Support theory for the small quantum group and the Springer resolution

TL;DR

This paper explores the structure of the small quantum group using support theory, establishing a connection with the Springer resolution and providing explicit spectra in type A.

Contribution

It introduces a support-theoretic framework linking the small quantum group to the Springer resolution, extending results across Dynkin types under certain conjectures.

Findings

Homeomorphism between Springer resolution and Balmer spectrum in type A

Explicit computation of the spectrum in type A_1 as the projectivized nilpotent cone

Extension of results to arbitrary Dynkin types assuming conjectures hold

Abstract

We consider the small quantum group u_q(G), for an almost-simple algebraic group G over the complex numbers and a root of unity q of sufficiently large order. We show that the Balmer spectrum for the small quantum group in type A admits a continuous surjection P(\~N) \to Spec(stab u_q(G)) from the (projectivized) Springer resolution. This surjection is shown to be a homeomorphism over a dense open subset in the spectrum. In type A_1 we calculate the Balmer spectrum precisely, where it is shown to be the projectivized nilpotent cone. Our results extend to arbitrary Dynkin type provided certain conjectures hold for the small quantum Borel. At the conclusion of the paper we touch on relations with geometric representation theory and logarithmic TQFTs, as represented in works of Arkhipov-Bezrukavnikov-Ginzburg and Schweigert-Woike respectively.