From complete to partial flags in geometric extension algebras

TL;DR

This paper explores the relationship between partial and complete flag cases in geometric extension algebras, revealing a recollement structure that connects modules in these settings, with applications to various algebraic structures.

Contribution

It establishes a recollement framework linking modules over partial and complete flag geometric extension algebras, advancing understanding of their structural relationships.

Findings

Modules over partial and complete flag algebras are related by a recollement.

Applications to parabolic affine nil Hecke algebras and Springer maps.

Provides examples involving parabolic quiver-graded Hecke algebras.

Abstract

A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g. a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundle over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated to parabolic Springer maps and an example of Reineke of a parabolic quiver-graded Hecke algebra.