For a symmetrizable Kac-Moody algebra the category of admissible representations is an analogue of the category of finite dimensional representations of a semisimple Lie algebra. The monoid associated to this category and the category of restricted duals by a generalized Tannaka-Krein reconstruction contains the Kac-Moody group as open dense unit group and has similar properties as a reductive algebraic monoid. In particular there are Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid, [M 1].
We determine the closure relations of the Bruhat and Birkhoff cells, which give extensions of the Bruhat order from the Weyl group to the Weyl monoid. We show that the Bruhat and Birkhoff cells are irreducible and principal open in their closures. We give product decompositions of the Bruhat and Birkhoff cells. We define extended length functions which are compatible with the extended Bruhat orders. We show a generalization of some of the Tits axioms for twinned BN-pairs.
In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer in J. Algebra, 163, 861--879 (1994). We also show that for a non-finite and non-cofinite biclosed set $B$ in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed $B$-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Final…
Starting with a highest weight representation of a Kac-Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highes…
Starting with a highest weight representation of a Kac-Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highes…
For any finite-dimensional complex semisimple Lie algebra two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group. The primary realization suggests an order on the Weyl group, which is useful for an exploration of the Bruhat order. The secondary realization is useful for finding for any element of the Weyl group all of its reduced expressions, which implies another realization of the Bruhat or…
We prove some injectivity results: that a Coxeter monoid $\mathbb{Z}$-algebra (or $0$-Hecke algebra) injects in the incidence $\mathbb{Z}$-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a right-angled Coxeter group injects in the Coxeter monoid $\mathbb{Z}[q,q^{-1}]$-algebra (and then in the incidence $\mathbb{Z}[q,q^{-1}]$-algebra of the corresponding Bruhat poset); that a right-angled Artin group injects in the group of invertible elements of the Hecke algebra of the corresponding Coxeter group (and then in the group of invertible elements of a Co…
We introduce and study a family of operators which act in the span of a Weyl group $W$ and provide a multi-parameter solution to the quantum Yang-Baxter equations of the corresponding type. Our operators generalize the "quantum Bruhat operators" that appear in the explicit description of the multiplicative structure of the (small) quantum cohomology ring of $G/B$. The main combinatorial applications concern the "tilted Bruhat order," a graded poset whose unique minimal element is an arbitrarily chosen element $w\in W$. (The ordinary Bruhat order corresponds to the case $w=1$.) Using the mixed …
For any Kac-Moody group $\mathbf{G}$, we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for $\mathbf{G}$ is strictly compatible with a $\mathbb{Z}$-valued length function. We conjecture in general and prove for $\mathbf{G}$ of affine ADE type that the Bruhat order is graded by this length function. We also formulate and discuss conjectures relating the length function to intersections of "double-affine Schubert varieties."
We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analogue of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the quantum Bruhat graph on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the quantum Bruhat graph to the parabolic one; this lemma compares paths between two vertices in the former graph. The results in this paper will be applied in a seco…
We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application…
For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory $\mathcal{F}(w)$ of a module category over a preprojective algebra of Dynkin type. This paper aims at studying categorical properties of $\mathcal{F}(w)$ via its connection with the root system. We show that by taking dimension vectors, simple objects in $\mathcal{F}(w)$ bijectively correspond to Bruhat inversion roots of $w$. As an application, we obtain a combinatorial criterion for $\mathcal{F}(w)$ to satisfy the Jordan-Hölder property (JHP). To achieve this, we develop a method to find simple o…
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 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 $W$.
In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations…
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…
By a generalized Tannaka-Krein reconstruction we associate to the admissible representations of the category O of a Kac-Moody algebra, and its category of admissible duals a monoid with a coordinate ring. The Kac-Moody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the Kac-Moody group is the algebra of strongly regular functions introduced by Kac and Peterson. This monoid has similar structural properties as a reductive algebraic monoid. In particular it is unit regular, its idempotents related to the faces of the Tits cone. It has Bruhat a…
The Casselman basis of Iwahori fixed vectors in a principal series representation of a p-adic group G is dual to the standard intertwining operators. To compute it one must compute a matrix m(u,v) indexed by pairs of Weyl group elements. This matrix is upper triangular with respect to the Bruhat order. In general this matrix is difficult to compute but it is shown that certain elements have a nice expression. This is also true of the inverse matrix to m(u,v). This leads to interesting conjectures regarding the Bruhat order.
Using truncated convolution of perverse sheaves on a flag variety Lusztig associated a monoidal category to a two sided cell in the Weyl group. We identify this category in the case which was not decided previously.
We construct a family of integrable Hamiltonian systems generalizing the relativistic periodic Toda lattice, which is recovered as a special case. The phase spaces of these systems are double Bruhat cells corresponding to pairs of Coxeter elements in the affine Weyl group. In the process we extend various results on double Bruhat cells in simple algebraic groups to the setting of Kac-Moody groups. We also generalize some fundamental results in Poisson-Lie theory to the setting of ind-algebraic groups, which is of interest beyond our immediate applications to integrable systems.
Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of $G$ are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on ${\mathbb{C}}$. It follows that all the Kogan-Zelevinsky integrable systems on $G$ have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of $SL(n, {\mathbb{C}})$. We in fact construct…
Motivated by the geometry of certain hyperplane arrangements, Manin and Schechtman defined for each positive integer n a hierarchy of finite partially ordered sets B(n, k), indexed by positive integers k, called the higher Bruhat orders. The poset B(n, 1) is naturally identified with the weak left Bruhat order on the symmetric group S_n, each B(n, k) has a unique maximal and a unique minimal element, and the poset B(n, k + 1) can be constructed from the set of maximal chains in B(n, k). Elias has demonstrated a striking connection between the posets B(n, k) for k = 2 and the diagrammatics of B…
Let $λ$ be a dominant weight of a finite dimensional simple Lie algebra and $W$ the Weyl group. The convex hull of $Wλ$ is defined as the weight polytope of $λ$. We provide a new proof that there is a natural bijection between the set of orbits of the nonempty faces of the weight polytope under the action of the Weyl group and the set of the connected subdiagrams of the extended Dynkin diagram that contain the extended node $\{-λ\}$. We show that each face of the polytope can be transformed to a standard parabolic face. We also show that a standard parabolic face is the convex hull of the orbi…
For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h and Weyl group W, we describe the category of perverse sheaves on h/W smooth w.r.t the natural stratification. The answer is given in terms of mixed Bruhat sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed Bruhat sheaves with the properties of various procedures of parabolic induction and restriction that connect different Levi subgroups in G.
It is shown that there is an order isomorphism $ϕ'$ from the poset $V$ of $B\times B$-orbits on the wonderful compactification of a semi-simple adjoint group $G$ with Weyl group $W$ to an interval in reverse Chevalley-Bruhat order on a non-canonically associated Coxeter group $\hat{W}$ (in general neither finite nor affine). Moreover, $ϕ'$ preserves the corresponding Kazhdan-Lusztig polynomials. Springer's (partly conjectural) construction of Kazhdan-Lusztig polynomials for the analogues of $V$ for general Coxeter groups $W$ is completed by reducing it by a similar order isomorphism to known r…
In this paper, we consider affine Deligne-Lusztig varieties $X_w(b)$ and their certain union $X(μ,b)$ inside the affine flag variety of a reductive group. Several important results in the study of affine Deligne-Lusztig varieties have been established under the so-called superregularity hypothesis. Such results include a description of generic Newton points in Iwahori double cosets of loop groups, covering relation in associated Iwahori-Weyl group and dimension formula for $X(μ,b)$. We show that one can considerably weaken the superregularity hypothesis and sometimes completely eliminate it, t…
By explicitly describing a cellular decomposition we determine the Borel invariant cycles that generate the Chow groups of the quotient of a reductive group by a Levi subgroup. For illustrations we consider the variety of polarizations $\mbf{SL}_n / \mbf{S}(\mbf{GL}_p\times \mbf{GL}_q)$, and we introduce the notion of a sect for describing its cellular decomposition. In particular, for $p=q$, we show that the Bruhat order on the sect corresponding to the dense cell is isomorphic, as a poset, to the rook monoid with the Bruhat-Chevalley-Renner order.
The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group $\tilde W$. The main discovery is a close connection between the Demazure product of an extended affine Weyl group and the quantum Bruhat graph of the finite Weyl group. As applications, we obtain explicit formulas on the generic Newton points and the Demazure products of elements in the lowest two-sided cell/shrunken Weyl chambers of $\tilde W$, and obtain an explicit formula …
We introduce semi-infinite Lakshmibai-Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of our previous result (which uses Littelmann's Lakshmibai-Seshadri paths), in which the level-zero dominant integral weight is assumed to be a posi…
An interesting open question is to determine the group of monoidal autoequivalences of the category of Yetter-Drinfeld modules over a finite group $G$, or equivalently the group of Bigalois objects over the dual of the Drinfeld double $DG$. In particular one would hope to decompose this group into terms related to monoidal autoequivalences for the group algebra, the dual group algebra and interaction terms. We report on our progress in this question: We first prove a decomposition of the group of Hopf algebra automorphisms of the Drinfeld double into three subgroups, which reduces in the case …
The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties. We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerati…
We study the $p$-adic loop group Iwahori-Hecke algebra $\mathcal{H}(G^+,I)$ constructed by Braverman, Kazhdan, and Patnaik and give positive answers to two of their conjectures. First, we algebraically develop the "double coset basis" of $\mathcal{H}(G^+,I)$ given by indicator functions of double cosets. We prove a generalization of the Iwahori-Matsumoto formula, and as a consequence, we prove that the structure coefficients of the double coset basis are polynomials in the order of the residue field. The basis is naturally indexed by a semi-group $\mathcal{W}_\mathcal{T}$ on which Braverman, K…
In this paper, we initiate the study of the twisted weak order associated to a twisted Bruhat order for a Coxeter group and explore the relationship between the lattice property of such order and the infinite reduced words. We show that for a 2 closure biclosed set $B$ in $Φ^+$, the $B$-twisted weak order is a non-complete meet semilattice if $B$ is the inversion set of an infinite reduced word and that the converse also holds in the case of affine Weyl groups.
There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in $\tilde{S}_n$ to the set of $\mathbb{N}$-weighted matchings in the $n$-element cycle graph. As an application, we compute a formula for the bivariate generating function counting the involutions in $\tilde S_n$ by length and absolute length. The 0-Hecke monoid of $\tilde{S}_n$ also acts on…
Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset generalizes many properties of the usual order on positive roots of W given by height. In this paper, a linear representation of the positive monoid of A is defined by use of the poset.
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…
Let $(π_{\mathbf{z}},V_{\mathbf{z}})$ be an unramified principal series representation of a reductive group over a nonarchimedean local field, parametrized by an element $\mathbf{z}$ of the maximal torus in the Langlands dual group. If $v$ is an element of the Weyl group $W$, then the standard intertwining integral $\mathcal{A}_v$ maps $V_{\mathbf{z}}$ to $V_{v\mathbf{z}}$. Letting $ψ^{\mathbf{z}}_w$ with $w\in W$ be a suitable basis of the Iwahori fixed vectors in $V_{\mathbf{z}}$, and $\widehatψ^{\mathbf{z}}_w$ a basis of the contragredient representation, we define $σ(u,v,w)$ (for $u,v,w\in…
In this paper we describe conjugacy classes of a Renner monoid $R$ with unit group $W$, the Weyl group. We show that every element in $R$ is conjugate to an element $ue$ where $u\in W$ and $e$ is an idempotent in a cross section lattice. Denote by $W(e)$ and $W_*(e)$ the centralizer and stabilizer of $e\in Λ$ in $W$, respectively. Let $W(e)$ act by conjugation on the set of left cosets of $W_*(e)$ in $W$. We find that $ue$ and $ve$ ($u, v\in W$) are conjugate if and only if $uW_*(e)$ and $vW_*(e)$ are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugac…
The standard Poisson structure on the rectangular matrix variety M_{m,n}(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T of GL_{m+n}(C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of M_{m,n}(C), isomorphic to intersections of dual Schubert cells in the full flag variety of GL_{m+n}(C). Three different presentations of the T-orbits of symplectic leaves in M_{m,n}(C) are obtained - (a) as pullbacks of Bruhat cells in GL_{m+n}(C) under a particular map; (b) in terms of rank conditions on rectangula…
We prove Schur--Weyl duality between the Brauer algebra $\mathfrak{B}_n(m)$ and the orthogonal group $O_{m}(K)$ over an arbitrary infinite field $K$ of odd characteristic. If $m$ is even, we show that each connected component of the orthogonal monoid is a normal variety; this implies that the orthogonal Schur algebra associated to the identity component is a generalized Schur algebra. As an application of the main result, an explicit and characteristic-free description of the annihilator of $n$-tensor space $V^{\otimes n}$ in the Brauer algebra $mathfrak{B}_n(m)$ is also given.
The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.
We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct act…
Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} 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.
