Unitary representations of cyclotomic rational Cherednik algebras (Q1655811)

The goals of this paper is towfold: first, to obtain the classification of irreducible unitary representations in category \(\mathcal{O}_c\) for the rational Cherednik algebras of type \(W=G(r, 1, n)\), and second, to give an explicit basis for each of them in terms of Specht-module-valued versions of non-symmetric Jack polynomials. The strategy is that of the author's appendix to [\textit{P. Etingof} and \textit{E. Stoica}, Represent. Theory 13, 349--370 (2009; Zbl 1237.20005)]. The tactics are somewhat different, requiring the development of tools that were previously unavailable for \(r>1\), and the answer is quite a bit more complicated. The irreducible objects in the category \(\mathcal{O}_c\) are indexed by the irreducible complex representations of the complex reflection group \(W\). Thus for \(W=S_n\) the symmetric group, they are indexed by integer partitions of \(n\), and for \(W=G(2,1,n)=W(B_n)\) the Weyl group of type \(B_n\) they are indexed by pairs \((\lambda^0, \lambda^1)\) of integer partitions with \(n\) total boxes. More generally, for the complex reflection group \(G(r,1,n)\) they are indexed by \(r\)-tuples \((\lambda^0, \dots, \lambda^{r-1})\) of integer partitions with \(n\) total boxes. In the case \(W=S_n\), the parameter \(c\) is a single real number; for the Weyl group of type \(B\) it is a pair \((c, d)\) consisting of two real numbers; and for the complex reflection group \(G(r,1,n)\) it is an \(r\)-tuple of real numbers. The author classifies the irreducible unitary modules in category \(\mathcal{O}_c\) for the rational Cherednik algebras of type \(G(r, 1, n)\) and give explicit combinatorial formulas for their graded characters. More precisely, he produces a combinatorial algorithm determining, for each \(r\)-partition \(\lambda^{\bullet}\) of \(n\), the closed semi-linear set of parameters \(c\) for which the contra-variant form on the irreducible representation \(L_c(\lambda^{\bullet})\) is positive definite. He uses this algorithm to give a closed form answer for the Cherednik algebra of the symmetric group (recovering a result of Etingof-Stoica and the author [loc. cit.]) and the Weyl groups of classical type.