Affiliaton: Department of Mathematics Kyoto University
Position: Professor
Research field: Geometric representation theory
Degree: Dr. (Math. Sci.) University of Tokyo 2003
	Advisor: Hisayosi Matumoto
	Dissertation: Equivariant bundles on group completions
Postal Address: Oiwake Kita-Shirakawa Sakyo Kyoto 606-8502 JAPAN
	Department of Mathematics, Kyoto University
E-mail:

Some papers:

1. A Borel-Weil-Bott type theorem for group completions
   • Published in the Journal of Algebra 259 no.2 572--580 (2003) Article
   • I computed the total cohomology of an arbitrary line bundle of the De-Concini-Procesi compactification of an adjoint semi-simple group over complex numbers.
   • The same result was obtained independently by Alexis Thoudjem. He subsequently extended this result in several directions. See [Comptes Rendes 334 2002][Ann. Sci. ENS (4) 37 2004][Bull. Soc. Math. Fr. 135 2007]
2. Equivariant vector bundles on group completions
   • Published in the Journal f\"ur die reine und angewandte Mathematik 581 71--116 (2005) Article arXiv
   • I described the category of bi-equivariant vector bundles of an arbitrary (bi-equivariant) toroidal compactification of a reductive group over complex numbers.
   • This is (a revised form of) my thesis.
3. An exotic Deligne-Langlands correspondence for symplectic groups
   • Published in the Duke Mathematical Journal 148 no.2 305--371 (2009) Article arXiv errata
   • I gave a variant of the Lusztig formalism of the Deligne-Langlands conjecture (for Hecke algebras), which exist only for symplectic group. As a consequence, I completed the classification of simple modules of affine Hecke algebras of classical types, except for the case roughly corresponding to "small root-of-unity parameter" case.
4. Deformations of nilpotent cones and Springer correspondences
   • Published in the American Journal of Mathematics 133 no.2 519--553 (2011) Article ArXiv
   • I determined our variant of Springer representations (introduced in [3]), and compared with the ordinary one. This gives computations of Joseph polonomials (up to scalar) in many cases which are previously unknown.
   • Dedicated to Ken-ichi Shinoda and Toshiaki Shoji on the occasion of their 60th birthdays
5. Tempered modules in exotic Deligne-Langlands correspondences
   • Published in the Advances in Mathematics 226 no.2 1538--1590 (2011) Article ArXiv
   • We determined the parameters corresponding to tempered modules within the framework of [3]. As a bonus, we completed the Heckman-Opdam theory on the classification of square-integrable solutions of an integrable system, called the Lieb-McGuire system.
   • This is a joint work with Dan Ciubotaru.
6. A homological study of Green polynomials
   • Published in the Annales Scientifiques de l'Ecole Normale Superieure 48 no. 5 1035--1074 (2015), arXiv.
   • We interpreted the orthogonality relation of Green functions (which are functions roughly govern the characters of finite groups of Lie types) in terms of homological algebra. In particular, we gave a new characterization of Green polynomials. Using this, we derived several properties of Green polynomials of type BC, including a transition formula along a parameter. The concept of Kostka systems (which is an abstraction of the above characterization) and its basic properties is designed to be applicable for complex reflection groups.
7. Equivariant K-theory of semi-infinite flag manifolds and Pieri-Chevalley type formula
   • Published in the Duke Mathematical Journal 169 no.13 2421--2500 (2020) Article arXiv
   • We provide a proof that the formal model of the semi-infinite flag manifolds have fundamental nice properties as predicted by Braverman-Finkelberg. This leads us to a proposal of "semi-infinite K-theory" of the semi-infinite flag manifolds, that admits a natural action of the nil-DAHA. This provides a natural framework for a version of the standard monomial theory of a semi-infinite flag manifold, that is described by a combinatorial formula (that particularly implies that the structure constants of such K-theory are manifestly positive).
   • This is a joint work with Satoshi Naito and Daisuke Sagaki
8. Loop structure on equivariant K-theory of semi-infinite flag manifolds
   • arXiv
   • Peterson discovered a relation between the (co)homology of the affine Grassmannian of a simple (simply connected) algebraic group G and the quantum cohomology of the flag variety of G. Study of automorphic forms and (local) geometric Langlands correspondence indicates that the geometry of affine Grassmannians and semi-infinite flag manifolds are essentially the same. We formulated and proved an assertion that (roughly says) equivariant geometries of an affine Grassmannian and a semi-infinite flag manifold, and equivariant quantum geometry of a flag manifold are numerically naturally equivalent. The main theorem yields two explicit ways to calculate the structure constants of the quantum K-group of the flag manifolds. One of them resolves conjectures by Lam-Li-Mihalcea-Shimozono. Our proof of the main result significantly differs from the (original) Peterson isomorphism by Lam-Shimozono, and do not require detailed knowledge of the structure constants (logically speaking, such "formulas" were conjectures prior to this paper).

Other papers can be found here.

Last update: October 1, 2020 by