×

Dirac induction for rational Cherednik algebras. (English) Zbl 1461.58009

This paper is well-written, well-organized and well-illustrated. It makes a valuable contribution in index theory.
The paper deals with development of an index theory for Dirac operators in the context of rational Cherednik algebras that are associated with finite-dimensional representations of complex reflection groups.
The authors introduced several new notions related to Dirac operators (for rational Cherednik algebras) as graded local theory, graded global theory and index polynomials (3.5). They were motivated by the works [the first autor, Sel. Math., New Ser. 22, No. 1, 111–144 (2016; Zbl 1383.20007); D. Barbasch et al., Acta Math. 209, No. 2, 197–227 (2012; Zbl 1276.20004); the first author and P. E. Trapa, Duke Math. J. 162, No. 2, 201–223 (2013; Zbl 1260.22012); the first author et al., J. Inst. Math. Jussieu 13, No. 3, 447–486 (2014; Zbl 1362.20004)].
Let \(G\) be a complex reflection group and \(G\to End(\mathrm{h})\) be a finite representation. Recall that \(\mathrm{H}_{t,c}(G,\mathrm{h})\) denotes the rational Cherednik algebra associated to \(G\to End(\mathrm{h})\) (see Section 2) and \(\mathcal{O}_{t,c}(G,\mathrm{h})\) denotes the category of left \(\mathrm{H}_{t,c}(G,\mathrm{h})\)-modules that are finitely generated and \(\mathrm{h}\)-locally nilpotent (see Definition 2.9).
For a certain class of graded modules of \(\mathcal{O}_{t,c}(G,\mathrm{h})\) (see Definition 3.4), the authors defined and studied \(G\)-invariant graded Dirac operators and the Dirac cohomology, and compute the index of such a class.
A major result of this paper is the localization formula Theorem 1.1, which gives a reformulation, in terms of Dirac theory, of the problem concerning computation of the graded \(G\)-character of a simple \(\mathrm{H}_{t,c}(G,\mathrm{h})\)-module that has been investigated by several authors (see for example [Y. Berest et al., Int. Math. Res. Not. 2003, No. 19, 1053–1088 (2003; Zbl 1063.20003); G. Bellamy, Bull. Lond. Math. Soc. 41, No. 2, 315–326 (2009; Zbl 1227.14008); I. Gordon, Bull. Lond. Math. Soc. 35, No. 3, 321–336 (2003; Zbl 1042.16017); U. Thiel, LMS J. Comput. Math. 18, 266–307 (2015; Zbl 1319.16036); U. Thiel, in: Representation theory – Current trends and perspectives. In part based on talks given at the last joint meeting of the priority program in Bad Honnef, Germany, in March 2015. Zürich: European Mathematical Society (EMS). 681–745 (2017; Zbl 1377.16022)]).
Another interesting result is Corollary 3.19, which connects the Dirac index polynomials with the graded Euler-Poincaré pairing.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
20C08 Hecke algebras and their representations
19K56 Index theory
PDFBibTeX XMLCite
Full Text: DOI arXiv