  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.
