The geometry of tilting composition series via Richardson varieties

TL;DR

This paper provides a geometric interpretation of tilting sheaves' multiplicities on flag varieties using Richardson varieties and Kazhdan--Lusztig polynomials, linking algebraic and geometric perspectives.

Contribution

It introduces a new geometric framework for understanding tilting sheaves' multiplicities via Richardson varieties and explicit formulas involving Kazhdan--Lusztig polynomials.

Findings

Multiplicities correspond to hypercohomology of sheaves on Richardson varieties.

Explicit formulas for multiplicities in terms of $$-Kazhdan--Lusztig polynomials.

Sheaves are described as tensor products of parity sheaves on Schubert varieties.

Abstract

We prove the (graded) Jordan--H\"{o}lder multiplicities of (mixed) tilting sheaves on flag varieties admit a geometric interpretation as the hypercohomology of certain sheaves on Richardson varieties in the Langlands dual flag variety. These sheaves are a motivic variant of geometric extensions, and may be described as a tensor product of parity sheaves on the Schubert and opposite Schubert varieties. We also provide an explicit formula for these multiplicities in terms of $\ell$-Kazhdan--Lusztig polynomials.