Product structure and regularity theorem for totally nonnegative flag varieties (Q6560720)

Let \(G\) be a Kac-Moody group split over \(\mathbb{R}\), and \(I\) be the set of simple roots. By a series of works by Lusztig, the theory of total positivity on the reductive groups and their flag varieties has been generalized to arbitrary Kac-Moody groups. Many fundamental properties are established for the full flag variety, but the closure relations among cells and their geometric properties were not well developed.\N\NIn this paper, the authors introduce a new notion of positivity given by a set \(J \subset I\), and investigate the positivity for \(J\)-Schubert cells, \(J\)-Richardson varieties. With this \(J\)-positivity, they study the projected Richardson varieties to a partial flag variety denoted by \(\mathcal{P}_K\) for \(K \subset I\), and get crucial results for them. If \(J = \emptyset\), they recover the results for the full flag variety. This includes a complete description of the closure relations and geometric properties, which are new. If the Weyl group \(W_J\) is finite, then it reduces to the ordinary positivity theory for finite dimensional reductive groups. However, unlike ordinary positivity, \(J\)-positivity mixes positive structures in a more complex way, and it gets more complicated for general \(J\).\N\NThe main results are as follows.\N\NLet \({U}_{\geqslant 0}^- = \{ h_1 \pi_J(h_2)^{-1} h_2 \mid h_1 \in U_{J, \geqslant 0}^-, h_2 \in U_{\leqslant 0}^- \}\), where \(\pi_J : P_J^- \to L_J\) is the projection from the opposite parabolic subgroup \(P_J^-\) associated with \(J\) to its Levi component \(L_J\). Note that \(h_1, h_2\) are coming from different parts of positive unipotent monoids, so that \(J\)-positivity is a mixture of positivities. Then \(J\)-totally nonnegative flag variety is defined to be \(\mathcal{B}_{\geqslant 0} := \overline{{U}_{\geqslant 0}g^- \cdot B^+} \subset \mathcal{B} = G/B^+\).\N\NThere is a notion of \(J\)-twisted Bruhat order denoted \(\leqslant\) on the Weyl group \(W\), and, for \(v, w \in W\) with \(v \leqslant w\), we call \(\mathcal{B}_{v,w} =^J{\#1}{B^+} w B^+/B^+ \cap^J{\#1}{B^-} v B^+/ B^+\) an open \(J\)-Richardson variety. Then the totally positive \(J\)-Richardson variety is defined as \(\mathcal{B}_{\#1,> 0}{v, w} = \mathcal{B}_{\geqslant 0} \cap\mathcal{B}_{v,w}\).\N\NThere are 4 main theorems for the totally positive \(J\)-Richardson varieties, which are summarized in Theorems A--D in \S 1. Theorem A states there are product structures: \(\mathcal{B}_{\#1,> 0}{w_1, w_2} \simeq \mathcal{B}_{\#1,> 0}{w_1, w_3} \times \mathcal{B}_{\#1,> 0}{w_3, w_2}\) for \(w_1\leqslant w_3\leqslant w_2\). The product structure in Theorem A is not only a novel feature but also provides a powerful inductive tool for analyzing the totally positive \(J\)-Richardson varieties \(\mathcal{B}_{\#1,> 0}{w_1, w_2}\).\N\NUsing the product structure, the closure is given by \(\overline{\#1}{\mathcal{B}_{\#1,> 0}{w_1, w_2}} = \bigsqcup_{w_1', w_2'} \mathcal{B}_{\#1,> 0}{w_1', w_2'}\), where the pair \(w_1', w_2'\) moves over under the condition \(w_1\leqslant w_1'\leqslant w_2\leqslant w_2\). In fact, \(\mathcal{B}_{\#1,> 0}s{w_1, w_2}\) is a cell and a connected component of \(\mathcal{B}_{w_1, w_2}(\mathbb{R})\), so the above decomposition of the closure also gives a cell decomposition (Theorem B). Theorem C states that a projected totally positive cell \(\mathcal{P}_{K, \alpha, > 0}\) for a partial flag variety is isomorphic to a totally positive cell \(J\)-Richardson variety \(\mathcal{B}_{\#1,> 0}{w_1, w_2}\) for a different Kac-Moody group and some choice of \(J\). So it is possible to study \(\mathcal{P}_{K, \alpha, > 0}\) inductively using Theorem C. Finally, Theorem D states the closure \(\overline{\#1}{\mathcal{P}_{K, \alpha, > 0}}\) and \(\overline{\#1}{\mathcal{B}_{\#1,> 0}s{w_1, w_2}}\) are both regular CW complexes homeomorphic to closed balls. Theorem D is called a regularity theorem, which confirms a conjecture of [\textit{P. Galashin} et al., J. Am. Math. Soc. 35, No. 2, 513--579 (2022; Zbl 1493.14074)]).\N\NIn summary, a newly defined \(J\)-positivity is a useful and sharp tool for the study of positivities in partial flag varieties, and even for the full flag variety it turns out to be powerful.