Injective and tilting resolutions and a Kazhdan-Lusztig theory for the general linear and symplectic group

TL;DR

This paper constructs explicit resolutions of induced modules for symplectic and general linear groups over fields of positive characteristic, linking module multiplicities to Kazhdan-Lusztig polynomials and establishing a Kazhdan-Lusztig theory for certain categories.

Contribution

It provides explicit tilting and injective resolutions of induced modules and demonstrates a Kazhdan-Lusztig theory in truncated categories for these groups.

Findings

Resolutions are expressed via Kazhdan-Lusztig polynomial coefficients.

Established a Kazhdan-Lusztig theory for truncated categories.

Connected module multiplicities with Kazhdan-Lusztig polynomials.

Abstract

Let k be an algebraically closed field of characteristic p>0 and let G be a symplectic or general linear group over k. We consider induced modules for G under the assumption that p is bigger than the greatest hook length in the partitions involved. We give explicit constructions of left resolutions of induced modules by tilting modules. Furthermore, we give injective resolutions for induced modules in certain truncated categories. We show that the multiplicities of the indecomposable tilting and injective modules in these resolutions are the coefficients of certain Kazhdan-Lusztig polynomials. We also show that our truncated categories have a Kazhdan-Lusztig theory in the sense of Cline, Parshall and Scott. This builds further on work of Cox-De Visscher and Brundan-Stroppel.