Unitary representations of type B rational Cherednik algebras and crystal combinatorics

TL;DR

This paper explores the relationship between crystal combinatorics of the level 2 Fock space and the classification of unitary irreducible representations of type B rational Cherednik algebras, revealing how unitarity is preserved under certain operations.

Contribution

It provides new insights into the structure of unitary representations, including a novel proof of their classification and the behavior of unitarity under crystal and parabolic restriction operations.

Findings

Finite-dimensional unitary irreducible representations are labeled by bipartitions with a rectangular partition.

Crystal operators that remove boxes preserve unitarity conditions.

Parabolic restriction functors send unitary representations to unitary representations.

Abstract

We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations.