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
E-mail:

The contents of some of my papers (occasionally updated):

1. 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.
2. 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.
3. 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
4. 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.
5. 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.
   • 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.
6. Loop structure on equivariant K-theory of semi-infinite flag manifolds
   • Accepted for publication in the Annals of Mathematics Article 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 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 (full) flag varieties is also established by this paper.
7. Frobenius splitting of Schubert varieties of semi-infinite flag manifolds
   • Published in the Forum of Mathematics, Pi 9 e5 56pp (2021) Article arXiv
   • 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.
8. A geometric realization of Catalan functions
   • arXiv
   • 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 X_Ψ and reduce the assertion to the analysis of these varieties. In short, X_Ψ is the variety whose Borel-Weil theory realizes Catalan polynomials.
   • The variety X_Ψ 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.
9. Categorification of k-Schur functions and refined Macdonald positivity
   • arXiv
   • k-Schur polynomials are introduced by LaPointe-Lascoux-Morse (2003) in order to formulate positivity behaviour of (transformed) Macdonald polynomial beyond the Schur positivity when the latter was about to be resolved by Haiman. In this paper, we establish a conjecture of Chen-Hamian that asserts that k-Schur polynomials are precisely the characters of simple modules of an additive category obtained by imposing homological orthogonality condition in the framework of [5]. This implies that the expansion coefficients of (transformed) Macdonald polynomials with respect to appropriate k-Schur polynomials are positive, and hence establishes the framework anticipated by LaPointe-Lascoux-Morse in the above paper by offering the reasoning why it must be true.
   • The existence of the Chen-Haiman conjecture was communicated to me by Thomas Lam around 2012.

Other papers can be found here.

Last update: June 19th 2025 by