Abstract
Abstract 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. 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 [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.
Highlights
The semi-infinite flag varieties are variants of affine flag varieties that encode the modular representation theory of a semi-simple Lie algebra, representation theory of a quantum group at roots of unity and representation theory of an affine Lie algebra at the critical level
In [51], we initiated the study of the formal model of a semi-infinite flag variety that follows the classical description of flag varieties [57, 69, 63, 58] more closely than the works already cited
We refer to this formal model of a semi-infinite flag variety as a ‘semi-infinite flag manifold’, since we hope to justify that it is ‘smooth’ in a sense
Summary
The semi-infinite flag varieties are variants of affine flag varieties that encode the modular representation theory of a semi-simple Lie algebra, representation theory of a quantum group at roots of unity and representation theory of an affine Lie algebra at the critical level They originate from the ideas of Lusztig [64] and Drinfeld, put forward by Feigin and Frenkel [24] and subsequently polished by the work of Braverman, Finkelberg, and their collaborators [29, 23, 2, 7, 8, 9, 10, 11]. Qrat K (K) (K(( )))/( (K) (K(( )))), which intertwines the natural (K(( )))⋉G (K)-actions on both sides, where G is the loop rotation
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