On unitary submodules in the polynomial representations of rational Cherednik algebras. (Q2909338)

In this article, the authors study unitary representations of the rational Cherednik algebra associated to a Weyl group \(W\). Each simple module in category \(\mathcal O\) of the rational Cherednik algebra has a unique (up to scalar) \(W\)-invariant, non-degenerate Hermitian form compatible, in the standard way, with the action of the algebra. The module is said to be unitary if this form is positive definite. The study of unitary representations of rational Cherednik algebras was initiated by \textit{P. Etingof} and \textit{E. Stoica} [Represent. Theory 13, 349-370 (2009; Zbl 1237.20005)].NEWLINENEWLINE The rational Cherednik algebra has a natural polynomial representation, which has a unique simple submodule \(S_c\). It was shown [in loc. cit.] that this representation is unitary provided that the integral NEWLINE\[NEWLINE\int_{\mathbb R^N}|f(x)|^2e^{-\frac{1}{2}|x|^2}\prod_{\alpha\in\mathcal R_+}|(\alpha,x)|^{-2c(\alpha)}dxNEWLINE\]NEWLINE is convergent for all \(f\in S_c\). This was shown [in loc. cit.] to be the case for all \(c\) when \(W\) is of type \(A\). In this article the authors use a method different from [loc. cit.] to show that the above integral converges in type \(A\), for many values in type \(B\) (including all equal parameter cases) and most values of \(c\) in type \(D\). It is explained that \(S_c\) is not unitary for some values of \(c\) in type \(D\). Partial results for other Weyl groups are also given in later sections.NEWLINENEWLINE By considering partial log resolutions of certain hyperplane arrangement singularities, the authors show that the function \(|f(x)|\prod_{\alpha\in\mathcal R_+}|(\alpha,x)|^{-c(\alpha)}\) is locally \(L^2\)-integrable on \(\mathbb R^N\) for \(f\in S_c\). This implies that the above mentioned integral converges. We also note that the authors construct several singular subspaces of the polynomial representation for various complex reflection groups. This allows them to describe several interesting submodules of the polynomial representation.