A coherent categorification of the based ring of the lowest two-sided cell

TL;DR

This paper develops a partial categorification of a specific algebraic structure related to affine Weyl groups, connecting it with coherent sheaves and actions on Schwartz space invariants, advancing understanding in geometric representation theory.

Contribution

It introduces a new partial coherent categorification of the based ring of the lowest two-sided cell in affine Weyl groups, with applications to Iwahori invariants and explicit constructions in low rank cases.

Findings

Constructs complexes lifting basis elements of the based ring.

Establishes a monoidal functor from coherent sheaves on the Steinberg variety.

Demonstrates actions on coherent categorifications of Schwartz space invariants.

Abstract

We give a partial coherent categorification of $J_0$, the based ring of the lowest two sided cell of an affine Weyl group, equipped with a monoidal functor from the category of coherent sheaves on the derived Steinberg variety. We show that our categorification acts on natural coherent categorifications of the Iwahori invariants of the Schwartz space of the basic affine space. In low rank cases, we construct complexes that lift the basis elements $t_w$ of $J_0$ and their structure constants.