Unitary representations of type B rational Cherednik algebras and crystal combinatorics

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. 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. Further, we find the supports of the unitary representations.

Unitary Representations of Cyclotomic Hecke Algebras at Roots of Unity: Combinatorial Classification and BGG Resolutions

“…Finally, for level = 2 with a noncylindric charge s ∈ Z , it is shown in [43] that all unitary representations of Cherednik algebras with full support can be reduced to the level = 1 case. Thus, for level 2, every unitary representation with full support of a Cherednik algebra (regardless of the charge) admits a BGG resolution.…”