On isomorphisms of certain functors for Cherednik algebras. (Q2836485)

This article proves a conjecture by \textit{R. Bezrukavnikov} and \textit{P. Etingof} on the properties of certain restriction and induction functors, that they defined [in Sel. Math., New Ser. 14, No. 3-4, 397-425 (2009; Zbl 1226.20002)], for category \(\mathcal O\) of a rational Cherednik algebra.NEWLINENEWLINE If \((\mathfrak h,W)\) is a complex reflection group, then associated to this pair is a family of non-commutative algebras \(H_c(\mathfrak h,W)\), depending on a deformation parameter \(c\). These are the rational Cherednik algebras. There is a natural category \(\mathcal O\), labeled \(\mathcal O(W)\), for each of these algebras. It is a highest weight category. If \(W'\) is a parabolic subgroup of \(W\), i.e. \(W'=W_b\) is the stabilizer of some point \(b\in\mathfrak h\), then there is a canonical \(W'\)-module decomposition \(\mathfrak h=\mathfrak h^{W'}\oplus\mathfrak h_{W'}\). By a theorem of Steinberg, \((\mathfrak h_{W'},W')\) is again a complex reflection group.NEWLINENEWLINE In the paper [loc. cit.] the authors construct, for each \(b\in\mathfrak h\) and \(\lambda\in\mathfrak h^*\) such that \(W'=W_b=W_\lambda\), a pair of exact restriction functors \(\mathrm{Res}_b,\mathrm{Res}_\lambda\colon\mathcal O(W)\to\mathcal O(W')\) and induction functors \(\mathrm{Ind}_b,\mathrm{Ind}_\lambda\colon\mathcal O(W')\to\mathcal O(W)\). These are not naive restriction and induction functors because it is not possible to embed \(H_c(\mathfrak h_{W'},W')\) in \(H_c(\mathfrak h,W)\).NEWLINENEWLINE Bezrukavnikov and Etingof conjectured that there are (non-canonical) isomorphisms of functors \(\mathrm{Res}_b\simeq\mathrm{Res}_\lambda\) and \(\mathrm{Ind}_b\simeq\mathrm{Ind}_\lambda\). In this article the author constructs these conjectured isomorphisms. The proof of the conjectures is by careful comparison of various different completions of the algebra \(H_c(\mathfrak h,W)\), and using these completions to construct a third functor \(\mathrm{Res}_{b,\lambda}\) that interpolates between \(\mathrm{Res}_b\) and \(\mathrm{Res}_\lambda\).NEWLINENEWLINE A corollary of the conjectures is that the induction functor \(\mathrm{Ind}_b\) is bi-adjoint to the restriction functor \(\mathrm{Res}_b\). A different proof of this corollary was given earlier by \textit{P. Shan} [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 1, 147-182 (2011; Zbl 1225.17019)], using Hecke algebras.