Syu Kato

Poincaré–Birkhoff–Witt bases and Khovanov–Lauda–Rouquier algebras

We generalize Lusztig’s geometric construction of the PBW bases of finite quantum groups of type ADE under the framework of [VaragnoloVasserot, J. reine angew. Math. 659 (2011)]. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the KLR-algebras. This enables us to prove Lusztig’s conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases. In addition, we verify Kashiwara’s problem on the finiteness of the global dimensions of the KLR-algebras of type ADE.We generalize Lusztig’s geometric construction of the PBW bases of finite quantum groups of type ADE under the framework of [VaragnoloVasserot, J. reine angew. Math. 659 (2011)]. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the KLR-algebras. This enables us to prove Lusztig’s conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases. In addition, we verify Kashiwara’s problem on the finiteness of the global dimensions of the KLR-algebras of type ADE.

An algebraic study of extension algebras

We present simple conditions which guarantee a geometric extension algebra to behave like a variant of quasi-hereditary algebras. In particular, standard modules of affine Hecke algebras of type

Deformations of nilpotent cones and Springer correspondences

\sf{BC}

Equivariant vector bundles on group completions

, and the quiver Schur algebras are shown to satisfy the Brauer-Humphreys type reciprocity and the semi-orthogonality property. In addition, we present a new criterion of purity of weights in the geometric side. This yields a proof of Shojis conjecture on limit symbols of type

A Borel–Weil–Bott type theorem for group completions☆

\sf{B}

Demazure character formula for semi-infinite flag varieties

[T. Shoji, {\it Adv. Stud. Pure Math.} 40 (2004)], and the purity of the exotic Springer fibers [S. Kato, {\it Duke Math. J.} 148 (2009)]. Using this, we describe the leading terms of the

An exotic Deligne-Langlands correspondence for symplectic groups

C^{\infty}

Tempered modules in exotic Deligne–Langlands correspondence

-realization of a solution of the Lieb-McGuire system in the appendix. In [S. Kato, {\it Duke Math. J.} 163 (2014)], we apply the results of this paper to the KLR algebras of type

On characters and formal degrees of discrete series of affine Hecke algebras of classical types

\sf{ADE}

An exotic Springer correspondence for symplectic groups

to establish Kashwaras problem and Lusztigs conjecture.