The half-quantum flag variety and representations for small quantum groups

TL;DR

This paper explores the structure of the half-quantum flag variety and its relation to small quantum group representations, establishing categorical embeddings and conjecturing a deep link with Springer resolution sheaves.

Contribution

It demonstrates a full embedding of small quantum group representations into the global sections of the half-quantum flag variety and relates geometric fibers to quantum Borel representations.

Findings

Full faithful embedding of small quantum group representations into global sections.

Fibers over G/B recover quantum Borel representation categories.

Conjecture linking dg sheaves on Springer resolution to the half-quantum flag variety.

Abstract

Consider an almost-simple algebraic group G and a choice of complex root of unity q. We study the category of quasi-coherent sheaves $\mathscr{X}_q$ on the half-quantum flag variety, which itself forms a sheaf of tensor categories over the classical flag variety G/B. We prove that the category of small quantum group representations for G at q embeds fully faithfully into the global sections of $\mathscr{X}_q$, and that the fibers of $\mathscr{X}_q$ over G/B recover the tensor categories of representations for the small quantum Borels. These relationships hold both at an abelian and derived level. Subsequently, reduction arguments, from the small quantum group to its Borels, appear algebrogeometrically as "fiber checking" arguments over $\mathscr{X}_q$. We conjecture that $\mathscr{X}_q$ also contains the category of dg sheaves over the Springer resolution as a full monoidal subcategory, at the derived level, and hence provides a monoidal correspondence between the Springer resolution and the small quantum group. We relate this conjecture to a known equivalence between dg sheaves on the Springer resolution and the principal block in the derived category of quantum group representations [ABG04, BL07].