Related papers: Affine Weyl groups in K-theory and representation …

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing…

We study the action of the Chevalley involution of a simple complex Lie group G on the set of its weight varieties (i.e. torus quotients of its flag manifolds). We find for torus quotients of Grassmannians (for GL(n)) that we obtain the…

In this note we describe a seemingly new approach to the complex representation theory of the wreath product $G\wr S_d$ where $G$ is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We…

Let $\mathbb{G}$ be a split connected reductive group scheme over the ring of integers $\mathfrak{o}$ of a finite extension $L|\mathbb{Q}_p$ and $\lambda\in X(\mathbb{T})$ an algebraic character of a split maximal torus…

The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In…

Let $G$ be a finite group, $X$ be a compact $G$-space. In this note we study the $(\mathbb{Z}_ + \times\mathbb{Z}/2\mathbb{Z})$-graded algebra $$\mathcal{F}^q_G(X) = \bigoplus_{n\geq0} q^n \cdot…

In the present notes we generalize the classical work of Demazure [Invariants sym\'etriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let G be a split semisemiple linear…

We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions:…

We prove an analogue of the Borel-Bott-Weil theorem in equivariant KK-theory by constructing certain canonical equivariant correspondences between minimal flag varieties G/B, with G a complex semisimple Lie group.

We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of…

Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other…

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are…

We show how the quantum Chevalley formula for G/B, as stated by Peterson and proved rigorously by Fulton and Woodward, combined with ideas of Fomin, S. Gelfand and Postnikov, leads to a formula which describes polynomial representatives of…

We show that if G is a split semisimple algebraic group over a model complete field K, then the groups G(K) and G(K)' (the commutator group which is a ``Chevalley group'' as for example the group PSL_2(K)) are model complete as well.

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum…

In this paper, we develop the foundations of the representation theory of quiver Hecke--Clifford superalgebras. We further construct a Schur--Weyl duality between quantum affine analogues of the queer Lie superalgebra and the quiver…

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We…

The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More…

Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_q$ and let $G = GL_{2/F}$. Let $\mathbf{q}$ be an indeterminate and let $H^{(1)}(\mathbf{q})$ be the generic pro-p Iwahori-Hecke algebra of the group $G(F)$. Let…

For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…