The Poisson degeneracy locus of a flag variety

Abstract

We present a comprehensive study of the degeneracy loci of the full flag varieties of all complex semisimple Lie groups equipped with the standard Poisson structures. The reduced Poisson degeneracy loci are shown to stratify under the action of the canonical maximal torus into open Richardson varieties $\mathcal{R}_v^w$ for pairs of Weyl group elements $v \leq w$ that extend the covering relation of the Bruhat order. Four different combinatorial descriptions of those pairs are given, and it is shown that their Bruhat intervals are power sets. The corresponding closed Richardson varieties $\overline{\mathcal{R}_v^m}$ are shown to be isomorphic to $(\mathbb{C}\mathbb{P}^1)^d$ for $d \geq 0$ in a compatible way with the stratification. As a consequence, we obtain that the reduced Poisson degeneracy loci of all full flag varieties are connected, and all of their irreducible components are isomorphic to $(\mathbb{C}\mathbb{P}^1)^n$ for some $n \geq 0$; they are not equidimensional in general. Using the framework of projected Richardson varieties, these results are extended to all partial flag varieties. The top dimension of irreducible components of the reduced Poisson degeneracy locus in the full flag case is proved to be equal to the cardinality of Kostant's cascade of roots and the reflective length of the longest Weyl group element. It is shown that the Poisson degeneracy loci of flag varieties are not reduced in general.