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In this paper, we give a complete classification of silted algebras for the quiver $\overrightarrow{\mathbb{A}}_{n}$ of type $\mathbb{A}_{n}$ with linear orientation and for the quiver obtained from $\overrightarrow{\mathbb{A}}_{n}$ by…

The paper relates character value of an irreducible representation of a compact connected Lie group at certain elements of finite order with the dimension of a representation on another group, up to some precise constants, which all have…

We construct certain absolutely irreducible Banach representations of the quaternion algebra of large canonical (Gelfand-Kirillov) dimension which yield counterexamples to a natural conjecture of Dospinescu-Schraen on the existence of an…

Elliptic stable envelopes are fundamental components in the geometric realization of quantum group representations. We present a formula for elliptic stable envelopes on type A Cherkis bow varieties, as a product of simple basic objects in…

The purpose of this work is to define a natural action of the cactus group on the set of Gelfand-Tsetlin patterns for orthogonal Lie algebras. These Gelfand-Tsetlin patterns are meant to index the Gelfand-Tsetlin basis in the irreducible…

We give a uniform construction of irreducible polynomial representations of all classical groups, including spin groups, using semistandard domino tableaux. We also give an explicit decomposition of the homogeneous coordinate ring of the…

We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional…

Let F be a non-archimedean local field with residue characteristic p. Let l be a prime number different from p. Let G be a connected reductive group which is split, semi-simple, and simply connected. On the one hand, we describe the…

In this paper, we study restricted Poisson algebras in characteristic 2 and their relationship with restricted Lie-Rinehart algebras, for which we develop a cohomology theory and investigate abelian extensions. We also construct a full…

By using characteristic cycles, we build a morphism from the canonical bases of integrable highest weight modules of quantum groups to the top Borel-Moore homology groups of Nakajima's quiver and tensor product varieties, and compare the…

Cuspidal systems parameterize KLR algebra representations via root partitions $\pi$, where simple modules $L(\pi)$ arise as heads of proper standard modules. Working in affine type A with an arbitrary convex preorder, we construct explicit…

Let $(\tau, V_{\tau})$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $\Gamma$ be a uniform torsion-free lattice in $G$. We obtain an infinitesimal…

We construct an element in a completion of the universal enveloping algebra of $\mathfrak{gl}_N$, which we call the Kirillov projector, that connects the topics of the title: on the one hand, it is defined using the evaluation homomorphism…

We generalise Gelfand-Graev characters to $\mathbb R/\mathbb Z$-graded Lie algebras and lift them to produce new test functions to probe the local character expansion in positive depth. We show that these test functions are well adapted to…

In this paper, we present an independent proof of the Cauchy identities for staircase matrices, originally discovered in arXiv:2411.03117, using the combinatorics of the Bruhat poset and the bubble-sort procedure. Additionally, we derive…

This is a contribution to the study of $\operatorname {Irr}(G)$ as an $\operatorname {Aut}(G)$-set for $G$ a finite quasi-simple group. Focusing on the last open case of groups of Lie type $\mathrm D$ and $^2\mathrm D$, a crucial property…

In this paper we study intertwining functors (Radon transforms) for twisted D-modules on partial flag varieties and their relation to the representations of semisimple Lie algebras. We show that certain intertwining functors give…

We develop the connection between the preprojective $K$-theoretic Hall algebra of a quiver $Q$ and the quantum loop group associated to $Q$ via stable envelopes of Nakajima quiver varieties.

We discuss degenerations of symplectic and orthogonal representations of symmetric quivers and algebras with self-dualities. As in the non-symmetric case, we define a partial ordering, that we call symmetric Ext-order which gives a…

We consider small quantum groups with root systems of Cartan, super and modular type, among others. These are constructed as Drinfeld doubles of finite-dimensional Nichols algebras of diagonal type. We prove a linkage principle for them by…