Total positivity in twisted flag varieties

TL;DR

This paper extends the theory of total positivity to twisted flag varieties in Kac-Moody groups, showing they decompose into cells with regular CW complex closures, and explores related geometric and combinatorial properties.

Contribution

It generalizes total positivity results from ordinary to twisted flag varieties, establishing cell decompositions and CW complex structures, and connects these to canonical bases and Bruhat cells.

Findings

Totally nonnegative parts of twisted flag varieties decompose into cells.

Closures of these cells form regular CW complexes.

Links of identities in nonnegative double Bruhat cells are regular CW complexes.

Abstract

Let $G$ be a Kac-Moody group, split over $\mathbb R$. The totally nonnegative part of $G$ and its (ordinary) flag variety $G/B^+$ was introduced by Lusztig. It is known that the totally nonnegative parts of $G$ and $G/B^+$ have remarkable combinatorial and topological properties. In this paper, we consider the totally nonnegative part of the $J$-twisted flag variety $G/{}^J B^+$, where ${}^J B^+$ is the Borel subgroup opposite to $B^+$ in the standard parabolic subgroup $P_J^+$ of $G$. The $J$-twisted flag varieties include the ordinary flag variety $G/B^+$ as a special case. Our main result show that the totally nonnegative part of $G/{}^J B^+$ decomposes into cells, and the closure of each cell is a regular CW complex. This generalizes the work of Galashin-Karp-Lam \cite{GKL22} and the joint work of Bao with the first author \cite{BH24} for ordinary flag varieties. As an application, we deduce that the totally nonnegative part of the double flag variety $G/B^+ \times G/B^-$ with respect to the diagonal $G$-action has similar nice properties. We also establish some connections between the totally nonnegative part of the double flag with the canonical basis of the tensor product of a lowest weight module with a highest weight module of $G$. As another application, we show that the link of identity in a totally nonnegative reduced double Bruhat cell of $G$ is a regular CW complex. This generalizes the work of Hersh \cite{Her14} on the link of $U_{\geq0}^-$ and gives a positive answer to an open question of Fomin and Zelevinsky.