---
_id: '14930'
abstract:
- lang: eng
text: 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.
acknowledgement: Special thanks go to Christof Geiss, Bernard Leclerc and Jan Schröer
for explaining their work but also for sharing some unpublished results with us.
We also thank the referee for many useful suggestions. We would like to thank Tommaso
Scognamiglio for pointing out a mistake in the proof of Proposition 5.17 in an earlier
version of the paper. We would like also to thank Alexander Beilinson, Bill Crawley-Boevey,
Joel Kamnitzer, and Peng Shan for useful discussions.
article_number: '20'
article_processing_charge: No
article_type: original
author:
- first_name: Tamás
full_name: Hausel, Tamás
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Emmanuel
full_name: Letellier, Emmanuel
last_name: Letellier
- first_name: Fernando
full_name: Rodriguez-Villegas, Fernando
last_name: Rodriguez-Villegas
citation:
ama: Hausel T, Letellier E, Rodriguez-Villegas F. Locally free representations of
quivers over commutative Frobenius algebras. Selecta Mathematica. 2024;30(2).
doi:10.1007/s00029-023-00914-2
apa: Hausel, T., Letellier, E., & Rodriguez-Villegas, F. (2024). Locally free
representations of quivers over commutative Frobenius algebras. Selecta Mathematica.
Springer Nature. https://doi.org/10.1007/s00029-023-00914-2
chicago: Hausel, Tamás, Emmanuel Letellier, and Fernando Rodriguez-Villegas. “Locally
Free Representations of Quivers over Commutative Frobenius Algebras.” Selecta
Mathematica. Springer Nature, 2024. https://doi.org/10.1007/s00029-023-00914-2.
ieee: T. Hausel, E. Letellier, and F. Rodriguez-Villegas, “Locally free representations
of quivers over commutative Frobenius algebras,” Selecta Mathematica, vol.
30, no. 2. Springer Nature, 2024.
ista: Hausel T, Letellier E, Rodriguez-Villegas F. 2024. Locally free representations
of quivers over commutative Frobenius algebras. Selecta Mathematica. 30(2), 20.
mla: Hausel, Tamás, et al. “Locally Free Representations of Quivers over Commutative
Frobenius Algebras.” Selecta Mathematica, vol. 30, no. 2, 20, Springer
Nature, 2024, doi:10.1007/s00029-023-00914-2.
short: T. Hausel, E. Letellier, F. Rodriguez-Villegas, Selecta Mathematica 30 (2024).
date_created: 2024-02-04T23:00:53Z
date_published: 2024-01-27T00:00:00Z
date_updated: 2024-02-05T12:58:21Z
day: '27'
department:
- _id: TaHa
doi: 10.1007/s00029-023-00914-2
intvolume: ' 30'
issue: '2'
language:
- iso: eng
month: '01'
oa_version: None
publication: Selecta Mathematica
publication_identifier:
eissn:
- 1420-9020
issn:
- 1022-1824
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Locally free representations of quivers over commutative Frobenius algebras
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 30
year: '2024'
...
---
_id: '14244'
abstract:
- lang: eng
text: "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 \r\n 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."
acknowledgement: We would like to thank Gergely Bérczy, Roger Bielawski, Philip Boalch,
Sergey Cherkis, Andrew Dancer, Brent Doran, Eloïse Hamilton, Frances Kirwan, Bernard
Leclerc, Emmanuel Letellier, Alessia Mandini, Maxence Mayrand, András Némethi, Szilárd
Szabó, and Daisuke Yamakawa for discussions related to the paper. We especially
thank the referee for an extensive list of very careful comments. At various stages
of this project, the authors were supported by the Advanced Grant “Arithmetic and
physics of Higgs moduli spaces” no. 320593 of the European Research Council, by
grant no. 153627 and NCCR SwissMAP, both funded by the Swiss National Science Foundation
as well as by EPF Lausanne and IST Austria. In the final stages of this project,
MLW was supported by SFB/TR 45 “Periods, moduli and arithmetic of algebraic varieties,”
subproject M08-10 “Moduli of vector bundles on higher-dimensional varieties.” DW
was also supported by the Fondation Sciences Mathématiques de Paris, as well as
public grants overseen by the Agence national de la recherche (ANR) of France as
part of the Investissements d'avenir program, under reference numbers ANR-10-LABX-0098
and ANR-15-CE40-0008 (Défigéo).
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Tamás
full_name: Hausel, Tamás
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Michael Lennox
full_name: Wong, Michael Lennox
last_name: Wong
- first_name: Dimitri
full_name: Wyss, Dimitri
last_name: Wyss
citation:
ama: Hausel T, Wong ML, Wyss D. Arithmetic and metric aspects of open de Rham spaces.
Proceedings of the London Mathematical Society. 2023;127(4):958-1027. doi:10.1112/plms.12555
apa: Hausel, T., Wong, M. L., & Wyss, D. (2023). Arithmetic and metric aspects
of open de Rham spaces. Proceedings of the London Mathematical Society.
Wiley. https://doi.org/10.1112/plms.12555
chicago: Hausel, Tamás, Michael Lennox Wong, and Dimitri Wyss. “Arithmetic and Metric
Aspects of Open de Rham Spaces.” Proceedings of the London Mathematical Society.
Wiley, 2023. https://doi.org/10.1112/plms.12555.
ieee: T. Hausel, M. L. Wong, and D. Wyss, “Arithmetic and metric aspects of open
de Rham spaces,” Proceedings of the London Mathematical Society, vol. 127,
no. 4. Wiley, pp. 958–1027, 2023.
ista: Hausel T, Wong ML, Wyss D. 2023. Arithmetic and metric aspects of open de
Rham spaces. Proceedings of the London Mathematical Society. 127(4), 958–1027.
mla: Hausel, Tamás, et al. “Arithmetic and Metric Aspects of Open de Rham Spaces.”
Proceedings of the London Mathematical Society, vol. 127, no. 4, Wiley,
2023, pp. 958–1027, doi:10.1112/plms.12555.
short: T. Hausel, M.L. Wong, D. Wyss, Proceedings of the London Mathematical Society
127 (2023) 958–1027.
date_created: 2023-08-27T22:01:18Z
date_published: 2023-10-01T00:00:00Z
date_updated: 2024-01-30T12:56:10Z
day: '01'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1112/plms.12555
ec_funded: 1
external_id:
arxiv:
- '1807.04057'
isi:
- '001049312700001'
file:
- access_level: open_access
checksum: 2af4d2d6a8ae42f7d3fba0188e79ae82
content_type: application/pdf
creator: dernst
date_created: 2024-01-30T12:56:00Z
date_updated: 2024-01-30T12:56:00Z
file_id: '14910'
file_name: 2023_ProcLondonMathSoc_Hausel.pdf
file_size: 651335
relation: main_file
success: 1
file_date_updated: 2024-01-30T12:56:00Z
has_accepted_license: '1'
intvolume: ' 127'
isi: 1
issue: '4'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: 958-1027
project:
- _id: 25E549F4-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '320593'
name: Arithmetic and physics of Higgs moduli spaces
- _id: 25E6C798-B435-11E9-9278-68D0E5697425
grant_number: '153627'
name: Arithmetic quantization of character and quiver varities
publication: Proceedings of the London Mathematical Society
publication_identifier:
eissn:
- 1460-244X
issn:
- 0024-6115
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Arithmetic and metric aspects of open de Rham spaces
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 127
year: '2023'
...
---
_id: '10704'
abstract:
- lang: eng
text: 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.
acknowledgement: We would like to thank Brian Collier, Davide Gaiotto, Peter Gothen,
Jochen Heinloth, Daniel Huybrechts, Quoc Ho, Joel Kamnitzer, Gérard Laumon, Luca
Migliorini, Alexander Minets, Brent Pym, Peng Shan, Carlos Simpson, András Szenes,
Fernando R. Villegas, Richard Wentworth, Edward Witten and Kōta Yoshioka for interesting
comments and discussions. Most of all we are grateful for a long list of very helpful
comments by the referee. We would also like to thank the organizers of the Summer
School on Higgs bundles in Hamburg in September 2018, where the authors and Richard
Wentworth were giving lectures and where the work in this paper started by considering
the mirror of the Lagrangian upward flows W+E investigated in [17]. The second author
wishes to thank EPSRC and ICMAT for support. Open access funding provided by Institute
of Science and Technology (IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Tamás
full_name: Hausel, Tamás
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Nigel
full_name: Hitchin, Nigel
last_name: Hitchin
citation:
ama: Hausel T, Hitchin N. Very stable Higgs bundles, equivariant multiplicity and
mirror symmetry. Inventiones Mathematicae. 2022;228:893-989. doi:10.1007/s00222-021-01093-7
apa: Hausel, T., & Hitchin, N. (2022). Very stable Higgs bundles, equivariant
multiplicity and mirror symmetry. Inventiones Mathematicae. Springer Nature.
https://doi.org/10.1007/s00222-021-01093-7
chicago: Hausel, Tamás, and Nigel Hitchin. “Very Stable Higgs Bundles, Equivariant
Multiplicity and Mirror Symmetry.” Inventiones Mathematicae. Springer Nature,
2022. https://doi.org/10.1007/s00222-021-01093-7.
ieee: T. Hausel and N. Hitchin, “Very stable Higgs bundles, equivariant multiplicity
and mirror symmetry,” Inventiones Mathematicae, vol. 228. Springer Nature,
pp. 893–989, 2022.
ista: Hausel T, Hitchin N. 2022. Very stable Higgs bundles, equivariant multiplicity
and mirror symmetry. Inventiones Mathematicae. 228, 893–989.
mla: Hausel, Tamás, and Nigel Hitchin. “Very Stable Higgs Bundles, Equivariant Multiplicity
and Mirror Symmetry.” Inventiones Mathematicae, vol. 228, Springer Nature,
2022, pp. 893–989, doi:10.1007/s00222-021-01093-7.
short: T. Hausel, N. Hitchin, Inventiones Mathematicae 228 (2022) 893–989.
date_created: 2022-01-30T23:01:34Z
date_published: 2022-05-01T00:00:00Z
date_updated: 2023-08-02T14:03:20Z
day: '01'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1007/s00222-021-01093-7
external_id:
arxiv:
- '2101.08583'
isi:
- '000745495400001'
file:
- access_level: open_access
checksum: a382ba75acebc9adfb8fe56247cb410e
content_type: application/pdf
creator: dernst
date_created: 2023-02-27T07:30:47Z
date_updated: 2023-02-27T07:30:47Z
file_id: '12687'
file_name: 2022_InventionesMahtematicae_Hausel.pdf
file_size: 1069538
relation: main_file
success: 1
file_date_updated: 2023-02-27T07:30:47Z
has_accepted_license: '1'
intvolume: ' 228'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 893-989
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Inventiones Mathematicae
publication_identifier:
eissn:
- 1432-1297
issn:
- 0020-9910
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
link:
- description: News on the ISTA Website
relation: press_release
url: https://ista.ac.at/en/news/the-tip-of-the-mathematical-iceberg/
scopus_import: '1'
status: public
title: Very stable Higgs bundles, equivariant multiplicity and mirror symmetry
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 228
year: '2022'
...
---
_id: '439'
abstract:
- lang: eng
text: "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\r\npossibility of a P =
W conjecture for a suitable wild Hitchin system."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Tamas
full_name: Hausel, Tamas
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Martin
full_name: Mereb, Martin
id: 43D735EE-F248-11E8-B48F-1D18A9856A87
last_name: Mereb
- first_name: Michael
full_name: Wong, Michael
last_name: Wong
citation:
ama: Hausel T, Mereb M, Wong M. Arithmetic and representation theory of wild character
varieties. Journal of the European Mathematical Society. 2019;21(10):2995-3052.
doi:10.4171/JEMS/896
apa: Hausel, T., Mereb, M., & Wong, M. (2019). Arithmetic and representation
theory of wild character varieties. Journal of the European Mathematical Society.
European Mathematical Society. https://doi.org/10.4171/JEMS/896
chicago: Hausel, Tamás, Martin Mereb, and Michael Wong. “Arithmetic and Representation
Theory of Wild Character Varieties.” Journal of the European Mathematical Society.
European Mathematical Society, 2019. https://doi.org/10.4171/JEMS/896.
ieee: T. Hausel, M. Mereb, and M. Wong, “Arithmetic and representation theory of
wild character varieties,” Journal of the European Mathematical Society,
vol. 21, no. 10. European Mathematical Society, pp. 2995–3052, 2019.
ista: Hausel T, Mereb M, Wong M. 2019. Arithmetic and representation theory of wild
character varieties. Journal of the European Mathematical Society. 21(10), 2995–3052.
mla: Hausel, Tamás, et al. “Arithmetic and Representation Theory of Wild Character
Varieties.” Journal of the European Mathematical Society, vol. 21, no.
10, European Mathematical Society, 2019, pp. 2995–3052, doi:10.4171/JEMS/896.
short: T. Hausel, M. Mereb, M. Wong, Journal of the European Mathematical Society
21 (2019) 2995–3052.
date_created: 2018-12-11T11:46:29Z
date_published: 2019-10-01T00:00:00Z
date_updated: 2023-08-24T14:24:49Z
day: '01'
department:
- _id: TaHa
doi: 10.4171/JEMS/896
ec_funded: 1
external_id:
arxiv:
- '1604.03382'
isi:
- '000480413600002'
intvolume: ' 21'
isi: 1
issue: '10'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1604.03382
month: '10'
oa: 1
oa_version: Preprint
page: 2995-3052
project:
- _id: 25E549F4-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '320593'
name: Arithmetic and physics of Higgs moduli spaces
publication: Journal of the European Mathematical Society
publication_identifier:
eissn:
- 1435-9855
publication_status: published
publisher: European Mathematical Society
publist_id: '7384'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Arithmetic and representation theory of wild character varieties
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 21
year: '2019'
...
---
_id: '6525'
abstract:
- lang: eng
text: 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:
- first_name: Tamás
full_name: Hausel, Tamás
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Anton
full_name: Mellit, Anton
id: 388D3134-F248-11E8-B48F-1D18A9856A87
last_name: Mellit
- first_name: Du
full_name: Pei, Du
last_name: Pei
citation:
ama: 'Hausel T, Mellit A, Pei D. Mirror symmetry with branes by equivariant verlinde
formulas. In: Geometry and Physics: Volume I. Oxford University Press;
2018:189-218. doi:10.1093/oso/9780198802013.003.0009'
apa: 'Hausel, T., Mellit, A., & Pei, D. (2018). Mirror symmetry with branes
by equivariant verlinde formulas. In Geometry and Physics: Volume I (pp.
189–218). Oxford University Press. https://doi.org/10.1093/oso/9780198802013.003.0009'
chicago: 'Hausel, Tamás, Anton Mellit, and Du Pei. “Mirror Symmetry with Branes
by Equivariant Verlinde Formulas.” In Geometry and Physics: Volume I, 189–218.
Oxford University Press, 2018. https://doi.org/10.1093/oso/9780198802013.003.0009.'
ieee: 'T. Hausel, A. Mellit, and D. Pei, “Mirror symmetry with branes by equivariant
verlinde formulas,” in Geometry and Physics: Volume I, Oxford University
Press, 2018, pp. 189–218.'
ista: 'Hausel T, Mellit A, Pei D. 2018.Mirror symmetry with branes by equivariant
verlinde formulas. In: Geometry and Physics: Volume I. , 189–218.'
mla: 'Hausel, Tamás, et al. “Mirror Symmetry with Branes by Equivariant Verlinde
Formulas.” Geometry and Physics: Volume I, Oxford University Press, 2018,
pp. 189–218, doi:10.1093/oso/9780198802013.003.0009.'
short: 'T. Hausel, A. Mellit, D. Pei, in:, Geometry and Physics: Volume I, Oxford
University Press, 2018, pp. 189–218.'
date_created: 2019-06-06T12:42:01Z
date_published: 2018-01-01T00:00:00Z
date_updated: 2021-01-12T08:07:52Z
day: '01'
department:
- _id: TaHa
doi: 10.1093/oso/9780198802013.003.0009
language:
- iso: eng
month: '01'
oa_version: None
page: 189-218
publication: 'Geometry and Physics: Volume I'
publication_identifier:
isbn:
- '9780198802013'
- '9780191840500'
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: 1
status: public
title: Mirror symmetry with branes by equivariant verlinde formulas
type: book_chapter
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
year: '2018'
...
---
_id: '1473'
abstract:
- lang: eng
text: 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:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Fernando
full_name: Rodríguez Villegas, Fernando
last_name: Rodríguez Villegas
citation:
ama: Hausel T, Rodríguez Villegas F. Cohomology of large semiprojective hyperkähler
varieties. Asterisque. 2015;2015(370):113-156.
apa: Hausel, T., & Rodríguez Villegas, F. (2015). Cohomology of large semiprojective
hyperkähler varieties. Asterisque. Societe Mathematique de France.
chicago: Hausel, Tamás, and Fernando Rodríguez Villegas. “Cohomology of Large Semiprojective
Hyperkähler Varieties.” Asterisque. Societe Mathematique de France, 2015.
ieee: T. Hausel and F. Rodríguez Villegas, “Cohomology of large semiprojective hyperkähler
varieties,” Asterisque, vol. 2015, no. 370. Societe Mathematique de France,
pp. 113–156, 2015.
ista: Hausel T, Rodríguez Villegas F. 2015. Cohomology of large semiprojective hyperkähler
varieties. Asterisque. 2015(370), 113–156.
mla: Hausel, Tamás, and Fernando Rodríguez Villegas. “Cohomology of Large Semiprojective
Hyperkähler Varieties.” Asterisque, vol. 2015, no. 370, Societe Mathematique
de France, 2015, pp. 113–56.
short: T. Hausel, F. Rodríguez Villegas, Asterisque 2015 (2015) 113–156.
date_created: 2018-12-11T11:52:13Z
date_published: 2015-01-01T00:00:00Z
date_updated: 2021-01-12T06:50:59Z
day: '01'
extern: 1
intvolume: ' 2015'
issue: '370'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1309.4914
month: '01'
oa: 1
page: 113 - 156
publication: Asterisque
publication_status: published
publisher: Societe Mathematique de France
publist_id: '5723'
quality_controlled: 0
status: public
title: Cohomology of large semiprojective hyperkähler varieties
type: review
volume: 2015
year: '2015'
...
---
_id: '1442'
abstract:
- lang: eng
text: 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.
acknowledgement: |-
The first author thanks the Royal Society for funding his research 2005-2012 in the form of a Royal Society University Research Fellowship as well as the Mathematical Institute and Wadham College in Oxford for a very productive environment. The second author is supported by Agence Nationale de la Recherche grant
ANR-09-JCJC-0102-01. The third author is supported by the NSF grant DMS-1101484 and a Research Scholarship from the Clay Mathematical Institute.
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Emmanuel
full_name: Letellier, Emmanuel
last_name: Letellier
- first_name: Fernando
full_name: Rodríguez Villegas, Fernando
last_name: Rodríguez Villegas
citation:
ama: Hausel T, Letellier E, Rodríguez Villegas F. Positivity for Kac polynomials
and DT-invariants of quivers. Annals of Mathematics. 2013;177(3):1147-1168.
doi:10.4007/annals.2013.177.3.8
apa: Hausel, T., Letellier, E., & Rodríguez Villegas, F. (2013). Positivity
for Kac polynomials and DT-invariants of quivers. Annals of Mathematics.
Princeton University Press. https://doi.org/10.4007/annals.2013.177.3.8
chicago: Hausel, Tamás, Emmanuel Letellier, and Fernando Rodríguez Villegas. “Positivity
for Kac Polynomials and DT-Invariants of Quivers.” Annals of Mathematics.
Princeton University Press, 2013. https://doi.org/10.4007/annals.2013.177.3.8.
ieee: T. Hausel, E. Letellier, and F. Rodríguez Villegas, “Positivity for Kac polynomials
and DT-invariants of quivers,” Annals of Mathematics, vol. 177, no. 3.
Princeton University Press, pp. 1147–1168, 2013.
ista: Hausel T, Letellier E, Rodríguez Villegas F. 2013. Positivity for Kac polynomials
and DT-invariants of quivers. Annals of Mathematics. 177(3), 1147–1168.
mla: Hausel, Tamás, et al. “Positivity for Kac Polynomials and DT-Invariants of
Quivers.” Annals of Mathematics, vol. 177, no. 3, Princeton University
Press, 2013, pp. 1147–68, doi:10.4007/annals.2013.177.3.8.
short: T. Hausel, E. Letellier, F. Rodríguez Villegas, Annals of Mathematics 177
(2013) 1147–1168.
date_created: 2018-12-11T11:52:02Z
date_published: 2013-01-01T00:00:00Z
date_updated: 2021-01-12T06:50:47Z
day: '01'
doi: 10.4007/annals.2013.177.3.8
extern: 1
intvolume: ' 177'
issue: '3'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1204.2375
month: '01'
oa: 1
page: 1147 - 1168
publication: Annals of Mathematics
publication_status: published
publisher: Princeton University Press
publist_id: '5754'
quality_controlled: 0
status: public
title: Positivity for Kac polynomials and DT-invariants of quivers
type: journal_article
volume: 177
year: '2013'
...
---
_id: '1443'
abstract:
- lang: eng
text: '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.'
alternative_title:
- Advanced Lectures in Mathematics
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
citation:
ama: 'Hausel T. Global topology of the Hitchin system. In: Handbook of Moduli:
Volume II. Vol 25. International Press; 2013:29-70.'
apa: 'Hausel, T. (2013). Global topology of the Hitchin system. In Handbook of
Moduli: Volume II (Vol. 25, pp. 29–70). International Press.'
chicago: 'Hausel, Tamás. “Global Topology of the Hitchin System.” In Handbook
of Moduli: Volume II, 25:29–70. International Press, 2013.'
ieee: 'T. Hausel, “Global topology of the Hitchin system,” in Handbook of Moduli:
Volume II, vol. 25, International Press, 2013, pp. 29–70.'
ista: 'Hausel T. 2013.Global topology of the Hitchin system. In: Handbook of Moduli:
Volume II. Advanced Lectures in Mathematics, vol. 25, 29–70.'
mla: 'Hausel, Tamás. “Global Topology of the Hitchin System.” Handbook of Moduli:
Volume II, vol. 25, International Press, 2013, pp. 29–70.'
short: 'T. Hausel, in:, Handbook of Moduli: Volume II, International Press, 2013,
pp. 29–70.'
date_created: 2018-12-11T11:52:03Z
date_published: 2013-03-15T00:00:00Z
date_updated: 2021-01-12T06:50:47Z
day: '15'
extern: 1
intvolume: ' 25'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1102.1717
month: '03'
oa: 1
page: 29 - 70
publication: 'Handbook of Moduli: Volume II'
publication_status: published
publisher: International Press
publist_id: '5753'
quality_controlled: 0
status: public
title: Global topology of the Hitchin system
type: book_chapter
volume: 25
year: '2013'
...
---
_id: '1469'
abstract:
- lang: eng
text: 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.
acknowledgement: During the preparation of this paper TH was supported by a Royal
Society University Research Fellowship at the University of Oxford. EL was supported
by ANR-09-JCJC-0102-01. FRV was supported by NSF grant DMS-0200605, an FRA from
the University of Texas at Austin, EPSRC grant EP/G027110/1, Visiting Fellowships
at All Souls and Wadham Colleges in Oxford and a Research Scholarship from the Clay
Mathematical Institute.
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Emmanuel
full_name: Letellier, Emmanuel
last_name: Letellier
- first_name: Fernando
full_name: Rodríguez Villegas, Fernando
last_name: Rodríguez Villegas
citation:
ama: Hausel T, Letellier E, Rodríguez Villegas F. Arithmetic harmonic analysis on
character and quiver varieties II. Advances in Mathematics. 2013;234:85-128.
doi:10.1016/j.aim.2012.10.009
apa: Hausel, T., Letellier, E., & Rodríguez Villegas, F. (2013). Arithmetic
harmonic analysis on character and quiver varieties II. Advances in Mathematics.
Academic Press. https://doi.org/10.1016/j.aim.2012.10.009
chicago: Hausel, Tamás, Emmanuel Letellier, and Fernando Rodríguez Villegas. “Arithmetic
Harmonic Analysis on Character and Quiver Varieties II.” Advances in Mathematics.
Academic Press, 2013. https://doi.org/10.1016/j.aim.2012.10.009.
ieee: T. Hausel, E. Letellier, and F. Rodríguez Villegas, “Arithmetic harmonic analysis
on character and quiver varieties II,” Advances in Mathematics, vol. 234.
Academic Press, pp. 85–128, 2013.
ista: Hausel T, Letellier E, Rodríguez Villegas F. 2013. Arithmetic harmonic analysis
on character and quiver varieties II. Advances in Mathematics. 234, 85–128.
mla: Hausel, Tamás, et al. “Arithmetic Harmonic Analysis on Character and Quiver
Varieties II.” Advances in Mathematics, vol. 234, Academic Press, 2013,
pp. 85–128, doi:10.1016/j.aim.2012.10.009.
short: T. Hausel, E. Letellier, F. Rodríguez Villegas, Advances in Mathematics 234
(2013) 85–128.
date_created: 2018-12-11T11:52:12Z
date_published: 2013-02-15T00:00:00Z
date_updated: 2021-01-12T06:50:57Z
day: '15'
doi: 10.1016/j.aim.2012.10.009
extern: 1
intvolume: ' 234'
month: '02'
page: 85 - 128
publication: Advances in Mathematics
publication_status: published
publisher: Academic Press
publist_id: '5724'
quality_controlled: 0
status: public
title: Arithmetic harmonic analysis on character and quiver varieties II
type: journal_article
volume: 234
year: '2013'
...
---
_id: '1470'
abstract:
- lang: eng
text: 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.
acknowledgement: Mark Andrea A. de Cataldo was partially supported by N.S.A. and N.S.F.
Tamás Hausel was supported by a Royal Society University Research Fellowship. Luca
Migliorini was partially supported by PRIN 2007 project "Spazi di moduli e teoria
di Lie"
author:
- first_name: Mark
full_name: De Cataldo, Mark A
last_name: De Cataldo
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Luca
full_name: Migliorini, Luca
last_name: Migliorini
citation:
ama: De Cataldo M, Hausel T, Migliorini L. Exchange between perverse and weight
filtration for the Hilbert schemes of points of two surfaces. Journal of Singularities.
2013;7:23-38. doi:10.5427/jsing.2013.7c
apa: De Cataldo, M., Hausel, T., & Migliorini, L. (2013). Exchange between perverse
and weight filtration for the Hilbert schemes of points of two surfaces. Journal
of Singularities. Worldwide Center of Mathematics. https://doi.org/10.5427/jsing.2013.7c
chicago: De Cataldo, Mark, Tamás Hausel, and Luca Migliorini. “Exchange between
Perverse and Weight Filtration for the Hilbert Schemes of Points of Two Surfaces.”
Journal of Singularities. Worldwide Center of Mathematics, 2013. https://doi.org/10.5427/jsing.2013.7c.
ieee: M. De Cataldo, T. Hausel, and L. Migliorini, “Exchange between perverse and
weight filtration for the Hilbert schemes of points of two surfaces,” Journal
of Singularities, vol. 7. Worldwide Center of Mathematics, pp. 23–38, 2013.
ista: De Cataldo M, Hausel T, Migliorini L. 2013. Exchange between perverse and
weight filtration for the Hilbert schemes of points of two surfaces. Journal of
Singularities. 7, 23–38.
mla: De Cataldo, Mark, et al. “Exchange between Perverse and Weight Filtration for
the Hilbert Schemes of Points of Two Surfaces.” Journal of Singularities,
vol. 7, Worldwide Center of Mathematics, 2013, pp. 23–38, doi:10.5427/jsing.2013.7c.
short: M. De Cataldo, T. Hausel, L. Migliorini, Journal of Singularities 7 (2013)
23–38.
date_created: 2018-12-11T11:52:12Z
date_published: 2013-01-01T00:00:00Z
date_updated: 2021-01-12T06:50:58Z
day: '01'
doi: 10.5427/jsing.2013.7c
extern: 1
intvolume: ' 7'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1012.2583
month: '01'
oa: 1
page: 23 - 38
publication: Journal of Singularities
publication_status: published
publisher: Worldwide Center of Mathematics
publist_id: '5725'
quality_controlled: 0
status: public
title: Exchange between perverse and weight filtration for the Hilbert schemes of
points of two surfaces
type: journal_article
volume: 7
year: '2013'
...
---
_id: '1471'
abstract:
- lang: eng
text: '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:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Christian
full_name: Pauly, Christian
last_name: Pauly
citation:
ama: Hausel T, Pauly C. Prym varieties of spectral covers. Geometry and Topology.
2012;16(3):1609-1638. doi:10.2140/gt.2012.16.1609
apa: Hausel, T., & Pauly, C. (2012). Prym varieties of spectral covers. Geometry
and Topology. University of Warwick. https://doi.org/10.2140/gt.2012.16.1609
chicago: Hausel, Tamás, and Christian Pauly. “Prym Varieties of Spectral Covers.”
Geometry and Topology. University of Warwick, 2012. https://doi.org/10.2140/gt.2012.16.1609.
ieee: T. Hausel and C. Pauly, “Prym varieties of spectral covers,” Geometry and
Topology, vol. 16, no. 3. University of Warwick, pp. 1609–1638, 2012.
ista: Hausel T, Pauly C. 2012. Prym varieties of spectral covers. Geometry and Topology.
16(3), 1609–1638.
mla: Hausel, Tamás, and Christian Pauly. “Prym Varieties of Spectral Covers.” Geometry
and Topology, vol. 16, no. 3, University of Warwick, 2012, pp. 1609–38, doi:10.2140/gt.2012.16.1609.
short: T. Hausel, C. Pauly, Geometry and Topology 16 (2012) 1609–1638.
date_created: 2018-12-11T11:52:13Z
date_published: 2012-08-01T00:00:00Z
date_updated: 2021-01-12T06:50:58Z
day: '01'
doi: 10.2140/gt.2012.16.1609
extern: 1
intvolume: ' 16'
issue: '3'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1012.4748
month: '08'
oa: 1
page: 1609 - 1638
publication: Geometry and Topology
publication_status: published
publisher: University of Warwick
publist_id: '5726'
quality_controlled: 0
status: public
title: Prym varieties of spectral covers
type: journal_article
volume: 16
year: '2012'
...
---
_id: '1472'
abstract:
- lang: eng
text: 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.
acknowledgement: Mark Andrea A. de Cataldo was partially supported by N.S.A. and N.S.F.
Tamás Hausel was supported by a Royal Society University Research Fellowship. Luca
Migliorini was partially supported by PRIN 2007 project "Spazi di moduli e teoria
di Lie"
author:
- first_name: Mark
full_name: De Cataldo, Mark A
last_name: De Cataldo
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Luca
full_name: Migliorini, Luca
last_name: Migliorini
citation:
ama: 'De Cataldo M, Hausel T, Migliorini L. Topology of hitchin systems and Hodge
theory of character varieties: The case A 1. Annals of Mathematics. 2012;175(3):1329-1407.
doi:10.4007/annals.2012.175.3.7'
apa: 'De Cataldo, M., Hausel, T., & Migliorini, L. (2012). Topology of hitchin
systems and Hodge theory of character varieties: The case A 1. Annals of Mathematics.
Princeton University Press. https://doi.org/10.4007/annals.2012.175.3.7'
chicago: 'De Cataldo, Mark, Tamás Hausel, and Luca Migliorini. “Topology of Hitchin
Systems and Hodge Theory of Character Varieties: The Case A 1.” Annals of Mathematics.
Princeton University Press, 2012. https://doi.org/10.4007/annals.2012.175.3.7.'
ieee: 'M. De Cataldo, T. Hausel, and L. Migliorini, “Topology of hitchin systems
and Hodge theory of character varieties: The case A 1,” Annals of Mathematics,
vol. 175, no. 3. Princeton University Press, pp. 1329–1407, 2012.'
ista: 'De Cataldo M, Hausel T, Migliorini L. 2012. Topology of hitchin systems and
Hodge theory of character varieties: The case A 1. Annals of Mathematics. 175(3),
1329–1407.'
mla: 'De Cataldo, Mark, et al. “Topology of Hitchin Systems and Hodge Theory of
Character Varieties: The Case A 1.” Annals of Mathematics, vol. 175, no.
3, Princeton University Press, 2012, pp. 1329–407, doi:10.4007/annals.2012.175.3.7.'
short: M. De Cataldo, T. Hausel, L. Migliorini, Annals of Mathematics 175 (2012)
1329–1407.
date_created: 2018-12-11T11:52:13Z
date_published: 2012-05-01T00:00:00Z
date_updated: 2021-01-12T06:50:59Z
day: '01'
doi: 10.4007/annals.2012.175.3.7
extern: 1
intvolume: ' 175'
issue: '3'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1004.1420
month: '05'
oa: 1
page: 1329 - 1407
publication: Annals of Mathematics
publication_status: published
publisher: Princeton University Press
publist_id: '5727'
quality_controlled: 0
status: public
title: 'Topology of hitchin systems and Hodge theory of character varieties: The case
A 1'
type: journal_article
volume: 175
year: '2012'
...
---
_id: '1467'
abstract:
- lang: eng
text: 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).
acknowledgement: |-
Hausel’s work was supported by National Science Foundation grants DMS-0305505 and DMS-0604775, by an Alfred Sloan Fellowship, and by a Royal Society University Research Fellowship. Letellier’s work supported by Agence Nationale de la Recherche grant ANR-09-JCJC-0102-01.
Rodriguez-Villegas’s work supported by National Science Foundation grant DMS-0200605, by an FRA from the University of Texas at Austin, by EPSRC grant EP/G027110/1, by visiting fellowships at All Souls and Wadham Colleges in Oxford, and by a Research Scholarship from the Clay Mathematical Institute.
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Emmanuel
full_name: Letellier, Emmanuel
last_name: Letellier
- first_name: Fernando
full_name: Rodríguez Villegas, Fernando
last_name: Rodríguez Villegas
citation:
ama: Hausel T, Letellier E, Rodríguez Villegas F. Arithmetic harmonic analysis on
character and quiver varieties. Duke Mathematical Journal. 2011;160(2):323-400.
doi:10.1215/00127094-1444258
apa: Hausel, T., Letellier, E., & Rodríguez Villegas, F. (2011). Arithmetic
harmonic analysis on character and quiver varieties. Duke Mathematical Journal.
Duke University Press. https://doi.org/10.1215/00127094-1444258
chicago: Hausel, Tamás, Emmanuel Letellier, and Fernando Rodríguez Villegas. “Arithmetic
Harmonic Analysis on Character and Quiver Varieties.” Duke Mathematical Journal.
Duke University Press, 2011. https://doi.org/10.1215/00127094-1444258.
ieee: T. Hausel, E. Letellier, and F. Rodríguez Villegas, “Arithmetic harmonic analysis
on character and quiver varieties,” Duke Mathematical Journal, vol. 160,
no. 2. Duke University Press, pp. 323–400, 2011.
ista: Hausel T, Letellier E, Rodríguez Villegas F. 2011. Arithmetic harmonic analysis
on character and quiver varieties. Duke Mathematical Journal. 160(2), 323–400.
mla: Hausel, Tamás, et al. “Arithmetic Harmonic Analysis on Character and Quiver
Varieties.” Duke Mathematical Journal, vol. 160, no. 2, Duke University
Press, 2011, pp. 323–400, doi:10.1215/00127094-1444258.
short: T. Hausel, E. Letellier, F. Rodríguez Villegas, Duke Mathematical Journal
160 (2011) 323–400.
date_created: 2018-12-11T11:52:11Z
date_published: 2011-01-01T00:00:00Z
date_updated: 2021-01-12T06:50:56Z
day: '01'
doi: 10.1215/00127094-1444258
extern: 1
intvolume: ' 160'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0810.2076
month: '01'
oa: 1
page: 323 - 400
publication: Duke Mathematical Journal
publication_status: published
publisher: Duke University Press
publist_id: '5728'
quality_controlled: 0
status: public
title: Arithmetic harmonic analysis on character and quiver varieties
type: journal_article
volume: 160
year: '2011'
...
---
_id: '1465'
abstract:
- lang: eng
text: 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.
acknowledgement: This work has been supported by a Royal Society University Research
Fellowship, NSF grants DMS-0305505 and DMS-0604775 and an Alfred Sloan Fellowship
2005-2007.
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
citation:
ama: Hausel T. Kac’s conjecture from Nakajima quiver varieties. Inventiones Mathematicae.
2010;181(1):21-37. doi:10.1007/s00222-010-0241-3
apa: Hausel, T. (2010). Kac’s conjecture from Nakajima quiver varieties. Inventiones
Mathematicae. Springer. https://doi.org/10.1007/s00222-010-0241-3
chicago: Hausel, Tamás. “Kac’s Conjecture from Nakajima Quiver Varieties.” Inventiones
Mathematicae. Springer, 2010. https://doi.org/10.1007/s00222-010-0241-3.
ieee: T. Hausel, “Kac’s conjecture from Nakajima quiver varieties,” Inventiones
Mathematicae, vol. 181, no. 1. Springer, pp. 21–37, 2010.
ista: Hausel T. 2010. Kac’s conjecture from Nakajima quiver varieties. Inventiones
Mathematicae. 181(1), 21–37.
mla: Hausel, Tamás. “Kac’s Conjecture from Nakajima Quiver Varieties.” Inventiones
Mathematicae, vol. 181, no. 1, Springer, 2010, pp. 21–37, doi:10.1007/s00222-010-0241-3.
short: T. Hausel, Inventiones Mathematicae 181 (2010) 21–37.
date_created: 2018-12-11T11:52:11Z
date_published: 2010-07-01T00:00:00Z
date_updated: 2021-01-12T06:50:56Z
day: '01'
doi: 10.1007/s00222-010-0241-3
extern: 1
intvolume: ' 181'
issue: '1'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0811.1569
month: '07'
oa: 1
page: 21 - 37
publication: Inventiones Mathematicae
publication_status: published
publisher: Springer
publist_id: '5730'
quality_controlled: 0
status: public
title: Kac's conjecture from Nakajima quiver varieties
type: journal_article
volume: 181
year: '2010'
...
---
_id: '1466'
abstract:
- lang: eng
text: 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:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Emmanuel
full_name: Letellier, Emmanuel
last_name: Letellier
- first_name: Fernando
full_name: Rodríguez Villegas, Fernando
last_name: Rodríguez Villegas
citation:
ama: Hausel T, Letellier E, Rodríguez Villegas F. Topology of character varieties
and representations of quivers. Comptes Rendus Mathematique. 2010;348(3-4):131-135.
doi:10.1016/j.crma.2010.01.025
apa: Hausel, T., Letellier, E., & Rodríguez Villegas, F. (2010). Topology of
character varieties and representations of quivers. Comptes Rendus Mathematique.
Elsevier. https://doi.org/10.1016/j.crma.2010.01.025
chicago: Hausel, Tamás, Emmanuel Letellier, and Fernando Rodríguez Villegas. “Topology
of Character Varieties and Representations of Quivers.” Comptes Rendus Mathematique.
Elsevier, 2010. https://doi.org/10.1016/j.crma.2010.01.025.
ieee: T. Hausel, E. Letellier, and F. Rodríguez Villegas, “Topology of character
varieties and representations of quivers,” Comptes Rendus Mathematique,
vol. 348, no. 3–4. Elsevier, pp. 131–135, 2010.
ista: Hausel T, Letellier E, Rodríguez Villegas F. 2010. Topology of character varieties
and representations of quivers. Comptes Rendus Mathematique. 348(3–4), 131–135.
mla: Hausel, Tamás, et al. “Topology of Character Varieties and Representations
of Quivers.” Comptes Rendus Mathematique, vol. 348, no. 3–4, Elsevier,
2010, pp. 131–35, doi:10.1016/j.crma.2010.01.025.
short: T. Hausel, E. Letellier, F. Rodríguez Villegas, Comptes Rendus Mathematique
348 (2010) 131–135.
date_created: 2018-12-11T11:52:11Z
date_published: 2010-02-01T00:00:00Z
date_updated: 2021-01-12T06:50:56Z
day: '01'
doi: 10.1016/j.crma.2010.01.025
extern: 1
intvolume: ' 348'
issue: 3-4
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0905.3491
month: '02'
oa: 1
page: 131 - 135
publication: Comptes Rendus Mathematique
publication_status: published
publisher: Elsevier
publist_id: '5731'
quality_controlled: 0
status: public
title: Topology of character varieties and representations of quivers
type: journal_article
volume: 348
year: '2010'
...
---
_id: '1468'
abstract:
- lang: eng
text: '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:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
citation:
ama: 'Hausel T. S-Duality in HyperkäHler Hodge Theory. In: The Many Facets of
Geometry: A Tribute to Nigel Hitchin. Oxford University Press; 2010. doi:10.1093/acprof:oso/9780199534920.003.0016'
apa: 'Hausel, T. (2010). S-Duality in HyperkäHler Hodge Theory. In The Many Facets
of Geometry: A Tribute to Nigel Hitchin. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199534920.003.0016'
chicago: 'Hausel, Tamás. “S-Duality in HyperkäHler Hodge Theory.” In The Many
Facets of Geometry: A Tribute to Nigel Hitchin. Oxford University Press, 2010.
https://doi.org/10.1093/acprof:oso/9780199534920.003.0016.'
ieee: 'T. Hausel, “S-Duality in HyperkäHler Hodge Theory,” in The Many Facets
of Geometry: A Tribute to Nigel Hitchin, Oxford University Press, 2010.'
ista: 'Hausel T. 2010.S-Duality in HyperkäHler Hodge Theory. In: The Many Facets
of Geometry: A Tribute to Nigel Hitchin. .'
mla: 'Hausel, Tamás. “S-Duality in HyperkäHler Hodge Theory.” The Many Facets
of Geometry: A Tribute to Nigel Hitchin, Oxford University Press, 2010, doi:10.1093/acprof:oso/9780199534920.003.0016.'
short: 'T. Hausel, in:, The Many Facets of Geometry: A Tribute to Nigel Hitchin,
Oxford University Press, 2010.'
date_created: 2018-12-11T11:52:12Z
date_published: 2010-09-01T00:00:00Z
date_updated: 2021-01-12T06:50:57Z
day: '01'
doi: 10.1093/acprof:oso/9780199534920.003.0016
extern: 1
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0709.0504
month: '09'
oa: 1
publication: 'The Many Facets of Geometry: A Tribute to Nigel Hitchin'
publication_status: published
publisher: Oxford University Press
publist_id: '5729'
quality_controlled: 0
status: public
title: S-Duality in HyperkäHler Hodge Theory
type: book_chapter
year: '2010'
...
---
_id: '1460'
abstract:
- lang: eng
text: '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.'
acknowledgement: The first author was supported by NSF grants DMS-0305505 and DMS-
0604775 an Alfred Sloan Fellowship and a Royal Society University Research Fellowship.
The second author was supported by an NSF grant DMS-0200605.
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Fernando
full_name: Rodríguez Villegas, Fernando
last_name: Rodríguez Villegas
citation:
ama: 'Hausel T, Rodríguez Villegas F. Mixed Hodge polynomials of character varieties:
With an appendix by Nicholas M. Katz. Inventiones Mathematicae. 2008;174(3):555-624.
doi:10.1007/s00222-008-0142-x'
apa: 'Hausel, T., & Rodríguez Villegas, F. (2008). Mixed Hodge polynomials of
character varieties: With an appendix by Nicholas M. Katz. Inventiones Mathematicae.
Springer. https://doi.org/10.1007/s00222-008-0142-x'
chicago: 'Hausel, Tamás, and Fernando Rodríguez Villegas. “Mixed Hodge Polynomials
of Character Varieties: With an Appendix by Nicholas M. Katz.” Inventiones
Mathematicae. Springer, 2008. https://doi.org/10.1007/s00222-008-0142-x.'
ieee: 'T. Hausel and F. Rodríguez Villegas, “Mixed Hodge polynomials of character
varieties: With an appendix by Nicholas M. Katz,” Inventiones Mathematicae,
vol. 174, no. 3. Springer, pp. 555–624, 2008.'
ista: 'Hausel T, Rodríguez Villegas F. 2008. Mixed Hodge polynomials of character
varieties: With an appendix by Nicholas M. Katz. Inventiones Mathematicae. 174(3),
555–624.'
mla: 'Hausel, Tamás, and Fernando Rodríguez Villegas. “Mixed Hodge Polynomials of
Character Varieties: With an Appendix by Nicholas M. Katz.” Inventiones Mathematicae,
vol. 174, no. 3, Springer, 2008, pp. 555–624, doi:10.1007/s00222-008-0142-x.'
short: T. Hausel, F. Rodríguez Villegas, Inventiones Mathematicae 174 (2008) 555–624.
date_created: 2018-12-11T11:52:09Z
date_published: 2008-12-01T00:00:00Z
date_updated: 2021-01-12T06:50:54Z
day: '01'
doi: 10.1007/s00222-008-0142-x
extern: 1
intvolume: ' 174'
issue: '3'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math/0612668
month: '12'
oa: 1
page: 555 - 624
publication: Inventiones Mathematicae
publication_status: published
publisher: Springer
publist_id: '5732'
quality_controlled: 0
status: public
title: 'Mixed Hodge polynomials of character varieties: With an appendix by Nicholas
M. Katz'
type: journal_article
volume: 174
year: '2008'
...
---
_id: '1461'
abstract:
- lang: eng
text: 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.
acknowledgement: The first author was partly supported by NSF grant DMS-0072675. The
second author was partly supported by a VIGRE postdoc under NSF grant number 9983660
to Cornell University.
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
- first_name: Edward
full_name: Swartz, Edward
last_name: Swartz
citation:
ama: Hausel T, Swartz E. Intersection forms of toric hyperkähler varieties. Proceedings
of the American Mathematical Society. 2006;134(8):2403-2409. doi:10.1090/S0002-9939-06-08248-7
apa: Hausel, T., & Swartz, E. (2006). Intersection forms of toric hyperkähler
varieties. Proceedings of the American Mathematical Society. American Mathematical
Society. https://doi.org/10.1090/S0002-9939-06-08248-7
chicago: Hausel, Tamás, and Edward Swartz. “Intersection Forms of Toric Hyperkähler
Varieties.” Proceedings of the American Mathematical Society. American
Mathematical Society, 2006. https://doi.org/10.1090/S0002-9939-06-08248-7.
ieee: T. Hausel and E. Swartz, “Intersection forms of toric hyperkähler varieties,”
Proceedings of the American Mathematical Society, vol. 134, no. 8. American
Mathematical Society, pp. 2403–2409, 2006.
ista: Hausel T, Swartz E. 2006. Intersection forms of toric hyperkähler varieties.
Proceedings of the American Mathematical Society. 134(8), 2403–2409.
mla: Hausel, Tamás, and Edward Swartz. “Intersection Forms of Toric Hyperkähler
Varieties.” Proceedings of the American Mathematical Society, vol. 134,
no. 8, American Mathematical Society, 2006, pp. 2403–09, doi:10.1090/S0002-9939-06-08248-7.
short: T. Hausel, E. Swartz, Proceedings of the American Mathematical Society 134
(2006) 2403–2409.
date_created: 2018-12-11T11:52:09Z
date_published: 2006-08-01T00:00:00Z
date_updated: 2021-01-12T06:50:54Z
day: '01'
doi: 10.1090/S0002-9939-06-08248-7
extern: 1
intvolume: ' 134'
issue: '8'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math/0306369
month: '08'
oa: 1
page: 2403 - 2409
publication: Proceedings of the American Mathematical Society
publication_status: published
publisher: American Mathematical Society
publist_id: '5733'
quality_controlled: 0
status: public
title: Intersection forms of toric hyperkähler varieties
type: journal_article
volume: 134
year: '2006'
...
---
_id: '1462'
abstract:
- lang: eng
text: 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.
acknowledgement: This work was supported by a Royal Society University Research Fellowship,
National Science Foundation Grant DMS-0305505, an Alfred P. Sloan Research Fellowship,
and a Summer Research Assignment of the University of Texas at Austin.
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
citation:
ama: Hausel T. Betti numbers of holomorphic symplectic quotients via arithmetic
Fourier transform. PNAS. 2006;103(16):6120-6124. doi:10.1073/pnas.0601337103
apa: Hausel, T. (2006). Betti numbers of holomorphic symplectic quotients via arithmetic
Fourier transform. PNAS. National Academy of Sciences. https://doi.org/10.1073/pnas.0601337103
chicago: Hausel, Tamás. “Betti Numbers of Holomorphic Symplectic Quotients via Arithmetic
Fourier Transform.” PNAS. National Academy of Sciences, 2006. https://doi.org/10.1073/pnas.0601337103.
ieee: T. Hausel, “Betti numbers of holomorphic symplectic quotients via arithmetic
Fourier transform,” PNAS, vol. 103, no. 16. National Academy of Sciences,
pp. 6120–6124, 2006.
ista: Hausel T. 2006. Betti numbers of holomorphic symplectic quotients via arithmetic
Fourier transform. PNAS. 103(16), 6120–6124.
mla: Hausel, Tamás. “Betti Numbers of Holomorphic Symplectic Quotients via Arithmetic
Fourier Transform.” PNAS, vol. 103, no. 16, National Academy of Sciences,
2006, pp. 6120–24, doi:10.1073/pnas.0601337103.
short: T. Hausel, PNAS 103 (2006) 6120–6124.
date_created: 2018-12-11T11:52:10Z
date_published: 2006-04-18T00:00:00Z
date_updated: 2021-01-12T06:50:55Z
day: '18'
doi: 10.1073/pnas.0601337103
extern: 1
intvolume: ' 103'
issue: '16'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math/0511163
month: '04'
oa: 1
page: 6120 - 6124
publication: PNAS
publication_status: published
publisher: National Academy of Sciences
publist_id: '5734'
quality_controlled: 0
status: public
title: Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform
type: journal_article
volume: 103
year: '2006'
...
---
_id: '1444'
abstract:
- lang: eng
text: 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.
alternative_title:
- Progress in Mathematics
author:
- first_name: Tamas
full_name: Tamas Hausel
id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
last_name: Hausel
citation:
ama: 'Hausel T. Mirror symmetry and Langlands duality in the non-Abelian Hodge theory
of a curve. In: Geometric Methods in Algebra and Number Theory. Vol 235.
Springer; 2005:193-217. doi:10.1007/0-8176-4417-2_9'
apa: Hausel, T. (2005). Mirror symmetry and Langlands duality in the non-Abelian
Hodge theory of a curve. In Geometric Methods in Algebra and Number Theory
(Vol. 235, pp. 193–217). Springer. https://doi.org/10.1007/0-8176-4417-2_9
chicago: Hausel, Tamás. “Mirror Symmetry and Langlands Duality in the Non-Abelian
Hodge Theory of a Curve.” In Geometric Methods in Algebra and Number Theory,
235:193–217. Springer, 2005. https://doi.org/10.1007/0-8176-4417-2_9.
ieee: T. Hausel, “Mirror symmetry and Langlands duality in the non-Abelian Hodge
theory of a curve,” in Geometric Methods in Algebra and Number Theory,
vol. 235, Springer, 2005, pp. 193–217.
ista: 'Hausel T. 2005.Mirror symmetry and Langlands duality in the non-Abelian Hodge
theory of a curve. In: Geometric Methods in Algebra and Number Theory. Progress
in Mathematics, vol. 235, 193–217.'
mla: Hausel, Tamás. “Mirror Symmetry and Langlands Duality in the Non-Abelian Hodge
Theory of a Curve.” Geometric Methods in Algebra and Number Theory, vol.
235, Springer, 2005, pp. 193–217, doi:10.1007/0-8176-4417-2_9.
short: T. Hausel, in:, Geometric Methods in Algebra and Number Theory, Springer,
2005, pp. 193–217.
date_created: 2018-12-11T11:52:03Z
date_published: 2005-01-01T00:00:00Z
date_updated: 2021-01-12T06:50:47Z
day: '01'
doi: 10.1007/0-8176-4417-2_9
extern: 1
intvolume: ' 235'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math/0406380
month: '01'
oa: 1
page: 193 - 217
publication: Geometric Methods in Algebra and Number Theory
publication_status: published
publisher: Springer
publist_id: '5752'
quality_controlled: 0
status: public
title: Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a
curve
type: book_chapter
volume: 235
year: '2005'
...