Differential operators on quantized flag manifolds at roots of unity. II. (English) Zbl 1311.14047

Let \(G\) be a connected, simply connected simple algebraic group over \(\mathbb{C}\) and \(\ell\) an odd integer greater than the Coxeter number of \(G\). Fixing a primite \(\ell\)th root of unity \(\zeta\), the De Concini-Kac quantized enveloping algebra of \(\mathrm{Lie} \;G\) is \(U_{\zeta}\). If \(\mathcal{B}_{\zeta}\) is the quantized flag manifold (a non-commutative space), it is possible to define the sheaf \(\mathcal{D}_{\mathcal{B}_{\zeta},t}\) of twisted differential operators on \(\mathcal{B}_{\zeta}\); the twist \(t\) belonging to \(H\), a maximal torus of \(G\). The Frobenius morphism \(\mathrm{Fr} : \mathcal{B}_{\zeta} \rightarrow \mathcal{B}\) is, in an appropriate sense, a finite morphism, where \(\mathcal{B}\) is the usual flag manifold. Then \(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t}\) is a genuine sheaf of non-commutative algebras on \(\mathcal{B}\). The category of coherent \(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t}\)-modules is denoted \(\mathrm{Mod}_{\mathrm{coh}}(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t})\). The Harish-Chandra center \(Z_{\zeta}\) of \(U_{\zeta}\) can be identified with \(\mathbb{C}[H/W]\). If \(t \in H\), we denote by \(U_{\zeta,t}\) the corresponding central quotient of \(U_{\zeta}\) and by \(\mathrm{Mod}_f (U_{\zeta,t})\) the category of finitely generated \(U_{\zeta,t}\)-modules.
Assume that \(t\) is regular. In this article the author states a conjectural derived equivalence between the bounded derived category of \(\mathrm{Mod}_f (U_{\zeta,t})\) and the bounded derived category of \(\mathrm{Mod}_{\mathrm{coh}}(\mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t})\). The derived equivalence should be given by the global sections functor \(R \Gamma(\mathcal{B}, - )\). The conjecture is motivated by the analogy between \(U_{\zeta}\) and the enveloping algebra of \(\mathrm{Lie} \;G\) in positive characteristic and the corresponding derived equivalence established in [R. Bezrukavnikov et al., Ann. Math. (2) 167, No. 3, 945–991 (2008; Zbl 1220.17009)]. Using the arguments of [loc. cit.], the author shows that the conjecture is true provided \(R \Gamma (\mathcal{B}, \mathrm{Fr}_* \mathcal{D}_{\mathcal{B}_{\zeta},t}) \simeq U_{\zeta,t}\). The author states a more general form of the conjecture, where the central quotients defined by \(t\) are replaced by generalized central characters.
In the final part of the article, the author states a different conjecture involving a certain induced \(U_{\zeta,t}\)-module. Remarkably, it is shown that this conjecture is equivalent to the above stated conjecture.
For Part I and III, see [the author, Adv. Math. 230, No. 4–6, 2235–2294 (2012; Zbl 1264.14067); Adv. Math. 392, Article ID 107990, 51 p. (2021; Zbl 1483.14086)].


14M15 Grassmannians, Schubert varieties, flag manifolds
20G05 Representation theory for linear algebraic groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14A22 Noncommutative algebraic geometry
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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