Related papers: Almost dominant generalized slices and convolution…

Braverman, Finkelberg and Nakajima introduced the generalized affine Grassmannian slices $\overline{\mathcal W}^{\lambda}_{\mu}$ and showed that they are Coulomb branches of $3d$ $\mathcal N=4$ gauge theories. We prove a conjecture of…

The generalized affine Grassmannian slices $\overline{\mathcal{W}}_\mu^\lambda$ are algebraic varieties introduced by Braverman, Finkelberg, and Nakajima in their study of Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories. We…

In this paper, we prove that for any semisimple simply connected algebraic group $G$, for any regular dominant character $\lambda$ of a maximal torus $T$ of $G$ and for any element $\tau$ in the Weyl group $W$, the character $e^{\rho}\cdot…

Let $\lambda$ be a dominant weight of a finite dimensional simple Lie algebra and $W$ the Weyl group. The convex hull of $W\lambda$ is defined as the weight polytope of $\lambda$. We provide a new proof that there is a natural bijection…

Let $G$ and $\tilde G$ be connected complex reductive Lie groups, $G$ semisimple. Let $\Lambda^+$ be the monoid of dominant weights for a positive root system $\Delta^+$, and let $l(w)$ be the length of a Weyl group element $w$. Let…

In this paper we study representations of skew group algebras $\Lambda G$, where $\Lambda$ is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field $k$ with characteristic $p…

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let ${\mathbf G}$ be an algebraic group of classical type with defining characteristic $p>0$, $\mu$ a dominant weight and $W$ the…

After some definitions, we review in the first part of this talk the construction and classification of classical $W$ (super)algebras symmetries of Toda theories. The second part deals with more recently obtained properties. At first, we…

Let $A$ be any commutative unital ring and let $\operatorname{GL}_{2,A}$ be the general linear group scheme on $A$ of rank $2$. We study the representation theory of $\operatorname{GL}_{2,A}$ and the symmetric powers…

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{C}$ and let $\rho :G\rightarrow GL(V_\lambda)$ be an irreducible representation of highest weight $\lambda$. Suppose that $\rho$ has finite kernel. Springer defined…

We study a relationship between the graded characters of generalized Weyl modules $W_{w \lambda}$, $w \in W$, over the positive part of the affine Lie algebra and those of specific quotients $V_{w}^- (\lambda) / X_{w}^- (\lambda)$, $w \in…

For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(\Lambda^{++})$ of strictly dominant weights $\Lambda^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any…

In this paper, we study gravitational symmetry algebras that live on 2-dimensional cuts $S$ of asymptotic infinity. We define a notion of wedge algebra $\mathcal{W}(S)$ which depends on the topology of $S$. For the cylinder $S=\mathbb{C}^*$…

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…

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation…

The Kronecker coefficient g_{\lambda \mu \nu} is the multiplicity of the GL(V)\times GL(W)-irreducible V_\lambda \otimes W_\mu in the restriction of the GL(X)-irreducible X_\nu via the natural map GL(V)\times GL(W) \to GL(V \otimes W),…

It was demonstrated recently that the $W_{1+\infty}$ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with…

Given a connected reductive complex algebraic group $G$ and a spherical subgroup $H \subset G$, the extended weight monoid $\widehat \Lambda^+_G(G/H)$ encodes the $G$-module structures on spaces of global sections of all $G$-linearized line…

For a simple Lie algebra $\mathfrak{g}$ of type $A_n,B_n,C_n$ or $D_n$, we give a characterization of the set of dominant integral weights $\lambda$ such that for any rational point $\mu$ in the fundamental Weyl chamber, $2\lambda-\mu$ is a…

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all…