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Affine flag varieties and quantum symmetric pairs

Zhaobing Fan, Chun-Ju Lai, Yiqiang Li, Li Luo, Weiqiang Wang

TL;DR

The paper develops a comprehensive geometric realization of eight quantum symmetric pairs of affine type C by stabilizing Lusztig–Schur algebra data from affine flag varieties. It constructs four Lusztig subalgebras and their idempotented versions for affine $\mathfrak{sl}_n$ and four corresponding coideal subalgebras for affine $\mathfrak{gl}_n$, yielding eight affine quantum symmetric-pair frameworks with monomial and canonical bases. A central advance is the demonstration of positivity properties of canonical bases under multiplication and comultiplication for the affine $\mathfrak{sl}_n$ coideals, alongside explicit stabilization algebras $\dot{\mathbf K}^{\mathfrak{c}}_n$ that relate to $\dot{\mathbf U}(\widehat{\mathfrak{gl}}_n)$. The work also develops multiple variants (including $\jmath\imath$, $\imath\jmath$, and $\imath\imath$) of quantum symmetric pairs, augmenting the geometric toolkit for modular representations and Langlands dual perspectives in classical types.

Abstract

The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$. In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type $C$. We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine $\mathfrak{sl}$ and $\mathfrak{gl}$ types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine $\mathfrak{sl}$ type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine $\mathfrak{gl}$ and its canonical basis.

Affine flag varieties and quantum symmetric pairs

TL;DR

The paper develops a comprehensive geometric realization of eight quantum symmetric pairs of affine type C by stabilizing Lusztig–Schur algebra data from affine flag varieties. It constructs four Lusztig subalgebras and their idempotented versions for affine and four corresponding coideal subalgebras for affine , yielding eight affine quantum symmetric-pair frameworks with monomial and canonical bases. A central advance is the demonstration of positivity properties of canonical bases under multiplication and comultiplication for the affine coideals, alongside explicit stabilization algebras that relate to . The work also develops multiple variants (including , , and ) of quantum symmetric pairs, augmenting the geometric toolkit for modular representations and Langlands dual perspectives in classical types.

Abstract

The quantum groups of finite and affine type admit geometric realizations in terms of partial flag varieties of finite and affine type . Recently, the quantum group associated to partial flag varieties of finite type is shown to be a coideal subalgebra of the quantum group of finite type . In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type . We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine and types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine and its canonical basis.

Paper Structure

This paper contains 65 sections, 146 theorems, 410 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The goal: affine type $C$
  4. An overview
  5. An overview of Part \ref{['part1']}
  6. An overview of Part \ref{['part2']}
  7. An overview of Part \ref{['part3']}
  8. The organization
  9. Affine flag varieties, Schur algebras, and Lusztig algebras
  10. Constructions in affine type $A$
  11. Lattice presentation of affine flag varieties of type $A$
  12. Monomial basis for quantum affine $\mathfrak{sl}_n$
  13. Algebras $\mathbf{U}_n$ and $\dot{\mathbf{U}}_n$
  14. Lattice presentation of affine flag varieties of type $C$
  15. Affine complete flag varieties of type $C$
  16. ...and 50 more sections

Key Result

Theorem A

The algebra $\mathbf{U}^{\mathfrak{c}}_{n,d}$ admits a monomial basis $\{\zeta_A \vert A \in \Xi_{n,d}^{{ap}} \}$ and a canonical basis $\{ \{A\}_d \vert A \in \Xi_{n,d}^{{ap}} \}$, which are compatible with the corresponding bases in ${\mathbf S}^{\mathfrak{c}}_{n,d}$ under the inclusion $\mathbf{

Figures (4)

  • Figure 1: Dynkin diagram of type $A^{(1)}_{2r+1}$ with involution of type $\jmath\jmath \equiv \mathfrak{c}$.
  • Figure 2: Dynkin diagram of type $A^{(1)}_{2r}$ with involution of type $\jmath\imath$.
  • Figure 3: Dynkin diagram of type $A^{(1)}_{2r}$ with involution of type $\imath\jmath$.
  • Figure 4: Dynkin diagram of type $A^{(1)}_{2r-1}$ with involution of type $\imath\imath$.

Theorems & Definitions (225)

  • Theorem A: Theorem \ref{['CB-Udn']}
  • Theorem B: Theorem \ref{['thm:QSP']}
  • Theorem C: Theorem \ref{['thm:iCB-Unc']}, Theorem \ref{['thm:positivity-Unc']}
  • Theorem D: Theorems \ref{['Kj-bases']}, \ref{['thm:Psi:c']}
  • Theorem E: Propositions \ref{['prop-cop-alg-hom']}, \ref{['prop-cop-coass']}, Remark \ref{['rem:KnQSP']}
  • Theorem F: Proposition \ref{['prop:sq']}, Theorems \ref{['Kji-sq']}, \ref{['Kij-bases']}, \ref{['Kii-bases']}
  • Definition 2 .2.1
  • Lemma 2 .2.2
  • proof
  • Proposition 2 .2.3
  • ...and 215 more