The Parity of Lusztig's Restriction Functor and Green's Formula

TL;DR

This paper proves that Lusztig's restriction functor satisfies Green's formula for all semisimple complexes, establishing a categorification of Green's formula in the context of quantum groups and quiver representations.

Contribution

It extends Green's formula to all semisimple complexes with Weil structure, providing a categorification in the setting of Lusztig's restriction functor.

Findings

Green's formula holds for all semisimple complexes with Weil structure.

The categorification of Green's formula is established.

Lusztig's restriction functor satisfies the quantum group comultiplication.

Abstract

Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [9], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula.