Remarks on quantum unipotent subgroup and dual canonical basis (Various Issues relating to Representation Theory and Non-commutative Harmonic Analysis)

Abstract

Quantum unipotent subgroup is the quantum coordinate ring of the unipotent subgroup associated with a finite subset which is defined by a Weyl group element of a symmetrizable Kac-Moody Lie algebra. I explain about the quantum coordinate ring of the pro-unipotent subgroup associated with a cofinite subset associated with a Weyl group element and its compatiblity with the dual canonical basis and the multiplicity-free property between the dual canonical basis element in the quantum unipotent subgroups and the one in the opposite. This is based on [Kim15].