@article{14930, abstract = {In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t2). Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q, R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra Fq[t]/(tr). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.}, author = {Hausel, Tamás and Letellier, Emmanuel and Rodriguez-Villegas, Fernando}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {2}, publisher = {Springer Nature}, title = {{Locally free representations of quivers over commutative Frobenius algebras}}, doi = {10.1007/s00029-023-00914-2}, volume = {30}, year = {2024}, } @article{14244, abstract = {In this paper, we determine the motivic class — in particular, the weight polynomial and conjecturally the Poincaré polynomial — of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank bundle on P1. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss–Leclerc–Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the four-dimensional cases are expected to be of type ALF.}, author = {Hausel, Tamás and Wong, Michael Lennox and Wyss, Dimitri}, issn = {1460-244X}, journal = {Proceedings of the London Mathematical Society}, number = {4}, pages = {958--1027}, publisher = {Wiley}, title = {{Arithmetic and metric aspects of open de Rham spaces}}, doi = {10.1112/plms.12555}, volume = {127}, year = {2023}, } @article{10704, abstract = {We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of C∗-actions on semiprojective varieties, C∗ characters of indices of C∗-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier–Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs bundles.}, author = {Hausel, Tamás and Hitchin, Nigel}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {893--989}, publisher = {Springer Nature}, title = {{Very stable Higgs bundles, equivariant multiplicity and mirror symmetry}}, doi = {10.1007/s00222-021-01093-7}, volume = {228}, year = {2022}, } @article{439, abstract = {We count points over a finite field on wild character varieties,of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma–Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P = W conjecture for a suitable wild Hitchin system.}, author = {Hausel, Tamas and Mereb, Martin and Wong, Michael}, issn = {1435-9855}, journal = {Journal of the European Mathematical Society}, number = {10}, pages = {2995--3052}, publisher = {European Mathematical Society}, title = {{Arithmetic and representation theory of wild character varieties}}, doi = {10.4171/JEMS/896}, volume = {21}, year = {2019}, } @inbook{6525, abstract = {This chapter finds an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL2 Higgs moduli space on a Riemann surface. On one side of the agreement, components of the Lagrangian brane of U(1,1) Higgs bundles, whose mirror was proposed by Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL2 Higgs moduli space, are present. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin’s proposal.}, author = {Hausel, Tamás and Mellit, Anton and Pei, Du}, booktitle = {Geometry and Physics: Volume I}, isbn = {9780198802013}, pages = {189--218}, publisher = {Oxford University Press}, title = {{Mirror symmetry with branes by equivariant verlinde formulas}}, doi = {10.1093/oso/9780198802013.003.0009}, year = {2018}, } @misc{1473, abstract = {In this paper we survey geometric and arithmetic techniques to study the cohomology of semiprojective hyperkähler manifolds including toric hyperkähler varieties, Nakajima quiver varieties and moduli spaces of Higgs bundles on Riemann surfaces. The resulting formulae for their Poincaré polynomials are combinatorial and representation theoretical in nature. In particular we will look at their Betti numbers and will establish some results and state some expectations on their asymptotic shape.}, author = {Tamas Hausel and Rodríguez Villegas, Fernando}, booktitle = {Asterisque}, number = {370}, pages = {113 -- 156}, publisher = {Societe Mathematique de France}, title = {{Cohomology of large semiprojective hyperkähler varieties}}, volume = {2015}, year = {2015}, } @article{1442, abstract = {We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomas-invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich-Soibelman. Thisis achieved by computing, via an arithmetic Fourier transform, the dimensions of the isotypical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincare polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties.}, author = {Tamas Hausel and Letellier, Emmanuel and Rodríguez Villegas, Fernando}, journal = {Annals of Mathematics}, number = {3}, pages = {1147 -- 1168}, publisher = {Princeton University Press}, title = {{Positivity for Kac polynomials and DT-invariants of quivers}}, doi = {10.4007/annals.2013.177.3.8}, volume = {177}, year = {2013}, } @inbook{1443, abstract = {Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory.}, author = {Tamas Hausel}, booktitle = {Handbook of Moduli: Volume II}, pages = {29 -- 70}, publisher = {International Press}, title = {{Global topology of the Hitchin system}}, volume = {25}, year = {2013}, } @article{1469, abstract = {We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on Cx × Cx, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.}, author = {Tamas Hausel and Letellier, Emmanuel and Rodríguez Villegas, Fernando}, journal = {Advances in Mathematics}, pages = {85 -- 128}, publisher = {Academic Press}, title = {{Arithmetic harmonic analysis on character and quiver varieties II}}, doi = {10.1016/j.aim.2012.10.009}, volume = {234}, year = {2013}, } @article{1470, abstract = {We show that a natural isomorphism between the rational cohomology groups of the two zero-dimensional Hilbert schemes of n-points of two surfaces, the affine plane minus the axes and the cotangent bundle of an elliptic curve, exchanges the weight filtration on the first set of cohomology groups with the perverse Leray filtration associated with a natural fibration on the second set of cohomology groups. We discuss some associated hard Lefschetz phenomena.}, author = {De Cataldo, Mark A and Tamas Hausel and Migliorini, Luca}, journal = {Journal of Singularities}, pages = {23 -- 38}, publisher = {Worldwide Center of Mathematics}, title = {{Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces}}, doi = {10.5427/jsing.2013.7c}, volume = {7}, year = {2013}, } @article{1471, abstract = {Given a possibly reducible and non-reduced spectral cover π: X → C over a smooth projective complex curve C we determine the group of connected components of the Prym variety Prym(X/C). As an immediate application we show that the finite group of n-torsion points of the Jacobian of C acts trivially on the cohomology of the twisted SL n-Higgs moduli space up to the degree which is predicted by topological mirror symmetry. In particular this yields a new proof of a result of Harder-Narasimhan, showing that this finite group acts trivially on the cohomology of the twisted SL n stable bundle moduli space.}, author = {Tamas Hausel and Pauly, Christian}, journal = {Geometry and Topology}, number = {3}, pages = {1609 -- 1638}, publisher = {University of Warwick}, title = {{Prym varieties of spectral covers}}, doi = {10.2140/gt.2012.16.1609}, volume = {16}, year = {2012}, } @article{1472, abstract = {For G = GL 2, PGL 2, SL 2 we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted G-Higgs bundles on a compact Riemann surface C agrees with the weight filtration on the rational cohomology of the twisted G character variety of C when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.}, author = {De Cataldo, Mark A and Tamas Hausel and Migliorini, Luca}, journal = {Annals of Mathematics}, number = {3}, pages = {1329 -- 1407}, publisher = {Princeton University Press}, title = {{Topology of hitchin systems and Hodge theory of character varieties: The case A 1}}, doi = {10.4007/annals.2012.175.3.7}, volume = {175}, year = {2012}, } @article{1467, abstract = {We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GLn(C) with fixed generic semisimple conjugacy classes at k punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of GLn(Fq). We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincaré polynomial of certain associated quiver varieties which we obtain using the character table of gln(Fq). In the last main result we prove that the Poincaré polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of GLn(Fq). As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, and typically wild, quivers and the representation theory of GLn(Fq).}, author = {Tamas Hausel and Letellier, Emmanuel and Rodríguez Villegas, Fernando}, journal = {Duke Mathematical Journal}, number = {2}, pages = {323 -- 400}, publisher = {Duke University Press}, title = {{Arithmetic harmonic analysis on character and quiver varieties}}, doi = {10.1215/00127094-1444258}, volume = {160}, year = {2011}, } @article{1465, abstract = {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 certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.}, author = {Tamas Hausel}, journal = {Inventiones Mathematicae}, number = {1}, pages = {21 -- 37}, publisher = {Springer}, title = {{Kac's conjecture from Nakajima quiver varieties}}, doi = {10.1007/s00222-010-0241-3}, volume = {181}, year = {2010}, } @article{1466, abstract = {In Hausel et al. (2008) [10] we presented a conjecture generalizing the Cauchy formula for Macdonald polynomial. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a punctured Riemann surface of genus g. We proved several results which support this conjecture. Here we announce new results which are consequences of those in Hausel et al. (2008) [10].}, author = {Tamas Hausel and Letellier, Emmanuel and Rodríguez Villegas, Fernando}, journal = {Comptes Rendus Mathematique}, number = {3-4}, pages = {131 -- 135}, publisher = {Elsevier}, title = {{Topology of character varieties and representations of quivers}}, doi = {10.1016/j.crma.2010.01.025}, volume = {348}, year = {2010}, } @inbook{1468, abstract = {This chapter surveys the motivations, related results, and progress made towards the following problem, raised by Hitchin in 1995: What is the space of L2 harmonic forms on the moduli space of Higgs bundles on a Riemann surface?}, author = {Tamas Hausel}, booktitle = {The Many Facets of Geometry: A Tribute to Nigel Hitchin}, publisher = {Oxford University Press}, title = {{S-Duality in HyperkäHler Hodge Theory}}, doi = {10.1093/acprof:oso/9780199534920.003.0016}, year = {2010}, } @article{1460, abstract = {We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n, q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of Mn: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.}, author = {Tamas Hausel and Rodríguez Villegas, Fernando}, journal = {Inventiones Mathematicae}, number = {3}, pages = {555 -- 624}, publisher = {Springer}, title = {{Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz}}, doi = {10.1007/s00222-008-0142-x}, volume = {174}, year = {2008}, } @article{1461, abstract = {This note proves combinatorially that the intersection pairing on the middle-dimensional compactly supported cohomology of a toric hyperkähler variety is always definite, providing a large number of non-trivial L 2 harmonic forms for toric hyperkähler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L 2 harmonic forms on complete hyperkähler manifolds of linear growth.}, author = {Tamas Hausel and Swartz, Edward}, journal = {Proceedings of the American Mathematical Society}, number = {8}, pages = {2403 -- 2409}, publisher = {American Mathematical Society}, title = {{Intersection forms of toric hyperkähler varieties}}, doi = {10.1090/S0002-9939-06-08248-7}, volume = {134}, year = {2006}, } @article{1462, abstract = {A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This technique in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence, simple unified proofs are obtained for formulas of Poincaré polynomials of toric hyperkähler varieties (recovering results of Bielawski-Dancer and Hausel-Sturmfels), Poincaré polynomials of Hubert schemes of points and twisted Atiyah-Drinfeld-Hitchin-Manin (ADHM) spaces of instantons on ℂ2 (recovering results of Nakajima-Yoshioka), and Poincaré polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable representations of a quiver is announced.}, author = {Tamas Hausel}, journal = {PNAS}, number = {16}, pages = {6120 -- 6124}, publisher = {National Academy of Sciences}, title = {{Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform}}, doi = {10.1073/pnas.0601337103}, volume = {103}, year = {2006}, } @inbook{1444, abstract = {The paper surveys the mirror symmetry conjectures of Hausel-Thaddeus and Hausel-Rodriguez-Villegas concerning the equality of certain Hodge numbers of SL(n, ℂ) vs. PGL(n, ℂ) flat connections and character varieties for curves, respectively. Several new results and conjectures and their relations to works of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan are explained. These use the representation theory of finite groups of Lie-type via the arithmetic of character varieties and lead to an unexpected conjecture for a Hard Lefschetz theorem for their cohomology.}, author = {Tamas Hausel}, booktitle = {Geometric Methods in Algebra and Number Theory}, pages = {193 -- 217}, publisher = {Springer}, title = {{Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve}}, doi = {10.1007/0-8176-4417-2_9}, volume = {235}, year = {2005}, }