Fullname: KATO Syu
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:

- 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]

- Equivariant vector bundles on group completions
- 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.

- 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

- 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.

- 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.

- Equivariant
*K*-theory of semi-infinite flag manifolds and Pieri-Chevalley type formula
- Published in the
*Duke Mathematical Journal* **169** (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

- 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: August 18, 2020 by