On the number of points of nilpotent quiver varieties over finite fields

Abstract

We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Finally we compute the number of points over a finite field of the various stratas of the Lusztig nilpotent variety involved in the geometric realization of the crystal graph.

45 pages, Latex

References in corpus (4)

• Lectures on Nakajima's Quiver Varieties
• Motivic Donaldson-Thomas invariants and McKay correspondence
• Counting Higgs bundles
• Fermionic forms and quiver varieties

Cited by in corpus (12)

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• Crystal Melting, BPS Quivers and Plethystics
• McKay correspondence, cohomological Hall algebras and categorification
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• Commuting matrices and volumes of linear stacks
• Motivic classes of Quot-schemes on surfaces
• Borcherds-Bozec algebras, root multiplicities and the Schofield construction
• Quantum Borcherds-Bozec algebras and their integrable representations
• Abstract crystals for quantum Borcherds-Bozec algebras
• Cohomological Hall algebras of quivers and Yangians
• Rational singularities for moment maps of totally negative quivers
• q-Whittaker functions, finite fields, and Jordan forms