Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\mathbb K$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind)scheme structure, its projective coordinate ring has a $\mathbb Z$-model, and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. {\bf 371} no.2 (2018)]) when $\mathsf{char} \, \mathbb K =0$ or $\gg 0$, and the higher cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$. Some particular cases of these results play crucial roles in our proof [K, arXiv:1805.01718] of a conjecture by Lam-Li-Mihalcea-Shimozono [J. Algebra {\bf 513} (2018)] that describes an isomorphism between affine and quantum $K$-groups of a flag manifold.Comment: 58pages, v4: added Appendix A on the Kempf type vanishing theorem, v5: corrected extremal condition of Theorem C, added Appendix B on the global sections of Richardson varieties, and many minor additions/improvement

Deformations of nilpotent cones and Springer correspondences

Let G = Sp (2n) be the symplectic group over Z. We present a certain kind of deformation of the nilpotent cone of G with G-action. This enables us to make direct links between the Springer correspondence of sp_{2n} over C, that over characteristic two, and our exotic Springer correspondence. As a by-product, we obtain a complete description of our exotic Springer correspondence.Comment: v6. 30pp, fixed 7.8 2) and 10.3 (v5), corrected typos, changed affiliation, and added thanks to Midori, to appear in Amer. J. Math.

PBW bases and KLR algebras

We generalize Lusztig's geometric construction of the PBW bases of finite quantum groups of type $\mathsf{ADE}$ under the framework of [Varagnolo-Vasserot, 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 the $\mathsf{ADE}$ case. In addition, we verify Kashiwara's problem on the finiteness of the global dimensions of the KLR-algebras of type $\mathsf{ADE}$.Comment: 35pp, v3: separate out arXiv:1207.4640 in order to stress the Shoji conjecture and "affine quasi-hereditary categories". v4: properties of the Saito reflection functors gathered (Theorem 3.6), v5: Modified arguments to correct an error in the proof of the desired property of the PBW basis in section 4. The last page contains more detailed descriptio