Quiver varieties and Demazure modules

Abstract

Using subvarieties, which we call Demazure quiver varieties, of the quiver varieties of Nakajima, we give a geometric realization of Demazure modules of Kac-Moody algebras with symmetric Cartan data. We give a natural geometric characterization of the extremal weights of a representation and show that Lusztig's semicanonical basis is compatible with the filtration of a representation by Demazure modules. For the case of affine sl_2, we give a characterization of the Demazure quiver variety in terms of a nilpotency condition on quiver representations and an explicit combinatorial description of the Demazure crystal in terms of Young pyramids.

14 pages, 2 figures; v2: Minor corrections and reference added; v3: Proofs of Proposition 6.1 and Theorem 8.1 corrected. This version incorporates an Erratum to the published version

References in corpus (2)

• Geometric and combinatorial realizations of crystal graphs
• Homological realization of Nakajima varieties and Weyl group actions

Cited by in corpus (6)

• A comparison of Newton-Okounkov polytopes of Schubert varieties
• Quiver grassmannians, quiver varieties and the preprojective algebra
• The canonical basis and the quantum Frobenius morphism
• Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal
• Folding procedure for Newton-Okounkov polytopes of Schubert varieties
• Parametrizations of canonical bases and irreducible components of nilpotent varieties