Related papers: Positivity for certain Weyl group representations

Let $c$ be the family of irreducible representations of a Weyl group $W$ corresponding to a two-sided cell of $W$. We define a subset $A_c$ of $c$ which contains the special representation of $W$ in $c$ and is in canonical bijection with…

We introduce $\Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $\Theta$-positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of…

Let G be a split semisimple algebraic group with trivial center. Let S be a compact oriented surface, with or without boundary. We define {\it positive} representations of the fundamental group of S to G(R), construct explicitly all…

Let $k$ be an infinite field of positive characteristic. We determine all homomorphisms between Weyl modules for $GLn(k)$, where one of the partitions is a hook. As a consequence we obtain a nonvanishing result concerning homomorphisms…

Categories of W*-bimodules are shown in an explicit and algebraic way to constitute an involutive W*-bicategory.

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 reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection…

The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove…

Let $\cH$ be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of $\cH$ and Lusztig's…

Let $W$ be an affine Weyl group, and let $\Bbbk$ be a field of characteristic $p>0$. The diagrammatic Hecke category $\mathcal{D}$ for $W$ over $\Bbbk$ is a categorification of the Hecke algebra for $W$ with rich connections to modular…

In this note we characterize those unitary one-parameter groups U^c which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary…

We define global and local Weyl modules for Lie superalgebras of the form $\mathfrak{g} \otimes A$, where $A$ is an associative commutative unital $\mathbb{C}$-algebra and $\mathfrak{g}$ is a basic Lie superalgebra or $\mathfrak{sl}(n,n)$,…

We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules…

Let W be a Weyl group, presented as a crystallographic reflection group on a Euclidean vector space V, and C an open Weyl chamber. In a recent paper, Waldspurger proved that the images (id-w)(C), for Weyl group elements w, are all disjoint,…

Let G be a simple reductive group over the complex numbers. Let W be the Weyl group of G. We propose a description of the Springer representations of W associated to various unipotent classes of G by a purely algebraic method involving the…

Let $\bH$ be the generic Iwahori--Hecke algebra associated with a finite Coxeter group $W$. Recently, we have shown that $\bH$ admits a natural cellular basis in the sense of Graham--Lehrer, provided that $W$ is a Weyl group and all…

We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several…

For extra-large Coxeter systems (m(s,r)>3), we construct a natural and explicit set of Soergel bimodules D={D_w}_{w\in W} such that each D_w contains as a direct summand (or is equal to) the indecomposable Soergel bimodule B_w. When…

We give a complete picture of when the tensor product of an induced module and a Weyl module is a tilting module for the algebraic group $SL_2$ over an algebraically closed field of characteristic $p$. Whilst the result is recursive by…

We give the first positive formulas for the weights of every simple highest weight module $L(\lambda)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we also express the weights of $L(\lambda)$ as an…