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Geometric Construction of Crystal Bases

Masaki Kashiwara·Yoshihisa Saito
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Abstract

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.

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Masaki Kashiwara, Yoshihisa Saito. 1996-06-08. Geometric Construction of Crystal Bases. https://arxiv.org/abs/q-alg/9606009

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