Research
My research falls in several related areas.
Geometric Langlands.
In joint work with Frenkel and Gaitsgory we establish the geometric Langlands conjecture for 
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.