  We realize the crystal associated to the quantized enveloping algebras with a
symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the
cotangent bundle of the quiver variety. As a by-product, we give a
counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of
the characteristic variety of the intersection cohomology sheaves associated
with the Schubert cells of type A and also to the similar problem asked by
Lusztig on the characteristic variety of the perverse sheaves corresponding to
canonical bases.

Quantum Algebra;Quantum Algebra