## 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)].

### MSC:

 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

### Citations:

Zbl 1220.17009; Zbl 1264.14067; Zbl 1483.14086
Full Text:

### References:

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