Representations of Infinitesimal Cherednik Algebras

TL;DR

This paper studies the algebraic structure and representation theory of infinitesimal Cherednik algebras, constructing their centers, classifying finite-dimensional modules, and exploring Poisson analogues to deepen understanding of their properties.

Contribution

It provides the first detailed analysis of the center, irreducible representations, and Poisson analogues of infinitesimal Cherednik algebras, extending known results to new cases.

Findings

Constructed the complete center for n=2 and identified a generator for general n.

Derived the formula for the Shapovalov determinant and criteria for irreducibility.

Classified all finite-dimensional irreducible representations and computed their characters.

Abstract

Infinitesimal Cherednik algebras, first introduced in [EGG], are continuous analogues of rational Cherednik algebras, and in the case of gl_n, are deformations of universal enveloping algebras of the Lie algebras sl_{n+1}. Despite these connections, infinitesimal Cherednik algebras are not widely-studied, and basic questions of intrinsic algebraic and representation theoretical nature remain open. In the first half of this paper, we construct the complete center of H_\zeta(gl_n) for the case of n=2 and give one particular generator of the center, the Casimir operator, for general n. We find the action of this Casimir operator on the highest weight modules to prove the formula for the Shapovalov determinant, providing a criterion for the irreducibility of Verma modules. We classify all irreducible finite dimensional representations and compute their characters. In the second half, we investigate Poisson-analogues of the infinitesimal Cherednik algebras and use them to gain insight on the center of H_\zeta(gl_n). Finally, we investigate H_\zeta(sp_{2n}) and extend various results from the theory of H_\zeta(gl_n), such as a generalization of Kostant's theorem.