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Register a new userQuiver varieties and Frenkel-Kac construction
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Authors
Kentaro Nagao
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An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type $A$ the former is obtained by Frenkel-Kac construction from the latter.
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We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.
We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GL_n(F_q) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gl_n(F_q), we calculate the E-polynomial of certain associated comet-shaped quiver varieties, the additive analogues of our character variety, and find that it is the pure part of our conjectured mixed Hodge polynomial. Finally, we observe that the pure part of our conjectured mixed Hodge polynomial also equals certain multiplicities in the tensor product of irreducible representations of GL_n(F_q). This implies a curious connection between the representation theory of GL_n(F_q) and Kac-Moody algebras associated with comet-shaped, typically wild, quivers.
We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.
In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.
We give an explicit description of the irreducible components of two-row Springer fibers in type A as closed subvarieties in certain Nakajima quiver varieties in terms of quiver representations. By taking invariants under a variety automorphism, we obtain an explicit algebraic description of the irreducible components of two-row Springer fibers of classical type. As a consequence, we discover relations on isotropic flags that describe the irreducible components.
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