Kari Vilonen
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Kari Vilonen has a total of 26 co-authors. Here are the top co-authors by number of shared publications:
- Ting Xue
School of Mathematics and Statistics, University of Melbourne, Parkville, Australia
14 shared publications - Tsao-Hsien Chen
Department of Mathematics, University of Chicago, Illinois, U.SA
5 shared publications - Wilfried Schmid
Harvard University
4 shared publications D. Gaitsgory
3 shared publicationsE. Frenkel
3 shared publicationsWilfried Schmid
3 shared publications
Latest Publications
- Open Access
Cuspidal character sheaves on graded Lie algebras II
In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the explicit classification of cuspidal character sheaves for graded Lie algebras.
Open AccessCuspidal character sheaves on graded Lie algebras II
In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the explicit classification of cuspidal character sheaves for graded Lie algebras.
- Open Access
Nearby Cycle Sheaves for Stable Polar Representations
We extend the results of [GVX] to the setting of a stable polar representation G|V (G connected, reductive over C), satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f : V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. Our main intended application is to the theory of character sheaves for graded Lie algebras over C.
Open AccessNearby Cycle Sheaves for Stable Polar Representations
We extend the results of [GVX] to the setting of a stable polar representation G|V (G connected, reductive over C), satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f : V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. Our main intended application is to the theory of character sheaves for graded Lie algebras over C.
- Open Access
Mixed Hodge modules and real groups
Let $G$ be a complex reductive group, $\theta \colon G \to G$ an involution, and $K = G^\theta$. In arXiv:1206.5547, W. Schmid and the second named author proposed a program to study unitary representations of the corresponding real form $G_\mathbb{R}$ using $K$-equivariant twisted mixed Hodge modules on the flag variety of $G$ and their polarizations. In this paper, we make the first significant steps towards implementing this program. Our first main result gives an explicit combinatorial formula for the Hodge numbers appearing in the composition series of a standard module in terms of the Lusztig-Vogan polynomials. Our second main result is a polarized version of the Jantzen conjecture, stating that the Jantzen forms on the composition factors are polarizations of the underlying Hodge modules. Our third main result states that, for regular Beilinson-Bernstein data, the minimal $K$-type of an irreducible Harish-Chandra module lies in the lowest piece of the Hodge filtration of the corresponding Hodge module. An immediate consequence of our results is a Hodge-theoretic proof of the signature multiplicity formula of arXiv:1212.2192, which was the inspiration for this work.Open AccessMixed Hodge modules and real groups
Let $G$ be a complex reductive group, $\theta \colon G \to G$ an involution, and $K = G^\theta$. In arXiv:1206.5547, W. Schmid and the second named author proposed a program to study unitary representations of the corresponding real form $G_\mathbb{R}$ using $K$-equivariant twisted mixed Hodge modules on the flag variety of $G$ and their polarizations. In this paper, we make the first significant steps towards implementing this program. Our first main result gives an explicit combinatorial formula for the Hodge numbers appearing in the composition series of a standard module in terms of the Lusztig-Vogan polynomials. Our second main result is a polarized version of the Jantzen conjecture, stating that the Jantzen forms on the composition factors are polarizations of the underlying Hodge modules. Our third main result states that, for regular Beilinson-Bernstein data, the minimal $K$-type of an irreducible Harish-Chandra module lies in the lowest piece of the Hodge filtration of the corresponding Hodge module. An immediate consequence of our results is a Hodge-theoretic proof of the signature multiplicity formula of arXiv:1212.2192, which was the inspiration for this work. - Open Access
Nearby Cycle Sheaves for Symmetric Pairs
We present a nearby cycle sheaf construction in the context of symmetric spaces. This construction can be regarded as a replacement for the Grothendieck-Springer resolution in classical Springer theory.
Open AccessNearby Cycle Sheaves for Symmetric Pairs
We present a nearby cycle sheaf construction in the context of symmetric spaces. This construction can be regarded as a replacement for the Grothendieck-Springer resolution in classical Springer theory.
- 17 October 2022
- Published by American Mathematical Society (AMS) in Representation Theory
- Vol. 26 (36) , 1097-1144
Character sheaves for classical symmetric pairs
We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.
- 17 October 2022
- Published by American Mathematical Society (AMS) in Representation Theory
- Vol. 26 (36) , 1097-1144
Character sheaves for classical symmetric pairs
We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.
- Open Access
Character sheaves for classical symmetric pairs
We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.
Open AccessCharacter sheaves for classical symmetric pairs
We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.
- 31 May 2021
- Published by Springer Nature in Inventiones Mathematicae
- Vol. 226 (1) , 139-193
Koszul duality for quasi-split real groups
- 31 May 2021
- Published by Springer Nature in Inventiones Mathematicae
- Vol. 226 (1) , 139-193
Koszul duality for quasi-split real groups
- Open Access
Koszul Duality for Quasi-split Real Groups
We establish a categorical version of Vogan duality for quasi-split real groups. This proves a conjecture of Soergel in the quasi-split case.
Open AccessKoszul Duality for Quasi-split Real Groups
We establish a categorical version of Vogan duality for quasi-split real groups. This proves a conjecture of Soergel in the quasi-split case.
- Open Access
Character Sheaves for Graded Lie Algebras: Stable Gradings
In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [GVX2].
Open AccessCharacter Sheaves for Graded Lie Algebras: Stable Gradings
In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [GVX2].
- Open AccessSpringer correspondence for the split symmetric pair in type $A$In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{SL}(N),\mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain results on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.Open AccessSpringer correspondence for the split symmetric pair in type $A$In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{SL}(N),\mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain results on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.