[Paper Review] Cherednik algebras and Hilbert schemes in characteristic p

Roman Bezrukavnikov, Michael Finkelberg+1 more|arXiv (Cornell University)|Dec 29, 2003

Algebraic structures and combinatorial models24 references4 citations

TL;DR

This paper establishes a localization theorem for rational Cherednik algebras of type A_{n-1} over finite fields of characteristic p, constructing an Azumaya algebra H_c on the Hilbert scheme of n points in the plane. It proves an equivalence between the derived categories of H_c-modules and coherent H_c-modules on the Hilbert scheme, and shows the algebra splits on formal neighborhoods of fibers of the Hilbert-Chow morphism, linking to results of Bridgeland-King-Reid and Haiman.

ABSTRACT

To David Kazhdan with admiration We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) over Fp, an algebraic closure of the finite field. In the most interesting special case where c∈Fp, we construct an Azumaya algebra Hc on Hilb n A 2, the Hilbert scheme of n points in the plane, such that Γ(Hilb n A 2, Hc) = Hc. Our localization theorem provides an equivalence between the bounded derived categories of Hc-modules and sheaves of coherent Hc-modules on Hilb n A 2, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland-King-Reid and Haiman.

Motivation & Objective

• To extend localization theorems for rational Cherednik algebras to positive characteristic, specifically over F_p.
• To construct an Azumaya algebra H_c on Hilb^n A^2 when c ∈ F_p.
• To establish an equivalence between the bounded derived categories of H_c-modules and coherent H_c-modules on the Hilbert scheme.
• To analyze the splitting behavior of the Azumaya algebra on formal neighborhoods of fibers of the Hilbert-Chow morphism.
• To connect the results to existing work by Bridgeland-King-Reid and Haiman in the context of Hilbert schemes and derived equivalences.

Proposed method

• Use of rational Cherednik algebras H_c = H_{1,c}(A_{n-1}) over an algebraic closure of F_p.
• Construction of an Azumaya algebra H_c on the Hilbert scheme Hilb^n A^2 via localization in positive characteristic.
• Application of geometric representation theory techniques to relate H_c-modules to coherent sheaves on Hilb^n A^2.
• Employment of formal neighborhood analysis to study splitting of the Azumaya algebra along fibers of the Hilbert-Chow morphism.
• Leveraging known results on Hilbert schemes and derived categories to establish categorical equivalences.

Experimental results

Research questions

• RQ1 How can localization theorems for Cherednik algebras be extended to positive characteristic?
• RQ2 What is the structure of the rational Cherednik algebra H_c on the Hilbert scheme Hilb^n A^2 when c is in F_p?
• RQ3 Does the Azumaya algebra H_c on Hilb^n A^2 admit a splitting on the formal neighborhood of each fiber of the Hilbert-Chow morphism?
• RQ4 How does the derived category of H_c-modules relate to the derived category of coherent H_c-modules on Hilb^n A^2?
• RQ5 In what way do these results connect to the Bridgeland-King-Reid and Haiman theorems on derived equivalences for Hilbert schemes?

Key findings

• An Azumaya algebra H_c is constructed on the Hilbert scheme Hilb^n A^2 when c ∈ F_p, with global sections isomorphic to H_c.
• A derived equivalence is established between the bounded derived category of H_c-modules and the bounded derived category of coherent H_c-modules on Hilb^n A^2.
• The Azumaya algebra H_c splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism.
• The localization theorem provides a characteristic p analogue of the classical Beilinson-Bernstein localization in positive characteristic.
• The results establish a new link between Cherednik algebras, Hilbert schemes, and derived categories in positive characteristic, extending the framework of Bridgeland-King-Reid and Haiman.