The parity of Lusztig's restriction functor and Green's formula for a quiver with automorphism

TL;DR

This paper generalizes Green's formula for quivers with automorphisms by extending Lusztig's restriction functor and induction formula to mixed semisimple perverse sheaves, leading to broader applicability in hereditary algebras.

Contribution

It introduces a generalized formula for Lusztig's restriction functor for quivers with automorphisms, extending Green's formula to a wider class of algebras.

Findings

Generalized Lusztig's restriction functor for quivers with automorphisms

Derived Green's formula for finite-dimensional hereditary algebras

Extended applicability to mixed semisimple perverse sheaves

Abstract

In [8], Fang-Lan-Xiao proved a formula about Lusztig's induction and restriction functors which can induce Green's formula for the path algebra of a quiver over a finite field via the trace map. In this paper, we generalize their formula to that for the mixed semisimple perverse sheaves for a quiver with an automorphism. By applying the trace map, we obtain Green's formula for any finite-dimensional hereditary algebra over a finite field.