One-W-type Modules for Rational Cherednik Algebra and Cuspidal Two-sided Cells

Abstract

We classify the simple modules for the rational Cherednik algebra H0,c that are irreducible when restricted to W , in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in the sense of Lusztig. We compute the Dirac cohomology of these modules and use the tools of Dirac theory to find nontrivial relations between the cuspidal Calogero-Moser cells, in the sense of Bellamy, and the cuspidal two-sided cells.

Cuspidal Calogero–Moser and Lusztig families for Coxeter groups

“…a family is said to be cuspidal if it contains a rigid module. By Theorem C, every cuspidal family in our sense is cuspidal in the sense of [14]. However, it is clear that for most complex reflection groups that are not of Coxeter type there exist many cuspidal families (in our sense) that are not cuspidal in the sense of loc.…”

“…Remark. While this paper was in preparation, the preprint [14] appeared, where rigid modules also play a key role (though the definition there is slightly different). Based on the analogy with affine Hecke algebras, they are called "one--type" modules in loc.…”

“…cit.. In the preprint [14] the author gives a different notion of cuspidal Calogero-Moser families. Namely, in loc.…”

“…The authors would like to thank Cédric Bonnafé and Meinolf Geck for many fruitful discussions. We also thank Dan Ciubotaru for informing us about his preprint [14] and his result that for 7 the cuspidal Lusztig family does not contain rigid modules. Moreover, we would like to thank Gunter Malle for commenting on a preliminary version of this article.…”

“…At = 0, the second author investigated rigid modules in [41]. Recently, they also played a prominent role in the work [14] of Ciubotaru on Dirac cohomology where they were called one--type modules. The terminology we adopt comes from the theory of module varieties, where it is standard.…”