Let G be a simple algebraic group over an algebraically closed field of characteristic zero or positive odd, good characteristic. Let B be a Borel subgroup of G. We show that the spherical conjugacy classes of G intersect only the double cosets of B in G corresponding to involutions in the Weyl group of G. This result is used to prove that for a spherical conjugacy class O with dense B-orbit v_0 contained in BwB there holds l(w)+rk(1-w)=dim(O) extending a characterization of spherical conjugacy classes obtained by N. Cantarini, M. Costantini and the author to the case of groups over fields of odd, good characteristic.
Let $\mathsf G$ be a connected reductive linear algebraic group defined over $\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf G(\mathbb R)$, $C(g)$ is conjugate to $g^{-1}$ in the group $\mathsf G(\mathbb R)$. Similar result on the Lie algebras is also obtained.
According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz' conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types; combined with previous work, thi…
Let G be a simple algebraic group over an algebraically closed field of good odd characteristic, and let theta be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical theta-twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. We generalize to good odd characteristic J-H. Lu's dimension formula for spherical twisted conjugacy classes.
Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that intersects every conjugacy class of involutions of G.
Letting tau denote the inverse transpose automorphism of GL(n,q), a formula is obtained for the number of g in GL(n,q) so that gg^{tau} is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg^{tau} behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n,q) and of symmetric fun…
Let G be a connected, reductive algebraic group over an algebraically closed field of characteristic zero or good and odd. We characterize the spherical conjugacy classes of G as those intersecting only Bruhat cells corresponding to involutions in the Weyl group of G.
Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.
Let W be a Weyl group. We introduce the notion of positive conjugacy class in W. This generalizes the notion of regular elliptic conjugacy class in the sense of Springer.
Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In this note we calculate the canonical distinguished involutions in the affine Weyl groups of rank <8. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams o…
We define a map from the set of conjugacy classes of a Weyl group W to the representation ring of W tensored with the ring of polynomials in one variable.
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.
We define and study a correspondence between the set of distinguished G^0-conjugacy classes in a fixed connected component of a reductive group G (with G^0 almost simple) and the set of (twisted) elliptic conjugacy classes in the Weyl group. We also prove a homogeneity property related to this correspondence.
Let G be a connected reductive algebraic group over an algebraic closed field. We define a (surjective) map from the set of conjugacy classes in the Weyl group to the set of unipotent classes of G.
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.
We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Psi from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.
Let N be the normalizer of a maximal torus T in a split reductive group over F_q and let w be an involution in the Weyl group N/T. We construct explicitly a lifting n of w in N such that the image of n under the Frobenius map is equal to the inverse of n.
Let G be a connected reductive group. We define a map from the set of unipotent classes in G to the set of conjugacy classes in the Weyl group (assuming that the characteristic is not bad). This map is a one sided inverse of a map in the opposite direction constructed in an earlier paper.
Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry.
It is proved that a Stein manifold acted on by a connected compact Lie group is spherical if and only if there exists an antiholomorphic involution preserving each orbit of the action. This involution can be chosen equivariant with respect to a Weyl involution of the group.
Let X be the group of weights of a maximal torus of a simply connected semisimple group over C and let W be the Weyl group. The semidirect product W(Q\otimes X/X) is called the extended Weyl group. There is a natural C(v)-algebra H called the extended Hecke algebra with basis indexed by the extended Weyl group which contains the usual Hecke algebra as a subalgebra. We construct an H-module with basis indexed by the involutions in the extended Weyl group. This generalizes a construction of the author and D.Vogan.
In a previous article we have defined an action of the Iwahori-Hecke algebra of a Coxeter group W on a free module with basis indexed by the involutions in W. In this paper we show that the specialization of this action at the parameter 0 has a simple description.
Let N be the normalizer of a maximal torus T in a split reductive group over F_q, and let w be an involution in the Weyl group N/T. We construct a section of W satisfying the braid relations, such that the image of the lift n of w under the Frobenius map is equal to the inverse of n.
For any two involutions y,w in a Weyl group (y\le w), let P_{y,w} be the polynomial defined in [KL]. In this paper we define a new polynomial P^σ_{y,w} whose i-th coefficient is a_i-b_i where the i-th coefficient of P_{y,w} is a_i+b_i (a_i,b_i are natural numbers). These new polynomials are of interest for the theory of unitary representations of complex reductive groups. We present an algorithm for computing these polynomials.
Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element. In this paper we define a map from the set of "twisted conjugay classes" in W to the set of unipotent G^0-conjugacy classes in D, generalizing an earlier construction which applied when G is connected.
For a connected complex semi-simple Lie group $G$ 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 non-empty, 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 $\mc \in W$ associated to $C$. We study properties of the elements $\mc$. For $G = SL(n+1, \Cset)$, we describe $\mc$ explicitly for every conjugacy class $C$, and for the case when $w \in W$ is an involution, we also giv…
The action of a Coxeter group $W$ on the set of left cosets of a standard parabolic subgroup deforms to define a module $\mathcal{M}^J$ of the group's Iwahori-Hecke algebra $\mathcal{H}$ with a particularly simple form. Rains and Vazirani have introduced the notion of a quasiparabolic set to characterize $W$-sets for which analogous deformations exist; a motivating example is the conjugacy class of fixed point free involutions in the symmetric group. Deodhar has shown that the module $\mathcal{M}^J$ possesses a certain antilinear involution, called the bar operator, and a certain basis invaria…
Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map Φ_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 Φ_p. First we show that Φ_p has a canonical one sided inverse. Next we show that Φ_0 =rΦ_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 Φ_0 with another natural map; we show that this map depends only on …
We study coadjoint $B$-orbits on $\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit $Ω_w$. We prove that, given basis involutions $σ$, $τ$ in $W$, if the orbit $Ω_σ$ is contained in the closure of the orbit $Ω_τ$ then $σ$ is less than or equal to $τ$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of $Ω_w$ and present a conjectural description of…
We consider the Chevalley involution in the context of real reductive groups. We show that if G(R) is the real points of a connected reductive group, there is an involution, unique up to conjugacy by G(R), taking any semisimple element to a conjugate of its inverse. As applications we give a condition for every irreducible representation of G(R) to be self-dual, and to the Frobenius Schur indicator for such groups.
Recently, Lusztig constructed for each reductive group a partition by unions of sheets of conjugacy classes, which is indexed by a subset of the set of conjugacy classes in the associated Weyl group. Sevostyanov subsequently used certain elements in each of these Weyl group conjugacy classes to construct strictly transverse slices to the conjugacy classes in these strata, generalising the classical Steinberg slice, and similar cross sections were built out of different Weyl group elements by He-Lusztig. In this paper we observe that He-Lusztig's and Sevostyanov's Weyl group elements share a ce…
Let W be an Iwahori-Weyl group of a connected reductive group G over a non-archimedean local field. I prove that if w is an element of W that does not act on the corresponding apartment of G by a translation then one can apply to w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w' for which there exists a simple reflection s such that l(sw's)>l(w'). Even for affine Weyl groups, a special case of Iwahori-Weyl groups and also an important subclass of Coxeter groups, this is a new fact about conjugacy classes. Further, there are impli…
Let $W_a$ be an affine Weyl group and $η:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $η(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold…
We define noncommutative deformations $W_q^s(G)$ of algebras of functions on certain (finite coverings of) transversal slices to the set of conjugacy classes in an algebraic group $G$ which play the role of Slodowy slices in algebraic group theory. The algebras $W_q^s(G)$ called q-W algebras are labeled by (conjugacy classes of) elements $s$ of the Weyl group of $G$. The algebra $W_q^s(G)$ is a quantization of a Poisson structure defined on the corresponding transversal slice in $G$ with the help of Poisson reduction of a Poisson bracket associated to a Poisson-Lie group $G^*$ dual to a quasit…
Kottwitz' conjecture is concerned with the intersections of Kazhdan--Lusztig cells with conjugacy classes of involutions in finite Coxeter groups. In joint work with Bonnafé, 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, w…
Let $G$ be a connected semisimple algebraic group over an algebraically closed field of characteristic zero, and let $þ$ be an automorphism of $G$. We give a characterization of $þ$-twisted spherical conjugacy classes in $G$ by a formula for their dimensions in terms of certain elements in the Weyl group of $G$, generalizing a result of N. Cantarini, G. Carnovale, and M. Costantini when $þ$ is the identity automorphism. For $G$ simple and $þ$ an outer automorphism of $G$, we also classify the Weyl group elements that appear in the dimension formula.
This is based on a talk given at the AMS/RMS joint meeting in June 2013, Alba Iulia. The main theme is that the conjugacy classes in the Lie group E_8 should be (roughly) parametrized by conjugacy classes in the corresponding Weyl group.
Let $G$ be a complex simply connected semisimple Lie group, and let $B_V$ be the canonical base of a Weyl module $V$ of $G$. We calculate explicitely the action of the longest element $w_0$ of the Weyl group on $B_V$ in terms of parametrizations. The method is based on results of Berenstein and Zelevinsky on the geometric lifting.
We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group.
We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at -1.
In [LV] the authors defined a Hecke algebra action and a bar involution on a vector space spanned by the involutions in a Weyl group. In this paper we give a new definition of the Hecke algebra action and the bar operator which, unlike the one in [LV], is completely elementary (does not use geometry) and in particular it makes sense for an arbitrary Coxeter group.
We define an imbedding of the Hecke algebra module carried by the involutions in a Weyl group W (defined by the author and Vogan) into a completion of the Hecke algebra. An analogous result is proved for any Coxeter group. A variant of the construction gives an imbedding of the biregular representation of a Hecke algebra of W in a completion of the Hecke algebra of W^2.
The special representations of a Weyl group can be regarded as the vertices of a graph with an involution i such that any edge e has the following property: either e or i(e) joins two vertices whose a-functions differ by 1.
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}.
Let G be an almost simple reductive group with Weyl group W. Let B be a Borel subgroup of G. Let C be an elliptic conjugacy class in W and let w be an element of minimal length of C. We investigate the existence of a semisimple class of G whose intersection with BwB has dimension dim(B). We show that in good characteristic such a semisimple class exists almost always.
Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set defined in terms of the Weyl group which is independent of the characteristic. In the case where $G$ is replaced by the corresponding loop group we define an analogous decomposition of the set of regular semisimple compact elements into countably many strata.
In 1972, R. Carter introduced admissible diagrams to classify conjugacy classes in a finite Weyl group W. We say that an admissible diagram Γis a Carter diagram if any edge {α, β} with inner product (α, β) > 0 (resp. (α, β) < 0) is drawn as dotted (resp. solid) edge. We construct an explicit transformation of any Carter diagram containing long cycles (with the number of vertices l > 4) into another Carter diagram containing only 4-cycles. Thus, all Carter diagrams containing long cycles can be eliminated from the classification list. There exist diagrams determining two conjugacy classes in W.…
Let $G$ be a simply connected semisimple group over $\mathbb{C}$. We show that a certain involution of an open subset of the affine Grassmannian of $G$, defined previously by Achar and the author, corresponds to the action of the nontrivial Weyl group element of $\mathrm{SL}(2)$ on the framed moduli space of $\mathbb{G}_m$-equivariant principal $G$-bundles on $\mathbb{P}^2$. As a result, the fixed-point set of the involution can be partitioned into strata indexed by conjugacy classes of homomorphisms $N\to G$ where $N$ is the normalizer of $\mathbb{G}_m$ in $\mathrm{SL}(2)$. When $G=\mathrm{SL…
We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no a…
We show that, for a connected reductive algebraic group G over an algebraically closed field of zero or good characteristic, the parts, called strata, in the partition of G recently introduced by Lusztig are unions of sheets of conjugacy classes. For G simple and adjoint we refine the parametrization of such sheets obtained in previous work with F. Esposito. We give a simple combinatorial description of strata containing spherical conjugacy classes, showing that Lusztig's correspondence induces a bijection between unions of spherical conjugacy classes and unions of classes of involutions in th…
For a Weyl group W, we give a simple closed formula (valid on elliptic conjugacy classes) for the character of the representation of W in each A-isotypic component of the full homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the W-character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of W and the Dirac operator for graded affine Hecke algebras play key roles.
Let A be a character sheaf on a reductive connected group G over an algebraically closed field. Assuming that the characteristic is not bad, we show that for certain conjugacy classes D in G the restriction of A to D is a local system up to shift; we also give a parametrization of unipotent cuspidal character sheaves on G in terms of restrictions to conjugacy classes. Without restriction on characteristic we define canonical bijections from the set of unipotent character sheaves on G (and from the set of unipotent representations of the corresponding split reductive group over a finite field) …
Let $G$ be the symplectic group, $Φ=C_n$ its root system, $B\subset G$ its standard Borel subgroup, $W$ the Weyl group of $Φ$. To each involution $σ\in W$ one can assign the $B$-orbit $Ω_σ$ contained in the dual space of the Lie algebra of the unipotent radical of $B$. We prove that $Ω_σ$ is contained in the Zariski closure of $Ω_τ$ if and only of $σ\leqτ$ with respect to the Bruhat--Chevalley order. We also prove that $\dimΩ_σ$ is equal to $l(σ)$, the length of $σ$ in $W$.
Let $G$ be a connected complex semisimple Lie group with a fixed maximal torus $T$ and a Borel subgroup $B \supset T$. For an arbitrary automorphism $θ$ of $G$, we introduce a holomorphic Poisson structure $π_θ$ on $G$ which is invariant under the $θ$-twisted conjugation by $T$ and has the property that every $θ$-twisted conjugacy class of $G$ is a Poisson subvariety with respect to $π_θ$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $π_θ$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $π_θ$. We give the lowest rank of $π_θ$ in any given $θ$-t…
