I am a Professor in the School of Mathematics and Statistics at the University of Melbourne.

I am a member of the Representation Theory Group and the Number Theory Group. My work is in the areas of real groups and the Langlands program. I have also worked on more foundational questions on perverse sheaves and D-modules including the microlocal point of view.

Here is my short CV.

My research falls in several related areas.

Geometric Langlands.

In joint work with Frenkel and Gaitsgory we establish the geometric Langlands conjecture for . This work consists of publications [21] and [22]; see also [14]. The conjecture originated in the work of Drinfeld and was made explicit in the work of Laumon.

In joint work [22] with Mirković, announced in [19], we establish the geometric Satake equivalence. It provides a construction of the dual group as a split group scheme over the integers from perverse sheaves on the affine Grassmannian.

Real Groups.

In joint work [12, 15, 18, 20]  with Schmid we prove a conjecture of Barbasch and Vogan and several related conjectures of Kashiwara. The Barbasch-Vogan conjecture postulates that two invariants of representations, one algebraic and one analytic, coincide under the Sekiguchi correspondence. On the way to the proof we establish two geometric character formulas for representations [18]. This work relies on Matsuki correspondence for sheaves, conjectured by Kashiwara and established in joint work [7] with Mirković  and Uzawa. The proof of the Barasch-Vogan conjecture amounts to showing that that microlocalization of the Matsuki correspondence yields the Sekiguchi correspondence. It is perhaps the first application of O-minimal structures outside of logic.

Vogan has established a remarkable duality between blocks of representations of a real group and a block of representations of a particular real form of the dual group. The statement is on the level of K-groups.  In joint work with Bezrukavnikov [37] we elevate it, as was conjectured by Soergel, to the level of categories of representations by establishing it as a Koszul duality when one of the real forms is quasi-split.

In more recent joint work [26,29] with Schmid we have formulated a series of conjectures which postulate the existence of an infinite dimensional Hodge structure on representations of real groups (with real infinitesimal character). This conjectural structure should arise from Saito’s Hodge modules.These conjectures have now been largely proved in a series of papers with Dougal Davis, preprints [5,6,7]. In particular we show that the unitarity of the representation can be read off from its Hodge structure.

Perverse sheaves and D-modules.

My earliest work, going back to my graduate student days, is the joint work with MacPherson on the structure of the category of perverse sheaves [3, 5]. A goal here is to give a microlocal description of perverse sheaves. In joint work [28] with Kashiwara, announced in [25], we prove the major conjecture, the codimension-three conjecture, in the subject. Combined with my earlier work with Gelfand and MacPherson, we are reduced to understanding interactions between Lagrangians in codimension two. Unfortunately this work still only exists as a preprint  [1].

Character sheaves.

Lusztig has introduced the remarkable notion of character sheaves which he consequently classified. In old joint work [6] with Mirković we show that the character sheaves of Lusztig are characterized by the property that their characteristic variety is nilpotent. In joint work with Xue our goal is to classify the character sheaves on a graded Lie algebra. We have completed this task in the case of symmetric spaces in [38,39]. We have also largely completed the classification of character sheaves for graded Lie algebras in preprints [41] and the preprint [4].

Research Support

My research is currently partially supported by Australian Research Council grants DP180101445 and FL200100141 and also by the Research Council of Finland grant 354948. Previously I was supported by Australian Research Council grant DP150103525 as well as grants from the Research Council of Finland. In the past I was also supported by the National Foundation of the United States, the National Security Agency of the United States, Defense Advanced Research Project Agency of the United States, and the United States Air Force Office of Sponsored Research.

Preprints

1. 1. D. Davis and K. Vilonen, Unitary representations of real groups and localization theory for Hodge modules. arXiv:2309.13215.
   2. D. Davis and K. Vilonen, Hodge filtrations of tempered Hodge modules. arXiv:2206.09091.
   3. D. Davis and K. Vilonen, Mixed Hodge modules and real groups. arXiv:2202.08797.
   4. K. Vilonen and T. Xue, Invariant systems and character sheaves for graded Lie algebras. arXiv:2111.08403.
   5. K. Vilonen and T. Xue, A note on Hessenberg varieties. arxiv: 2101.08652.
   6. M. Grinberg, K. Vilonen, and T. Xue, Nearby Cycle Sheaves for Stable Polar Representations, arXiv:2012.14522.
   7. S. Gelfand, R. MacPherson, and K. Vilonen, Microlocal Perverse Sheaves, arXiv:0509440.

Publications

1. 1. K. Vilonen, and T. Xue, Character sheaves for graded Lie algebras: stable gradings, Adv. Math. 417 (2023), arXiv:2012.08111.
   2. M. Grinberg, K. Vilonen, and T. Xue, Nearby cycle sheaves for symmetric pairs. Amer. J. Math. 145 (2023), no.1, 1-63. arXiv.
   3. K. Vilonen and T. Xue, Character sheaves for symmetric pairs: special linear groups.  Trans. Amer. Math. Soc. 376 (2023) no. 2, 837-853. arXiv.
   4. K. Vilonen, and T. Xue, Character sheaves for classical symmetric pairs. With an appendix by Dennis Stanton. Represent. Theory 26 (2022), 1097-1144.
   5. R. Bezrukavnikov and K. Vilonen, Koszul Duality for Quasi-split Real Groups, Inventiones Mathematicae, 226 (2021),139-193, arXiv:1510.08343.
   6. T.-H. Chen, K. Vilonen and T. Xue, Springer correspondence, Hyperelliptic curves, and cohomology of Fano varieties. Math. Res. Lett. 27 (2020), No. 5, 1281-1323. arXiv.
   7. T.-H. Chen, K. Vilonen and T. Xue, Hessenberg varieties, intersections of quadrics, and the Springer correspondence, Trans. Amer. Math. Soc. 373 (2020), No. 4, 2427-2461. arXiv. MR4069224
   8. J. Taskinen and K. Vilonen, Cartan theorems for Stein manifolds over a discrete valuation base, J. Geom. Anal. 29 (2019), No. 1, 577-615. arXiv. MR3897027
   9. T. -H. Chen, K. Vilonen and T. Xue, Springer correspondence for the split symmetric pair in type A, Compos. Math. 154 (2018), No. 11, 2403-2425. arXiv. MR3867304
   10. I. Mirkovic and K. Vilonen, Erratum for “Geometric Langlands duality and representations of algebraic groups over commutative rings”, Ann. of Math. (2) 188 (2018), No. 3, 1017-1018. MR3866890
   11. T.-H. Chen, K. Vilonen and T. Xue, On the cohomology of Fano varieties and the Springer correspondence, With an appendix by Dennis Stanton. Adv. Math. 318 (2017), 515-533. arXiv. MR3689749
   12. T. Xue and K. Vilonen, The null-cone and cohomology of vector bundles on flag varieties, Represent. Theory 20 (2016), 482-498. MR3589334
   13. W. Schmid and K. Vilonen, Hodge theory and unitary representations, Representations of reductive groups, Progr. Math. 312, Birkhäuser/Springer, Cham (2015), 443-453. MR3495806
   14. M. Kashiwara and K. Vilonen, Microdifferential systems and the codimension-three conjecture, Ann. of Math. (2) 180 (2014), 573-620. MR3224719
   15. K. Vilonen and G. Williamson, Characteristic cycles and decomposition numbers, Math. Res. Lett. 20 (2013), 359-366. MR3151652
   16. W. Schmid and K. Vilonen, Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences, International Press (2011), 397-420, arXiv. MR3050836
   17. M. Kashiwara and K. Vilonen, On the codimension-three conjecture, Proc. Japan, Acad., Ser. A 86 (2010), 154-158. MR2780007
   18. I. Mirkovic and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Annals of Math., 166 (2007), 95-143. MR2342692
   19. M. Emerton, D. Nadler, and K. Vilonen, A geometric Jacquet functor, Duke Math. J. 125 (2004), No. 2, 267-278. arXiv. MR2096674
   20. E. Frenkel, D.Gaitsgory, and K. Vilonen, On the geometric Langlands conjecture, Journal of Amer. Math. Soc., 15 (2002), No. 2, 364-417. MR1887638
   21. E. Frenkel, D. Gaitsgory and K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Annals of Math. 153 (2001), 699-748. arXiv. MR1836286
   22. W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive groups, Annals of Math. 151 (2000), 1071-1118. arXiv. MR1779564
   23. I. Mirkovic and K. Vilonen, Perverse Sheaves on Affine Grassmannians and Langlands Duality, Mathematical Research Letters 7 (2000), 13-24. MR1748284
   24. W. Schmid and K. Vilonen, On the geometry of nilpotent orbits, Asian J. Math., vol 8, no 1 (1999), 233-274, A Special Volume in Honour of Sir Michael Atiyah.
       Reprinted in – Surveys in differential geometry, Surv. Differ. Geom., VII, Int. Press (2000), 565-623. MR1919437
   25. K. Vilonen, Geometric methods in representation theory, Representation theory of Lie groups, Amer. Math. Soc. (1998), 241-290. MR1737730
   26. K. Vilonen, Topological methods in representation theory, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 595–604. MR1648108
   27. W. Schmid and K. Vilonen, Two geometric character formulas for reductive Lie groups, Journal of AMS, 11 (1998), No. 2, 799-876. MR1612634
   28. E. Frenkel, D.Gaitsgory, D. Kazhdan, and K. Vilonen, Geometric realization of Whittaker Functions and the Langlands Conjecture, Journal of AMS, 11 (1998), No. 2, 451-484. MR1484882
   29. S. Gelfand, R. MacPherson, and K. Vilonen, Perverse sheaves and quivers, Duke Math. Journal 83 (1996), 621-643. MR1390658
   30. W. Schmid and K. Vilonen, Characteristic cycles of constructible sheaves, Invent. Math. 124 (1996), 451-502. MR1369425
   31. W. Schmid and K. Vilonen, Characters, characteristic cycles, and nilpotent orbits. Geometry, Topology & Physics (1995), 329-340. MR1358623
   32. W. Schmid and K. Vilonen, Weyl group actions on Lagrangian cycles and Rossmann’s formula. Noncompact Lie groups and some of their applications (San Antonio, TX, 1993), 243–250, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 429, Kluwer Acad. Publ., Dordrecht, 1994. MR1306528
   33. K. Vilonen, Perverse sheaves and finite-dimensional algebras, Trans. Amer. Math. Soc. 341 (1994), 665-676. MR1135104
   34. W. Schmid and K. Vilonen, Characters, fixed points and Osborne’s conjecture, Representation theory of groups and algebras, Amer. Math. Soc. (1993), 287-303. MR1216196
   35. I. Mirkovic, T. Uzawa and K. Vilonen, Matsuki correspondence for sheaves, Invent. Math. 109 (1992), No. 2, 231-245. MR1172690
   36. I. Mirkovic and K. Vilonen, Characteristic varieties of character sheaves, Invent. Math. 93 (1988), No. 2, 405-418. MR0948107
   37. R. MacPherson and K. Vilonen, Perverse sheaves with singularities along the curve , Comment. Math. Helv. 63 (1988), No. 1, 89-102. MR0928029
   38. R. Mirollo and K. Vilonen, Bernstein-Gel’fand-Gel’fand reciprocity on perverse sheaves, Ann. Sci. Ecole Norm. Sup. (4) 20 (1987), No. 3, 311-323. MR0925719
   39. R. MacPherson and K. Vilonen, Elementary construction of perverse sheaves, Invent. Math. 84 (1986), No. 2, 403-435. MR0833195
   40. K. Vilonen, Intersection homology D-module on local complete intersections with isolated singularities, Invent. Math. 81 (1985), No. 1, 107-114. MR0796193
   41. R. MacPherson and K. Vilonen, Elementary construction of perverse sheaves (French), C.R.Acad. Sci. Paris Ser. I. Math. 299 (1984), No, 10, 443-446. MR0768077

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Peter Hall Building
School of Mathematics & Statistics
University of Melbourne
Parkville, Victoria, 3010
Australia

Tel.: +61 3 8344 1100
E-mail: kari.vilonen@unimelb.edu.au

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