Geometric Representation Theory of Affine Quantum Groups and Crystal Bases via Quiver Varieties

This monograph establishes a rigorous geometric realization of the crystal bases for integrable highest weight modules of affine quantum groups Uq(g"0362g) via the geometry of Nakajima quiver varieties. We analyze the cotangent bundle of the moduli space of framed quiver representations, specifically focusing on the Lagrangian subvarieties L(v, w) defined by the zero locus of the moment map. We demonstrate that the set of irreducible components of these Lagrangian subvarieties admits a crystal structure isomorphic to the Kashiwara crystal B() of the negative part of the quantum group Uq-(g"0362g). Furthermore, we elucidate the correspondence between the singular support of Lusztig's canonical basis elements in the category of perverse sheaves and these irreducible components. The study extends to the convolution algebra formalism, providing a geometric categorification of the quantum enveloping algebra and establishing a bridge between the algebraic combinatorics of crystals and the symplectic geometry of hyper-Kähler manifolds.