Semi-infinite Lakshmibai--Seshadri paths and level-zero extremal weight modules over twisted quantum affine algebras

TL;DR

The paper extends the semi-infinite Lakshmibai--Seshadri path model to general twisted affine types and proves that the crystal of level-zero extremal weight modules $\mathcal{B}(\lambda)$ is isomorphic to the crystal of semi-infinite LS paths $\tfrac{\infty}{2}\mathrm{LS}(\lambda)$ for all $\lambda\in P^0_+$. It darkens the untwisted-proven framework by adapting the semi-infinite length, Peterson coset machinery, and crystal operators to twisted types, including dual untwisted types and the special $A_{2\ell}^{(2)}$ case (via $D_{\ell+1}^{(2)}$). Key steps include defining twisted LS paths, establishing the projection and stability lemmas, and proving a Sublemma that characterizes when nontrivial $a$-chains exist in the twisted semi-infinite Bruhat graph. By reducing to known untwisted results and carefully handling parity conditions for short roots, the authors derive the twisted-type isomorphism and lay groundwork for connections to Demazure submodules and nonsymmetric Macdonald polynomials in this setting.

Abstract

In this paper, we study level-zero extremal weight modules over twisted quantum affine algebras. To this end, we introduce semi-infinite Lakshmibai--Seshadri paths associated with a level-zero dominant integral weight $λ$. We then show that the set $\tfrac{\infty}{2}\mathrm{LS}(λ)$ of semi-infinite LS paths of shape $λ$ is isomorphic, as a crystal, to the crystal basis $\mathcal{B}(λ)$ of the corresponding level-zero extremal weight module $V(λ)$.

Theorems & Definitions (61)

• Definition 3.1.1: $a$-chain of shape $\lambda$
• Definition 3.1.2: semi-infinite LS paths
• Theorem 3.2.1: Crystal structure, INS
• Lemma 3.2.2: Projection lemma, INS
• Lemma 3.2.4: Comparison Lemma 2
• Lemma 3.2.5: Stability Lemma, INS
• Theorem 3.3.1: Isomorphism Theorem
• Lemma 3.3.2: Connected components of $\tfrac{\infty}{2}\mathrm{LS}(\lambda)$, INS
• Lemma 3.3.3: Special elements, INS
• Lemma 3.3.5: Identification of identity components, INS