Lusztig sheaves and integrable highest weight modules in the symmetrizable case · arXivDesk
2411.09188Nov 14, 2024In this version, we include a new section that provides a geometric realization of the tensor product of integrable highest weight modules
Lusztig sheaves and integrable highest weight modules in the symmetrizable case
This paper continues the work of fang2023lusztigsheavesintegrablehighest and fang2023lusztigsheavestensorproducts. For a symmetrizable generalized Cartan matrix C and the corresponding quantum group U, we consider an associated quiver Q equipped with an admissible automorphism a. We construct a category Q/N
Nearby in the stack
obtained from localizations of Lusztig sheaves for the corresponding framed and
2
-framed quivers with automorphism. The Grothendieck groups of these categories realize the integrable highest weight module
L(λ)
and the tensor product
L(λ1)⊗L(λ2)
of integrable highest weight
U
-modules. After quotienting by traceless objects, Lusztig sheaves yield the signed canonical bases of
L(λ)
and
L(λ1)⊗L(λ2)
. As applications, we recover symmetrizable crystal structures on Nakajima quiver varieties, Nakajima tensor product varieties, and Lusztig nilpotent varieties of preprojective algebras.