Representation theory of rectangular finite $W$-algebras

TL;DR

This paper classifies all finite-dimensional irreducible representations of rectangular finite W-algebras, which are associated with symplectic or orthogonal Lie algebras and specific nilpotent elements, advancing understanding in representation theory.

Contribution

It provides a complete classification of finite-dimensional irreducible representations for a specific class of finite W-algebras, filling a gap in the literature.

Findings

Classification of irreducible representations achieved

Explicit description for symplectic and orthogonal cases

Enhanced understanding of W-algebra representation theory

Abstract

We classify the finite dimensional irreducible representations of rectangular finite $W$-algebras, i.e., the finite $W$-algebras $U(\mathfrak{g}, e)$ where $\mathfrak{g}$ is a symplectic or orthogonal Lie algebra and $e \in \mathfrak{g}$ is a nilpotent element with Jordan blocks all the same size.