Related papers: Computing the Lusztig--Vogan Bijection

The two main theorems proved here are as follows: If $A$ is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of $A$ is invariant under derived equivalence.…

We prove that a smooth and connected algebraic group $G$ is affine if and only if any invertible sheaf on any normal $G$-variety is $G$-invariant. For the proof, a key ingredient is the following result: if $G$ is a connected and smooth…

We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group…

Let $ \pi $ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ AV(\pi) $. If $ AV(\pi) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus…

We define a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. We study commutativity, cancellativity and associativity of this operation. We focus on rigid irreducible components and discuss…

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$. We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N \subset \mathfrak{g}$…

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…

Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $…

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and…

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we…

Let $\mathfrak g$ be a finite simple Lie algebra, and let $r$ denote the ratio of the square length of long roots to that of short roots. Let $\wp>2r$ be an integer and $\zeta$ a primitive $\wp$-th root of unity. Denote by $\mathcal…

We consider the space (weighted Fourier algebra) of Banach algebra valued functions $A^q_{\omega}(\Gamma,\cX),$ which consists of all Fourier transforms of functions in $L^q_\omega(G,\cX)$. Here $\omega$ is a Beurling-Domar type weight on a…

Let $V$ be a vertex operator algebra and $g$ an automorphism of finite order. We construct an associative algebra $A_g(V)$ and a pair of functors between the category of $A_g(V)$-modules and a certain category of admissible $g$-twisted…

This article concerns the study of a new invariant bilinear form $\mathcal B$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathcal B$ using the asymptotics maps from…

Let $G$ be a connected reductive group, $T$ a maximal torus of $G$, and $N$ the normalizer of $T$ in $G$. In this paper we study the connection between the derived category of l-adic sheaves on the stack $[Lie(T)/N]$ and the derived…

When ${\frak g}$ is a complex semisimple Lie algebra, we study the variety ${\mathcal L}$ of subalgebras of ${\frak g}\oplus{\frak g}$ that are maximally isotropic with respect to $K_1 - K_2$, where $K_i$ is the Killing form on the ith…

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible…

For $G$ an algebraic group definable over a model of $\operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $\widehat{G}$ of $G$, as introduced by Loeser and the second author.…

Let g be a real form of a simple complex Lie algebra. Based on ideas of Djokovic and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant-Sekiguchi correspondence. Our algorithms are…

In this paper, we present a uniform formula of Lusztig's $ \mathbf{a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand-Kirillov dimensions of simple highest weight modules of classical Lie algebras,…