in Mathematics 215(2):469-503 · November 2007

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DOI: 10.1016/j.aim.2007.04.005 · Source: arXiv

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We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570–600].

- ... We then use the " partial conjugation " method introduced in [11] to reduce to a similar problem for a finite Coxeter group, which is proved in [8] and [9] via a case-by-case analysis and later in [15] by a case-free argument. This leads to a case-free proof which works for all cases, including the exceptional affine Weyl groups, which seems very difficult via the approach in [12]. ...... (2) For any minimal length element˜w of O, the centralizer of˜w is obtained via cyclic shift in the sense of Lusztig [23]. Property (1) was due to Geck and Pfeiffer [9] and the first author [11]. Property (2) was first conjectured and verified by Lusztig [23, 1.2] for untwisted classical groups and proved in general in [15]. ...... The following result is proved in [8], [7] and [11] via a case-by-case analysis with the aid of computer for exceptional types. A case-free proof which does not rely on computer calculation was recently obtained in [15]. ...ArticleFull-text available
- Dec 2011

Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We then introduce the "class polynomials" for affine Hecke algebra $H$ and prove that $T_{w_\co}$, where $\co$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \cite{L4}. - ... Here d is the order of w and w i is the longest element of the parabolic subgroup of W generated by Π i . It was proved in [8], [7] and [11] that for any conjugacy class of W , there exists a good minimal length element. In [12], the second and third-named authors gave a general proof, which also provides an explicit construction of good minimal length elements. ...... Now one computes that λ ∨ 1 = ρ ∨ 1 = (0, 3, 3, 5, 3, 0) and its dominant conjugate is λ ∨ 1 = (3, 5, 6, 9, 6, 3). Note that ρ ∨ = (8,11,15,21,15,8). Then the full sequence gives 11,16,21,30,21,11)). ...... Note that ρ ∨ = (8,11,15,21,15,8). Then the full sequence gives 11,16,21,30,21,11)). ...Preprint
- Mar 2019

Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple conjugacy classes of $G$. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case. - ... The inequalities ( * ) are just2, 3, 4, 1, 1, 1, 1, 1). Case 2. w 1 = s 3 s 4 s [8,1] ı and I(J, w 1 , δı) = ∅. The inequalities ( * ) are just qm 1 −m 4 > 0, qm 2 −m 1 > 0, qm 3 −m 2 > 0, qm 4 −m 3 −m 4 −m 5 > 0, qm 5 − m 6 > 0, qm 6 − m 7 > 0 and qm 7 − m 8 > 0. So we may take (m 1 , m 2 , m 3 , m 4 , m 5 , m 6 , m 7 , m 8 ) = (5, 3, 2, 9, 1, 1, 1, 1). ...... The inequalities ( * ) are just qm 1 −m 4 > 0, qm 2 −m 1 > 0, qm 3 −m 2 > 0, qm 4 −m 3 −m 4 −m 5 > 0, qm 5 − m 6 > 0, qm 6 − m 7 > 0 and qm 7 − m 8 > 0. So we may take (m 1 , m 2 , m 3 , m 4 , m 5 , m 6 , m 7 , m 8 ) = (5, 3, 2, 9, 1, 1, 1, 1). Case 3. w 1 = s 4 s 5 s 3 s 4 s [8,1] ı and I(J, w 1 , δı) = ∅. The inequalities ( * ) are just qm 1 −m 5 > 0, qm 2 −m 1 > 0, qm 3 −m 2 −m 4 > 0, qm 4 −m 3 > 0,(5, 3, 3, 2, 4, 1, 1, 1). ...... The inequalities ( * ) are just qm 1 −m 5 > 0, qm 2 −m 1 > 0, qm 3 −m 2 −m 4 > 0, qm 4 −m 3 > 0,(5, 3, 3, 2, 4, 1, 1, 1). Case 4. w 1 = s 2 s 4 s 3 s 5 s 4 s [8,1] ı, I(J, w 1 , δı) = {3, 4}, Ad(w 1 ) is of order 2 on I(J, w 1 , δı) and v = s 3 or s 3 s 4 s 3 . The inequalities ( * ) are just ...Article
- Aug 2007
- J ALGEBRA

We prove that the Deligne-Lusztig variety associated to minimal length elements in any δ-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in [10]. 1.1 Notations. Let k be an algebraic closure of the finite prime field Fp and G be a connected reductive algebraic group over k with an endomorphism F: G → G such that some power F d of F is the Frobenius endomorphism relative to a rational structure over a finite subfield k0 of k. Let q be the positive number with q d = |k0|. We fix a F-stable Borel subgroup B and a F-stable maximal torus T ⊂ B. Let Φ be the set of roots and (αi)i∈I be the set of simple roots corresponding to (B, T). For i ∈ I, let ø ∨ i be the corresponding fundamental coweight. Let W = N(T)/T be the Weyl group and (si)i∈I be the set of simple reflections. For w ∈ W, let l(w) be the length of w. Since (B, T) is F-stable, F induces a bijection on I and an automorphism on W. We denote the induced maps on W and I by δ. Now δ also induces isomorphisms on the set of characters X = Hom(T, Gm) and the set of cocharacters X ∨ = Hom(Gm, T) which we also denote by δ. Then it is easy to see that F ∗ µ = q · δı(µ) for µ ∈ X ∨. For J ⊂ I, let ΦJ be the set of roots generated by {αj}j∈J and WJ be the subgroup of W generated by {sj}j∈J. Let W J be the set of minimal length coset representatives for W/WJ. The unique maximal element in W will be denoted by w0 and the unique maximal element in WJ will be denoted by w J 0. Let α0 = ∑ i∈I niαi be the highest root and n0 = ∑ i∈I ni. 1.2. Let B be the set of Borel subgroups of G. For w ∈ W, let O(w) = { ( g B, g ˙w B); g ∈ G} be the G-orbit on B × B that corresponding to w. - ... Since δ(H) = H and δ(B) = B, δ acts on the set Γ of simple roots and on W such that δ(w) = δ • w • δ −1 : ∆ → ∆. Minimal length elements in δ-twisted conjugacy classes in W have been studied in [15] [16] ...... It is well-known that elements in M ′ correspond to special subsets of the set Γ of simple roots. Indeed, minimal or maximal length elements in conjugacy classes of involutions in W have been studied (see, for example, [14] [16] [18] [19] [20] [22] and especially [14, Remark 3.2.13] for minimal length elements, [19, Theorem 1.1] for maximal length elements, and [16, Lemma 3.6] for minimal length elements in twisted conjugacy classes). ...For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B – ) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB – is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution m C ∈ 2 W associated to C. We prove that the element m C is the unique maximal length element in its conjugacy class in W, and we classify all such elements in W. For G = SL(n + 1; k), we describe m C explicitly for every conjugacy class C, and when w ∈ W ≌ Sn+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.
- ... We first recall some properties about " partial conjugation action " of W oñ W . Although the results were proved for affine Weyl groups in [11], it is easy to see that they also hold for extended affine Weyl groups. Let J ⊂ ˜ S. We consider the conjugation action of W oñ ...... Our strategy is as follows. First, we use the Proposition 4.2 below to reduce to problem to elements iñ W S , using the technique of " partial conjugation action " introduced in [11]. Then in Proposition 4.4 we reduce the elements iñ W S with quasi-regular translation part to some elements for which the nonemptiness is already known. ...We discuss some connections between the closure of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne-Lusztig varieties Xw(1) in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of Xw(1) if the translation part of w is quasi-regular. As a by-product, we give a new proof of the explicit description of, first obtained in He ["Unipotent variety in the group compactification." Advances in Mathematics 203 (2006): 109-31].
- ... In the special case where λ is a cominuscule coweight, θ gives an order-reversing bijection between Q J and Adm(λ). The proof relies on properties of the Demazure, or monoidal product of Coxeter groups, studied for example by He, and He and Lu in [4] [5] [6]. ...... In the special case where λ is a cominuscule coweight, θ gives an order-reversing bijection between Q J and Adm(λ). The proof relies on properties of the Demazure, or monoidal product of Coxeter groups, studied for example by He, and He and Lu in [4,5,6]. ...Let $G$ be a complex simple algebraic group and $G/P$ be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of $G/P$. We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of $G$. Furthermore, we compare the torus-equivariant cohomology and $K$-theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of type $A$.
- ... We say that O is cuspidal (or elliptic) if it has empty intersection with every proper F -stable parabolic subgroup of W . The following result is proven in [15] [14] [17] (see also [18] for a case-free proof). ...Article
- Oct 2013
- P LOND MATH SOC

We study the decomposition matrices for the unipotent $\ell$-blocks of finite special unitary groups SU$_n(q)$ for unitary primes $\ell$ larger than $n$. Up to some unknow entries, we give a complete solution for $n=2,\ldots,10$. We formulate a strong positivity conjecture on characters of certain intersection cohomology complexes, which then predicts most of the remaining missing entries. We also prove a general result for two-column partitions when $\ell$ divides $q+1$. This is achieved using projective modules coming from the $\ell$-adic cohomology of Deligne--Lusztig varieties. - ... The closures of the sets [v 1 , v 2 ] A,C in G 1 × G 2 are described in §5. We point out that in a recent paper [9], X. He studies the (R A , R C )-stable pieces in G 1 × G 2 from the point of view of Coxeter groups. ...We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1 and A_2 are subgraphs of the Dynkin graph \Gamma of G and a : A_1 \to A_2 is an isomorphism. The partitions of \bar{G} of Springer and Lusztig correspond respectively to the triples (\emptyset, \emptyset, \id) and (\Gamma, \Gamma, \id).
- ... In [51], Lusztig in- troduced G-stable pieces for reductive groups over algebraically closed fields. The closure relation between G-stable pieces was determined in [23] and a more sys- tematic approach using the " partial conjugation action" technique was given later in [24]. The notion and the closure relation of G-stable pieces also found appli- cation in arithmetic geometry, e.g. in the work of Pink, Wedhorn and Ziegler on algebraic zip data [58]. ...Preprint
- Jul 2018

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned. - ... The proof of Mazur's inequality is based on two properties of the admissible sets: • The additivity of the admissible sets (Theorem 5.1), proved by Zhu's global Schubert varieties [26]. • The compatibility of admissible sets (Theorem 6.1), proved by the " partial conjugation method " in [8] ...Article
- Aug 2014
- ANN SCI ECOLE NORM S

In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X(\mu, b)_K$ for any tamely ramified group $G$ and its parahoric subgroup $K$. We show that $X(\mu, b)_K \neq \emptyset$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of $\sigma$-conjugacy classes of a twisted loop group. - ... This is what we will do in this section. Another important technique is the " partial conjugation " method introduced in [5], which will be discussed in the next section. ...This is a continuation of the sequence of papers \cite{HN2}, \cite{H99} in the study of the cocenters and class polynomials of affine Hecke algebras $\ch$ and their relation to affine Deligne-Lusztig varieties. Let $w$ be a $P$-alcove element, as introduced in \cite{GHKR} and \cite{GHN}. In this paper, we study the image of $T_w$ in the cocenter of $\ch$. In the process, we obtain a Bernstein presentation of the cocenter of $\ch$. We also obtain a comparison theorem among the class polynomials of $\ch$ and of its parabolic subalgebras, which is analogous to the Hodge-Newton decomposition theorem for affine Deligne-Lusztig varieties. As a consequence, we present a new proof of \cite{GHKR} and \cite{GHN} on the emptiness pattern of affine Deligne-Lusztig varieties.
- ArticleFull-text available
- Dec 2019

We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic. - ArticleFull-text available
- Oct 2011

In this notes, we will give an exposition of some results on the method of partial conjugation action. We first discuss the partial conjugation action of a parabolic subgroup of a Coxeter group. We then discuss some applications to Lusztig's $G$-stable pieces and its affine generalization. We also discuss some recent work on the $\s$-conjugacy classes of loop groups and affine Deligne-Lusztig varieties. - Article
- Aug 2011
- DUKE MATH J

We give a geometric proof that minimal length elements in a (twisted) conjugacy class of a finite Coxeter group $W$ have remarkable properties with respect to conjugation, taking powers in the associated Braid group and taking centralizer in $W$. - Article
- Apr 2011
- Represent Theor

Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map \Phi_p from the set \underline{W} of conjugacy classes in the Weyl group W to the set of unipotent classes in G. Here we prove three results about \Phi_p. First we show that \Phi_p has a canonical one sided inverse. Next we show that \Phi_0 =r\Phi_p for a unique map r. Finally we construct a natural surjective map from \underline{W} to the set of special representations of W which is the composition of \Phi_0 with another natural map; we show that this map depends only on the Coxeter group structure of W. We also define the special conjugacy classes in W (in 1-1 correspondence with the special representations of W) and describe them explicitly for each simple type. - Article
- Oct 2010
- DOC MATH

An algebraic zip datum is a tuple $\CZ := (G,P,Q,\phi)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\phi\colon P/R_uP\to Q/R_uQ$. We study the action of the group $E := \{(p,q)\in P{\times}Q | \phi(\pi_{P}(p)) =\pi_Q(q)\}$ on $G$ given by $((p,q),g)\mapsto pgq^{-1}$. We define certain smooth $E$-invariant subvarieties of $G$, show that they define a stratification of $G$. We determine their dimensions and their closures and give a description of the stabilizers of the $E$-action on $G$. We also generalize all results to non-connected groups. We show that for special choices of $\CZ$ the algebraic quotient stack $[E \backslash G]$ is isomorphic to $[G \backslash Z]$ or to $[G \backslash Z']$, where $Z$ is a $G$-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of $G$ and where $Z'$ has been defined by Moonen and the second author in their classification of $F$-zips. In these cases the $E$-invariant subvarieties correspond to the so-called "$G$-stable pieces" of $Z$ defined by Lusztig (resp. the $G$-orbits of $Z'$). - Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y in exactly one point. In this paper this is generalized to the case where w is replaced by any elliptic element in the Weyl group of minimal length d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to an affine space of dimension d and Y is replaced by a unipotent class Y' of codimension d in such a way that the intersection of V' and Y' is finite.
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- Dec 2010

Let w be an elliptic element of the Weyl group of a connected reductive group G. Let X be the set of pairs (g,B) where g is an element of G, B is a Borel subgroup of G and B,gBg^{-1} are in relative position w. Then G acts naturally on X. Assume that w has minimal length in its conjugacy class. We show that the set of G-orbits in X has a well defined structure of an affine algebraic variety V. When G is a classical group we show that this variety is an affine space modulo the action of a finite diagonalizable group. In this case we also construct some nontrivial automorphisms of X. - Article
- Dec 2010
- ADV MATH

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification. - ArticleFull-text available
- Apr 2010

We study the minimal length elements in an integral conjugacy class of a classical extended affine Weyl group and we show that these elements are quite "special" in the sense of Geck and Pfeiffer \cite{GP93}. We also discuss some application on extended affine Hecke algebras and loop groups. - ArticleFull-text available
- Oct 2009

In this paper, we discuss some partitions of affine flag varieties. These partitions include as special cases the partition of affine flag variety into affine Deligne-Lusztig varieties and the affine analogue of the partition of flag varieties into $\cb_w(b)$ introduced by Lusztig in \cite{L1} as part of the definition of character sheaves. Among other things, we give a formula for the dimension of affine Deligne-Lusztig varieties for classical loop groups in terms of degrees of class polynomials of extended affine Hecke algebra. We also prove that any simple $GL_n(\FF_q((\e)))$-module occurs as a subquotient of the cohomology of affine Deligne-Lusztig variety $X_w(1)$ for some $w$ in the extended affine Weyl group $\ZZ^n \rtimes S_n$ must occurs for some $w$ in the finite Weyl group $S_n$. Similar result holds for $Sp_{2n}$. Comment: Preliminary version. Comments are welcome - Let G be a connected semi-simple algebraic group of adjoint type over an algebraically closed field and let be the wonderful compactification of G. For a fixed pair (B,B−) of opposite Borel subgroups of G, we look at intersections of Lusztig’s G-stable pieces and the B−×B-orbits in , as well as intersections of B×B-orbits and B−×B−-orbits in . We give explicit conditions for such intersections to be nonempty, and in each case, we show that every nonempty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the nonempty intersections form a strongly admissible partition of Ḡ.
- ArticleFull-text available
- Oct 2007

We will use the combinatorics of the $G$-stable pieces to describe the closure relation of the partition of partial flag varieties in \cite[section 4]{L3}. - This paper studies affine Deligne-Lusztig varieties $X_{\tw}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{\tw}(b)$ for a minimal length element $\tw$ in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in \cite{HL} to the affine case. We then provide a reduction method that relates the structure of $X_{\tw}(b)$ for arbitrary elements $\tw$ in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of G\"ortz, Haines, Kottwitz and Reuman in \cite{GHKR}.
- Article
- Jun 2012
- J LIE THEORY

Kottwitz' conjecture is concerned with the intersections of Kazhdan--Lusztig cells with conjugacy classes of involutions in finite Coxeter groups. In joint work with Bonnaf\'e, we have recently found a way to prove this conjecture for groups of type $B_n$ and $D_n$. The argument for type $D_n$ relies on two ingredients which were used there without proof: (1) a strengthened version of the "branching rule" and (2) the consideration of "$\diamond$-twisted" involutions where $\diamond$ is a graph automorphism. In this paper we deal with (1), (2) and complete the argument for type $D_n$; moreover, we establish Kottwitz' conjecture for $\diamond$-twisted involutions in all cases where $\diamond$ is non-trivial. - Article
- Aug 2008
- J ALGEBRA

We prove that the Deligne–Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik–Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Broué's conjectures. - Article
- Feb 2015

In this paper, we give explicit descriptions of the centers and cocenters of $0$-Hecke algebras associated to finite Coxeter groups. - Preprint
- Sep 2019

We investigate qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the dimension is maximal possible. The model case for the first alternative is the Lubin-Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases. - Preprint
- Mar 2019

We show that the totally nonnegative part of a partial flag variety $G/P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov. - Preprint
- Jan 2019

We prove a character formula for some closed fine Deligne-Lusztig varieties. We apply it to compute fixed points for fine Deligne-Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport-Zink spaces arising from the arithmetic Gan-Gross-Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic. - PreprintFull-text available
- Dec 2018

Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, yielding an analogue of Matsumoto's theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. - Article
- Aug 2016
- J ALGEBRA

Suppose $G$ is a reductive algebraic group, $T$ is a Cartan subgroup, $N=\text{Norm}(T)$, and $W=N/T$ is the Weyl group. If $w\in W$ has order $d$, it is natural to ask about the orders lifts of $w$ to $N$. It is straightforward to see that the minimal order of a lift of $w$ has order $d$ or $2d$, but it can be a subtle question which holds. We first consider the question of when $W$ itself lifts to a subgroup of $N$ (in which case every element of $W$ lifts to an element of $N$ of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of $w$ are conjugate, and therefore have the same order. We also consider the twisted case. - Article
- Nov 2015

This survey article, is written as an extended note and supplement of my lectures in the current developments in mathematics conference in 2015. We discuss some recent developments on the conjugacy classes of affine Weyl groups and $p$-adic groups, and some applications to Shimura varieties and to representations of affine Hecke algebras. - Article
- Sep 2015
- MANUSCRIPTA MATH

In the paper four stratifications in the reduction modulo $p$ of a general Shimura variety are studied: the Newton stratification, the Kottwitz-Rapoport stratification, the Ekedahl-Oort stratification and the Ekedahl-Kottwitz-Oort-Rapoport stratification. We formulate a system of axioms and show that these imply non-emptiness statements and closure relation statements concerning these various stratifications. These axioms are satisfied in the Siegel case. - Article
- Sep 2012

Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y in exactly one point. In this paper this is generalized to the case where w is replaced by any elliptic element in the Weyl group of minimal length d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to an affine space of dimension d and Y is replaced by a unipotent class Y' of codimension d in such a way that the intersection of V' and Y' is finite. - Article
- Dec 2009
- J ALGEBRA

Let (W,I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 13–88] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type. - Article
- Nov 2014
- COMPOS MATH

Let \$W\$ be an extended affine Weyl group. We prove that the minimal length elements \$w_{{\mathcal{O}}}\$ of any conjugacy class \${\mathcal{O}}\$ of \$W\$ satisfy some nice properties, generalizing results of Geck and Pfeiffer [On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), 79–94] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some \$p\$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra \$H\$. We prove that \$T_{w_{{\mathcal{O}}}}\$, where \${\mathcal{O}}\$ ranges over all the conjugacy classes of \$W\$, forms a basis of the cocenter \$H/[H,H]\$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne–Lusztig varieties He [Geometric and cohomological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404]. - ArticleFull-text available
- Nov 2013

This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and \cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties. - Fundamental elements are certain special elements of affine Weyl groups introduced by Görtz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne–Lusztig varieties. In this paper, we obtain characterizations of the fundamental elements and their natural generalizations. We also derive an inverse to a version of “Newton-Hodge decomposition” in affine flag varieties. As an application, we obtain a group-theoretic generalization of Oort’s results on minimal \(p\) -divisible groups, and we show that, in certain good reduction of PEL Shimura datum, each Newton stratum contains a minimal Ekedahl–Oort stratum. This generalizes a result of Viehmann and Wedhorn.
- ArticleFull-text available
- Aug 2013

This note is based on my talk at ICCM 2013, Taipei. We give an exposition of the group-theoretic method and recent results on the questions of non-emptiness and dimension of affine Deligne-Lusztig varieties in affine flag varieties. - Article
- Mar 2013
- ADV MATH

We provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of the pin cover $\wti W$, a certain double cover of the Weyl group $W$, and an extended Dirac operator for graded Hecke algebras. Our approach leads to a new and uniform construction of the irreducible genuine $\wti W$-characters. In the process, we give a construction of the action by an outer automorphism of the Dynkin diagram on the cohomology groups of Springer theory, and we also introduce a $q$-elliptic pairing for $W$ with respect to the reflection representation $V$. These constructions are of independent interest. The $q$-elliptic pairing is a generalization of the elliptic pairing of $W$ introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product $A_W=\bC[W]\ltimes S(V)$. - Article
- Aug 2012
- J PURE APPL ALGEBRA

We determine a basis of the (twisted) cocenter of graded affine Hecke algebras with arbitrary parameters. In this setting, we prove that the kernel of the (twisted) trace map is the commutator subspace (Density theorem) and that the image is the space of good forms (trace Paley-Wiener theorem). - Preprint
- Oct 2019

In this paper we study the geometry of reduction modulo $p$ of the Kisin-Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR strata on these Shimura varieties, using the theories of $G$-zips and mixed characteristic local $\mathcal{G}$-Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties. - ArticleFull-text available
- Sep 2007

We use $G$-stable pieces to construct some equidimensional varieties and as a consequence, obtain Lusztig's dimension estimates \cite[section 4]{L2}. This is a generalization of \cite{HL}.

- Article
- Oct 2001
- ALGEBR COLLOQ

This note is a sequel to part I [Trans. Am. Math. Soc. 353, No. 7, 2725–2739 (2001; Zbl 0996.20003)]. We establish the minimal basis theory for the centralizers of parabolic subalgebras of Iwahori-Hecke algebras associated to finite Coxeter groups of any type, generalizing the approach introduced in [J. Algebra 221, No. 1, 1–28 (1999; Zbl 0940.20011)] from centers to the centralizer case. As a pre-requisite, we prove a reducibility property in the twisted J-conjugacy classes in finite Coxeter groups, which is a generalization of results of M. Geck and R. Rouquier [Prog. Math. 141, 251–272 (1997; Zbl 0868.20013] and part I [loc. cit.]. - Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy
- Jan 1981
- 5-171

- A A Beilinson
- J Bernstein
- P Deligne

A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Soc. Math. France, Paris, 1982, pp. 5–171. - Article
- Aug 1982

In this paper we give an elementary method for classifying conjugacy classes of involutions in a Coxeter group (W, S). The classification is in terms of (W-equivalence classes of certain subsets of S). - Article
- Jul 2000
- J Algebra

Let W be a finite Coxeter group and let F be an automorphism of W that leaves the set of generators of W invariant. We establish certain properties of elements of minimal length in the F-conjugacy classes of W that allow us to define character tables for the corresponding twisted Iwahori–Hecke algebras. These results are extensions of results obtained by Geck and Pfeiffer in the case where F is trivial. - Article
- Dec 2006
- ADV MATH

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in [G. Lusztig, Parabolic character sheaves, II, Mosc. Math. J. 4 (4) (2004) 869–896]. We also prove some properties of the character sheaves on the group compactification. - Article
- Sep 2006
- TRANSFORM GROUPS

If G is a group and an automorphism of G, one has the twisted conjugation action of G on itself This paper collects a number of results — more or less well known — for the case that G is a simply connected semisimple group. - Article
- Apr 1997
- ADV MATH

this paper to prove a similar statement and to describe a similar algorithm for Weyl groups and their Hecke algebras of any given type. Our approach which is completely elementary can be described entirely within the Weyl group itself, as follows - Let (W,S) be a finite Coxeter system and B = B(W) the corresponding Artin-Tits braid group. The natural map B → W has a canonical section r:W → B defined by the condition that if w ∈ W is written as a reduced expression in the generators in S then r(w)is the corresponding product taken in B. The main result of the present paper is as follows. Let C be a conjugacy class in W whose elements have order d say. Then there exists an element w ∈ C of minimal length in C such that r(w)d is a product of terms of the form $r(w_I)^{d_I}$ where the following hold: dI is a non-negative even integer, I runs over a sequence of subsets of S which decreases (which implies that the terms commute), and wI is the longest element in the corresponding parabolic subgroup of W. Such an element will be called a ‘good’ element in C. The result is proved case by case, using the classification of irreducible finite Coxeter groups and the knowledge of representatives of minimal length from the article by Geck and Pfeiffer in Advances in Math. 102 (1993) 79–94. The main application of this result concerns the problem of calculating character values of Iwahori-Hecke algebras. The generic Iwahori-Hecke algebra H associated with (W,S) is a quotient of the group algebra of B by the ideal generated by quadratic relations of the form (s-q)(s+1) where s ∈ S and q is an indeterminate. Thus, H is an algebra over a suitable field of rational functions in the variable ∼q. The above result implies that if w is a good element in the class C of W, then the eigen values of the standard basis element Tw of H in an irreducible representation of H are roots of unity times fractional powers of ∼q, and the fractional powers occurring can be explicitly determined from the ordinary character table of W. This result is used to compute the character table of the Iwahori-Hecke algebra of type E8. To determine the roots of unity, we use additional relations coming from the modular representation theory of ∼H. This completes the program of determining the character tables of Iwahori-Hecke algebras. 1991 Mathematics Subject Classification: 20C20, 20F36.
- We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1 and A_2 are subgraphs of the Dynkin graph \Gamma of G and a : A_1 \to A_2 is an isomorphism. The partitions of \bar{G} of Springer and Lusztig correspond respectively to the triples (\emptyset, \emptyset, \id) and (\Gamma, \Gamma, \id).
- Article
- Jun 2006
- ADV MATH

We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves. - Let $G$ be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification $\bar{G}$ of $G$ into finite many $G$-stable pieces, which were introduced by Lusztig. In this paper, we will investigate the closure of any $G$-stable piece in $\bar{G}$. We will show that the closure is a disjoint union of some $G$-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many $G$-orbits.
- Article
- Aug 2003

The main theme of this paper is establishing the "generalized Springer correspondence" in complete generality that is, for not necessarily connected reductive algebraic groups. - Article
- Sep 2002

These are notes for the Aisenstadt lectures given in may/june 2002 at CRM, Montreal. The main object is the study of Iwahori-Hecke algebras arising from reductive groups over finite or p-adic fields. We try to extend various results known in the equal parameter case to the case of not necessarily equal parameters.ai - Article
- Jan 2005
- Represent Theor

We define and study convolution of parabolic character sheaves. As an application we attach to any parabolic character sheaf the orbit of a tame local system on the maximal torus under a subgroup of the Weyl group. - Article
- Apr 2004
- Represent Theor

We continue the study of character sheaves on a not necessarily connected reductive group. We prove orthogonality formulas for certain characteristic functions. - Article
- Mar 2003
- MOSC MATH J

We study a class of perverse sheaves on the variety of pairs (P,gU_P) where P runs through a conjugacy class of parabolics in a connected reductive group G and gU_P runs through G/U_P. This is a generalization of the theory of character sheaves.