Weights of mixed tilting sheaves and geometric Ringel duality

TL;DR

This paper develops methods to compute weights of mixed tilting sheaves, introduces a non-cancellation property, and proves a geometric Ringel duality, linking tilting sheaves to Kazhdan-Lusztig polynomials on flag varieties.

Contribution

It introduces a new non-cancellation property for mixed tilting sheaves and establishes a geometric Ringel duality, connecting tilting and projective objects via Radon transforms.

Findings

Weights of mixed tilting sheaves relate to Kazhdan-Lusztig polynomials.

The non-cancellation property ensures uniqueness and computability of weights.

A geometric analogue of Ringel duality is proved for certain varieties.

Abstract

We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called "non-cancellation property" which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their weights effectively. When we have a certain Radon transform, we prove a geometric analogue of Ringel duality which sends tilting objects to projective objects. We apply these methods to (partial) flag varieties and affine (partial) flag varieties and show that the weight polynomials of mixed tilting sheaves on flag and affine flag varieties are essentially given by Kazhdan-Lusztig polynomials. This verifies a mixed geometric analogue of a conjecture by W.Soergel in \cite{Sg1}.