Weyl Group Element Research Articles

Curve neighborhoods of Schubert varieties

A previous result of the authors with Chaput and Perrin states that the closure of the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space $G/P$ is again a Schubert variety. In this paper we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization.We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety.

Schubert Eisenstein Series

We define Schubert Eisenstein series as sums like usual Eisenstein series but with the summation restricted to elements of a particular Schubert cell, indexed by an element of the Weyl group. They are generally not fully automorphic. We will develop some results and methods for ${\rm GL}_3$ that may be suggestive about the general case. The six Schubert Eisenstein series are shown to have meromorphic continuation and some functional equations. The Schubert Eisenstein series $E_{s_1s_2}$ and $E_{s_2s_1}$ corresponding to the Weyl group elements of order three are particularly interesting: at the point where the full Eisenstein series is maximally polar, they unexpectedly become (with minor correction terms added) fully automorphic and related to each other.

Fundamental elements of an affine Weyl group

Fundamental elements are certain special elements of affine Weyl groups introduced by Gortz, 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.

Distinguished conjugacy classes and elliptic Weyl group elements

We define and study a correspondence between the set of distinguished G 0 G^0 -conjugacy classes in a fixed connected component of a reductive group G G (with G 0 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.

GROSSBERG-KARSHON TWISTED CUBES AND BASEPOINT-FREE DIVISORS

Let $G$ be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible $G$-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e. closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e. each lattice point appears precisely once in the formula, with coefficient $+1$. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $\lambda$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $\lambda$ and $\underline{w}$.

Abstract simplicity of locally compact Kac–Moody groups

AbstractIn this paper, we establish that complete Kac–Moody groups over finite fields are abstractly simple. The proof makes essential use of Mathieu and Rousseau’s construction of complete Kac–Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.

Representatives of elliptic Weyl group elements in algebraic groups

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Quantum cluster algebras of type A and the dual canonical basis

The article concerns the subalgebra Uv+(w) of the quantized universal enveloping algebra of the complex Lie algebra 𝔰𝔩n+1 associated with a particular Weyl group element of length 2n. We verify that Uv+(w) can be endowed with the structure of a quantum cluster algebra of type An. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiß–Leclerc–Schröer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis under Kashiwara's bilinear form.

On Schubert Varieties

In this article we prove that a Schubert variety X(w) is a toric variety if and only if the Weyl group element w is a product of distinct simple reflections.

Cyclic elements in semisimple lie algebras

We develop a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, we classfiy all nilpotents giving rise to semisimple and regular semisimple cyclic elements. As an application, we obtain an explicit construction of all regular elements in Weyl groups.

Reducibility of certain induced representations of [formula omitted] and [formula omitted

We give equivalent conditions for certain parabolically induced representations of E6 and E7 to be irreducible by calculating the action of certain Weyl group element and detecting poles of the first two Langlands–Shahidi L-functions at s=0.

Cluster structures on quantum coordinate rings

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac–Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory $${\fancyscript{C}_{w}}$$ of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities that can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.

On orbits of order ideals of minuscule posets

An action on order ideals of posets considered by Fon-Der-Flaass is analyzed in the case of posets arising from minuscule representations of complex simple Lie algebras. For these minuscule posets, it is shown that the Fon-Der-Flaass action exhibits the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White. A uniform proof is given by investigation of a bijection due to Stembridge between order ideals of minuscule posets and fully commutative Weyl group elements. This bijection is proven to be equivariant with respect to a conjugate of the Fon-Der-Flaass action and an arbitrary Coxeter element. If $P$ is a minuscule poset, it is shown that the Fon-Der-Flaass action on order ideals of the Cartesian product $P \times [2]$ also exhibits the cyclic sieving phenomenon, only the proof is by appeal to the classification of minuscule posets and is not uniform.

A Markov Chain on the Symmetric Group That Is Schubert Positive?

We study a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the stationary distribution of this Markov chain can be expressed in terms of positive sums of Schubert polynomials. This Markov chain is a multivariate generalization of a Markov chain introduced by the first author in the study of random affine Weyl group elements.

Elliptic Weyl group elements and unipotent isometries with 𝑝=2

Let G G be a classical group over an algebraically closed field of characteristic 2 2 and let C C be an elliptic conjugacy class in the Weyl group. In a previous paper the first named author associated to C C a unipotent conjugacy class Φ ( C ) \Phi (C) of G G . In this paper we show that Φ ( C ) \Phi (C) can be characterized in terms of the closure relations between unipotent classes. Previously, the analogous result was known in odd characteristic and for exceptional groups in any characteristic.

A Proof of the Goodearl–Lenagan Polynormality Conjecture

The quantum nilpotent algebras U^w_-(g), defined by De Concini-Kac-Procesi and Lusztig, are large classes of iterated skew polynomial rings with rich ring theoretic structure. In this paper, we prove in an explicit way that all torus invariant prime ideals of the algebras U^w_-(g) are polynormal. In the special case of the algebras of quantum matrices, this construction yields explicit polynormal generating sets consisting of quantum minors for all of their torus invariant prime ideals. This gives a constructive proof of the Goodearl-Lenagan polynormality conjecture. Furthermore we prove that Spec U^w_-(g) is normally separated for all simple Lie algebras g and Weyl group elements w, and deduce from it that all algebras U^w_-(g) are catenary.

Quantum unipotent subgroup and dual canonical basis

Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical basis $S^{*}$. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras to the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$, which contains quantum flag minors and satisfies a factorization property with respect to the `$q$-center' of $O_{q}[N(w)]$. This generalizes Caldero's results from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.

Maximal Newton polygons via the quantum Bruhat graph

This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety. Cet article étudie une relation surprenante entre la cohomologie quantique de la variété de drapeaux complets et l'ensemble partiellement ordonné de polygones de Newton associé à un élément du groupe de Weyl affine. L’élément clé pour établir cette connexion est le fait que les chemins dans le graphe de Bruhat quantique, qui est un graphe orienté pondéré dont les sommets sont indexés par des éléments du groupe de Weyl fini, encodent des chaînes saturées dans l'ordre de Bruhat fort sur le groupe de Weyl affine. Cette correspondance est aussi fondamentale dans les travaux de Lam et Shimonozo qui établissent l'isomorphisme de Peterson entre la cohomologie quantique de la variété de drapeaux finie et l'homologie de la Grassmannienne affine. De plus, en utilisant la géométrie associée à l'ensemble partiellement ordonné des polygones de Newton, on obtient des preuves indépendantes pour plusieurs assertions combinatoires sur les chemins dans le graphe de Bruhat quantiques et les symétries de ce graphe, qui ont été originellement démontrées par Postnikov en utilisant l'ordre de Bruhat incliné. Une application géométrique importante de ce travail est une inégalité qui donne une condition nécessaire pour que certaines variétés de Deligne-Lusztig affines dans la variété de drapeaux affine soient non-vides.

Casselman’s Basis of Iwahori Vectors and the Bruhat Order

AbstractW. Casselman defined a basis fu of Iwahori fixed vectors of a spherical representation of a split semisimple p-adic group G over a nonarchimedean local field F by the condition that it be dual to the intertwining operators, indexed by elements u of the Weyl group W. On the other hand, there is a natural basis , and one seeks to find the transition matrices between the two bases. Thus, let and . Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then , where z are the Langlands parameters for the representation and α runs through the set S(u, v) of positive coroots (the dual root systemof G) such that with rα the reflection corresponding to α. The condition is conjecturally always satisfied if G is simply-laced and the Kazhdan–Lusztig polynomial Pw0v,w0u = 1 with w0 the long Weyl group element. There is a similar formula for conjecturally satisfied if Pu,v = 1. This leads to various combinatorial conjectures.

A Note on Bessel Functions for Supercuspidal Representations of GL(2) over a p-Adic Field

We compute the Bessel functions which give the action of the Weyl group element in the Kirillov model of supercuspidal representations of GL 2(K), where Kis a p-adic field. Together with the known action of the Borel subgroup, this gives the full action of GL 2(K) for such representations. We consider supercuspidal representations constructed by Jacquet and Langlands using the Weil representation. When the residual characteristic is odd, these are all the supercuspidal representations, hence we get a full description of all supercuspidal representations in this case.