Symmetric Lie superalgebras and deformed quantum Calogero-Moser problems

TL;DR

This paper explores the representation theory of symmetric Lie superalgebras and their connection to deformed quantum Calogero-Moser systems, establishing a bijection between certain modules and eigenspaces of integrals.

Contribution

It introduces a novel correspondence between spherically typical modules of symmetric Lie superalgebras and eigenspaces of Calogero-Moser integrals in a specific symmetric pair case.

Findings

Established a bijection between projective covers and eigenspaces.

Linked representation theory with integrable quantum systems.

Provided new insights into spherical functions of Lie superalgebras.

Abstract

The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g=gl(n,2m), k=osp(n,2m) we establish a natural bijection between projective covers of spherically typical irreducible g-modules and the finite dimensional generalised eigenspaces of the algebra of Calogero-Moser integrals D_{n,m} acting on the corresponding Laurent quasi-invariants A_{n,m}.