Related papers: Generalized exponents of small representations. I
This article develops an alcove geometric approach to the representation theory of certain affine Hecke algebra quotients generalizing the blob algebra; and gives an exposition of some new representations of these algebras.
We study rational Cherednik algebras over an algebraically closed field of positive characteristic. We first prove several general results about category O, and then focus on rational Cherednik algebras associated to the general and special…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the…
Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the…
In this paper we give an intimate connection between the characteristic zero representation theories of the Additive and Heisenberg groups, and their characteristic p >0 theories when p is much larger than the dimension a representation. In…
We extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby…
We give explicit expressions for \vSapovalov elements in Type A Lie algebras and superalgebras. Explicit expressions were already given in arXiv:1710.10528 Section 9, using non-commutative determinants, and in fact our first main results,…
An elementary approach to the construction of Coxeter group representations is presented.
The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…
We propose an explicit construction of a weighted generalised Grassmannian. For a weighted Grassmannian (i.e., for series A) we obtain an effective parametrisation of possible $\mathbb{Z}$-gradings on Pl\"{u}cker coordinates, and provide…
Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy…
A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of…
We review some definitions and basic notions relating to generalised spin structures and introduce the notion of reducibility. We discuss connections on these structures, define a covariant Lie derivative for associated bundles and develop…
We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to $U_q(\mathfrak{s}\mathfrak{l}_2)$ at roots of unity and we show that…
Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ of the finite field of prime order $p$ and let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism with $G = \mathbf{G}^F$…
We present some results about the irreducible representations appearing in the exterior algebra $\Lambda \mathfrak{g}$, where $ \mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$. For Lie algebras of type $B$, $C$ or $D$ we prove that…
Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural "Coxeter…