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On quantum $K$-groups of partial flag manifolds

Syu Kato

Abstract

We show that the equivariant small quantum $K$-group of a partial flag manifold is a quotient of that of the full flag manifold in a way it respects the Schubert basis. This is a $K$-theoretic analogue of the parabolic version of Peterson's theorem [Lam-Shimozono, Acta Math. {\bf 204} (2010)] that exhibits different shape from the case of quantum cohomology. Our quotient maps send some of the Novikov variables to $1$, and its geometric meaning is unclear in quantum $K$-theory. This paper can be seen as a continuation of [K, Ann. of Math. (to appear), and Forum of Math., Pi {\bf 9} (2021)].

On quantum $K$-groups of partial flag manifolds

Abstract

We show that the equivariant small quantum -group of a partial flag manifold is a quotient of that of the full flag manifold in a way it respects the Schubert basis. This is a -theoretic analogue of the parabolic version of Peterson's theorem [Lam-Shimozono, Acta Math. {\bf 204} (2010)] that exhibits different shape from the case of quantum cohomology. Our quotient maps send some of the Novikov variables to , and its geometric meaning is unclear in quantum -theory. This paper can be seen as a continuation of [K, Ann. of Math. (to appear), and Forum of Math., Pi {\bf 9} (2021)].

Paper Structure

This paper contains 12 sections, 26 theorems, 82 equations.

Table of Contents

  1. Preliminaries
  2. Groups, root systems, and Weyl groups
  3. Quasi-map spaces
  4. Graph and map spaces and their line bundles
  5. Equivariant quantum $K$-group of $\mathscr B_{\mathtt{J}}$
  6. A description of the quantum $K$-groups
  7. $K$-groups of semi-infinite partial flag manifolds
  8. $\mathscr{Q}_{\mathtt{J}} ( \beta, w )$ has at worst rational singularities
  9. Comparison of equivariant $K$-groups
  10. Comparison between equivariant quantum $K$-groups
  11. Comparison with affine Grassmannians
  12. Examples: $G = \mathop{SL} ( 3 )$

Key Result

Theorem A

There exists a surjective morphism of algebras that sends a Schubert basis to a Schubert basis. Moreover, if $B \subset P' \subset P$ is an intermediate standard parabolic subgroup, then the above algebra map factors through $qK_H ( G / P' )$.

Theorems & Definitions (42)

  • Theorem A: $\doteq$ Theorem \ref{['qKquot']}
  • Definition 1.1: Drinfeld-Plücker data
  • Theorem 1.2: Drinfeld, see FM99BFGM and Kat18d
  • Theorem 1.3: Braverman-Finkelberg BF14aBF14bBF14c, see also Kat18c § 4.1
  • Theorem 1.4: IMT15 and ACT18
  • proof
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3: Kat18d Corollary C and Appendix A
  • Theorem 2.4: KNS17 Corollary 4.31 and Kat18d Appendix A
  • ...and 32 more