Affiliation: 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
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I described the category of bi-equivariant vector bundles of
an arbitrary (bi-equivariant) toroidal compactification of a reductive
group over complex numbers.
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.
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
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.
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.
The framework presented in this paper leads to a solution to the conjecture of Shoji on Kostka polynomials of G(l,1,m) posed in 2004.
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 K-group 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
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 that) equivariant geometries of an affine Grassmannian and a semi-infinite flag manifold, and equivariant quantum geometry of a flag manifold are numerically 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 a priori knowledge of the structure constants themselves (logically speaking, all of such "formulas" were conjectures prior to this paper). Last but not the least, the finiteness conjecture on the quantum K-groups of flag varieties is also originally established in this paper.
We have developed the basic theory of the formal model of the semi-infinite flag manifolds and its parabolic analogue over the ring of integers (with 2 inverted). This reveals that the scheme constructed in a similar spirit to [7] is the most natural algebraic variety that represents semi-infinite flag manifolds, and such a naturality is guaranteed by some property in representation theory (that we also proved in a suitable generality). Such a tight connection between nice properties hold in much greater generality, but I think this paper is the first to tell such a storyline. Both properties are rather abstract, and hence it might be interesting to appreciate that guises of such properties turns into numerical consequences.
Global sections of numerically effective line bundles over the cotangent bundle of the flag varieties of general linear groups are known to represent Hall-Littlewood polynomials by the works of Brylinski and Broer around 1990. In an attempt to generalize this, a series of conjectures are posed by Shimozono and Weyman around 2000. After that, these conjectures are refined and pursued by Chen and Blasiak-Morse-Pun (among others), and particularly the latter established the numerical portion of the Shimozono-Weyman conjecture. In particular, their works reveal that the polynomials arising from these considerations (called Catalan polynomials) forms a very nice family. In this paper, we have established the vanishing part (that connects the numerical part to the geometric description) of the Shimozono-Weyman conjecture and its generalizations, that were practically untouched in the above line of works, by introducing varieties called XΨ and reduce the assertion to the analysis of these varieties. In short, XΨ is the variety whose Borel-Weil theory realizes Catalan polynomials.
The variety XΨ looks to be a better object than I have originally thought. For example, we showed that their Betti cohomology naturally realize the chromatic symmetric function of a unit interval graph in this paper.
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