Affine Weyl Group Research Articles

A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters

We prove Lusztig's conjectures ${\bf P1}$--${\bf P15}$ for the affine Weyl group of type $\tilde{G}_2$ for all choices of parameters. Our approach to compute Lusztig's $\mathbf{a}$-function is based on the notion of a "balanced system of cell representations" for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the $\mathbf{a}$-function and we explicitly construct such a system in type $\tilde{G}_2$ for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in type $\tilde{G}_2$, allowing us to prove ${\bf P1}$ and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types $G_2$ and $A_1$, along with some explicit computations for the finite cells.

Investigating relations between discrete Painlevé equations: The multistep approach

We show how, starting from a mapping where the independent variable advances one step at a time, one can obtain versions of the mapping corresponding to a multi-step evolution. The same procedure is applied to discrete Painlevé equations, and we proceed to establish Miura relations between the single-step and the multi-step versions (in the present study “multi” referring to double, triple, and quintuple). These Miura relations are discrete Painlevé equations on their own right. We show that, while in some cases it is impossible to obtain a multi-step equation for a single variable, deriving a Miura system is still possible. We perform our analysis for equations associated with the affine Weyl groups E8(1), E7(1), E6(1), and A4(1).

Quantization of Hamiltonian loop group spaces

We prove a Fredholm property for spin-c Dirac operators $$\mathsf {D}$$ on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group $$K\ltimes \Gamma $$ , with K compact and $$\Gamma $$ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing $$[{\mathcal {X}}] \cap [\mathsf {D}]$$ yields an element of the formal completion $$R^{-\infty }(T)$$ of the representation ring of a maximal torus $$T \subset H$$ ; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating $$[{\mathcal {X}}] \cap [\mathsf {D}]$$ corresponds to an element of the ring of projective positive energy representations of the loop group.

Canonical Left Cells and the Lowest Two-sided Cell in an Affine Weyl Group

We give some discussions to the relations between canonical left cells and the lowest two-sided cell of an affine Weyl group. In particular, we use the relations to construct irreducible modules attached to the lowest two-sided cell and some one dimensional representations of an affine Hecke algebra.

DISCRETE SOLITONS IN INFINITE REDUCED WORDS

We consider a discrete dynamical system where the roles of the states and the carrier are played by translations in an affine Weyl group of type A. The Coxeter generators are enriched by parameters, and the interactions with the carrier are realized using Lusztig’s braid move (a, b, c) ↦ (bc/(a+c), a+c, ab/(a+c)). We use wiring diagrams on a cylinder to interpret chamber variables as τ-functions. This allows us to realize our systems as reductions of the Hirota bilinear difference equation and thus obtain N-soliton solutions.

Moments about the mean of the size of a self-conjugate [formula omitted]-core partition

Johnson proved that if s,t are coprime integers, then the rth moment of the size of an (s,t)-core is a polynomial of degree 2r in t for fixed s. After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of S˜t∕St and t-cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an (s,t)-core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an (s,t)-core and the first nine moments about the mean of the size of an (s,s+1)-core using Maple. To get the moments about the mean of the size of a self-conjugate (s,t)-core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if s,t are coprime integers, then the rth moment about the mean of the size of a self-conjugate (s,t)-core is a quasipolynomial of period 2 and degree 2r in t for fixed odd s. Then, based on a bijection of Ford, Mai and Sze between self-conjugate (s,t)-cores and lattice paths in s2×t2 rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate (s,t)-cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate (s,t)-core.

Cohomology in Singular Blocks for a Quantum Group at a Root of Unity

Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra mathfrak {g} and a root of unity ζ. When L, L′ are irreducible Uζ-modules having regular highest weights, the dimension of text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of Uζ. This paper shows for L, L′ irreducible modules in a singular block that dim text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra.

Poincaré duality and Langlands duality for extended affine Weyl groups

In this paper we construct an equivariant Poincare duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group C ∗ -algebras of extended affine Weyl groups at the level of K -theory.

Singular and Calabi–Yau varieties linked with billiard trajectories and diffusion operators

The images under a certain map of periodic billiard trajectories inside the fundamental region of the affine Weyl group of the root system $$A_{2}$$ are closed curves and configurations of lines described by a one-parameter family of polynomials. The polynomials are related with eigenvectors of symmetric diffusion operators connected with the deltoid. Several associated singular varieties and Calabi–Yau threefolds defined over the rationals are constructed.

Commuting involution graphs for

ABSTRACTIn this article we consider commuting graphs of involution conjugacy classes in the affine Weyl group of type . We show that where the graph is connected the diameter is at most n+2.

Parabolic Double Cosets in Coxeter Groups

Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and double quotients $W / W_I$ and $W_I \backslash W / W_J$, appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this article we look at a less studied object: the set of all double cosets $W_I w W_J$ for $I, J \subseteq S$. Double cosets are not uniquely presented by triples $(I,w,J)$. We describe what we call the lex-minimal presentation, and prove that there exists a unique such object for each double coset. Lex-minimal presentations are then used to enumerate double cosets via a finite automaton depending on the Coxeter graph for $(W,S)$. As an example, we present a formula for the number of parabolic double cosets with a fixed minimal element when $W$ is the symmetric group $S_n$ (in this case, parabolic subgroups are also known as Young subgroups). Our formula is almost always linear time computable in $n$, and we show how it can be generalized to any Coxeter group with little additional work. We spell out formulas for all finite and affine Weyl groups in the case that $w$ is the identity element.

On Iwahori-Hecke algebras for p-adic loop groups: double coset basis and Bruhat order

We study the $p$-adic loop group Iwahori-Hecke algebra ${\cal H}(G^+,I)$ constructed by Braverman, Kazhdan, and Patnaik and give positive answers to two of their conjectures. First, we algebraically develop the “double coset basis” of ${\cal H}(G^+,I)$ 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 $\cal{W}_{\cal{T}}$ on which Braverman, Kazhdan, and Patnaik define a preorder. Their preorder is a natural generalization of the Bruhat order on affine Weyl groups, and they conjecture that the preorder is a partial order. We define another order on $\cal{W}_{\cal{T}}$ which carries a length function and is manifestly a partial order. We prove the two definitions coincide, which implies a positive answer to their conjecture. Interestingly, the length function seems to naturally take values in ${\Bbb Z}\oplus{\Bbb Z}\varepsilon$ where $\varepsilon$ is “infinitesimally” small.

Tilting modules and the p-canonical basis

In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GLn using the theory of 2-Kac–Moody actions. Finally, we prove that the diagrammatic Hecke category of a general crystallographic Coxeter group may be described in terms of parity complexes on the flag variety of the corresponding Kac–Moody group.

A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E7(1)

Recently a certain $q$-Painlev\'e type system has been obtained from a reduction of the $q$-Garnier system. In this paper it is shown that the $q$-Painlev\'e type system is associated with another realization of the affine Weyl group symmetry of type $E_7^{(1)}$ and is different from the well-known $q$-Painlev\'e system of type $E_7^{(1)}$ from the point of view of evolution directions. We also study a connection between the $q$-Painlev\'e type system and the $q$-Painlev\'e system of type $E_7^{(1)}$. Furthermore determinant formulas of particular solutions for the $q$-Painlev\'e type system are constructed in terms of the terminating $q$-hypergeometric function.

Some left cells in the affine Weyl group Ẽ6

ABSTRACTThe aim of the present paper is to describe all the left cells L of the affine Weyl group with a(L)≤11. We find a representative set of those left cells which occur as the vertices of the corresponding left cell graphs. The main technical tools are the algorithm designed by the first-named author and the various right primitive pairs. Some interesting empirical phenomenon is observed concerning isomorphic left cell graphs in different two-sided cells of .

Multiplicative equations related to the affine Weyl group E8

We derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated with the affine Weyl group E8(1). Five such systems are obtained, three of which turn out to be linearisable and the remaining two are integrable in terms of elliptic functions. In the case of the linearisable mappings, we derive non-autonomous forms which contain a free function of the independent variable and we present the linearisation in each case. The two remaining systems are deautonomised to new discrete Painlevé equations. We show that these equations are in fact special forms of much richer systems associated with the affine Weyl groups E7(1) and E8(1), respectively.

A uniform classification of discrete series representations of affine Hecke algebras

We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra $\mathcal{H}_{\mathbf{v}}$, in terms of a canonically defined basis $\mathcal{B}_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\mathcal{H}$, and to all $\mathbf{v}\in\mathcal{Q}$, where $\mathcal{Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\mathcal{H}$. By analytic Dirac induction we define for each $b\in \mathcal{B}_{gm}$ a continuous (in the sense of [OS2]) family $\mathcal{Q}^{reg}_b:=\mathcal{Q}_b\backslash\mathcal{Q}_b^{sing}\ni\mathbf{v}\to\operatorname{Ind}_{D}(b;\mathbf{v})$, such that $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$ (for some $\epsilon(b;\mathbf{v})\in\{\pm 1\}$) is an irreducible discrete series character of $\mathcal{H}_{\mathbf{v}}$. Here $\mathcal{Q}^{sing}_b\subset\mathcal{Q}$ is a finite union of hyperplanes in $\mathcal{Q}$. In the non-simply laced cases we show that the families of virtual discrete series characters $\operatorname{Ind}_{D}(b;\mathbf{v})$ are piecewise rational in the parameters $\mathbf{v}$. Remarkably, the formal degree of $\operatorname{Ind}_{D}(b;\mathbf{v})$ in such piecewise rational family turns out to be rational. This implies that for each $b\in \mathcal{B}_{gm}$ there exists a universal rational constant $d_b$ determining the formal degree in the family of discrete series characters $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$. We will compute the canonical constants $d_b$, and the signs $\epsilon(b;\mathbf{v})$. For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.

The lowest two-sided cell of a Coxeter group with complete graph

In this paper we give a description of the structure of the based ring of the lowest two-sided cell for a weighted Coxeter group with complete graph. It is proved that this ring is generated, in a simple way, by some subrings which are isomorphic to the based rings of the lowest two-sided cells for affine Weyl groups of type A˜1 or A˜2.

On simple polynomial GrT-modules

Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for G=GLn, we consider polynomial representations of GrT for an arbitrary closed reductive subgroup scheme G⊆GLn and a maximal torus T of G in positive characteristic. We give sufficient conditions on G making a classification of simple polynomial GrT-modules similar to the case G=GLn possible and apply this to recover the corresponding result for GLn with a different proof, extending it to symplectic similitude groups, Levi subgroups of GLn and, in a weaker form, to odd orthogonal similitude groups. We also consider orbits of the affine Weyl group and give a condition for equivalence of blocks of polynomial representations for GrT in the case G=GLn.

Strange expectations and simultaneous cores

Let $$\gcd (a,b)=1$$gcd(a,b)=1. J. Olsson and D. Stanton proved that the maximum of in a (a, b)-core is $$\begin{aligned} \max _{\lambda \in {\mathrm {core}}(a,b)} (\mathsf{size}(\lambda )) = \frac{(a^2-1)(b^2-1)}{24} \end{aligned}$$maxźźcore(a,b)(size(ź))=(a2-1)(b2-1)24and that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected of in a (a, b)-core is $$\begin{aligned} \mathop {\mathbb {E}}\limits _{\lambda \in {\mathrm {core}}(a,b)}\left( \mathsf{size}(\lambda )\right) = \frac{(a-1)(b-1)(a+b+1)}{24}. \end{aligned}$$Eźźcore(a,b)size(ź)=(a-1)(b-1)(a+b+1)24.We extend Johnson's method to compute the variance to be $$\begin{aligned} \mathop {\mathbb {V}}\limits _{\lambda \in {\mathrm {core}}(a,b)}\left( \mathsf{size}(\lambda )\right) = \frac{ab(a-1)(b-1)(a+b)(a+b+1)}{1440}, \end{aligned}$$Vźźcore(a,b)size(ź)=ab(a-1)(b-1)(a+b)(a+b+1)1440,and also prove polynomiality of all moments. By extending the definitions of simultaneous cores and number of boxes to affine Weyl groups, we give uniform generalizations of all three formulae above to simply laced affine types. We further explain the appearance of the 24 using the strange formula of H. Freudenthal and H. de Vries.