On the Knutson-Woo-Yong maps and some poisson homogeneous spaces

Abstract

Richardson varieties and Schubert varieties associated to a connected complex semisimple Lie group $G$ are certain closed subvarieties of the flag variety $G/B$ of $G$, and they play very important roles in Lie theory. In a paper in 2013, A. Knutson, A. Woo, and A. Yong introduced certain maps that relate the singularities of Richardson varieties with that of Schubert varieties. In this thesis, the Knutson-Woo-Yong maps were studied from the points of view of Poisson geometry, and applications of the Knutson-Woo-Yong maps to Poisson geometry were obtained.

In the first part of the thesis, the Knutson-Woo-Yong maps were generalized to certain homogeneous space $G/Q$ of $G$ and were shown to be Poisson isomorphisms with respect to a naturally defined Poisson structure $\pi_{\sG/\sQ}$ on $G/Q$. In the second part of the thesis, the generalized Knutson-Woo-Yong maps for $G/Q$ were used to define coordinate charts on $G/Q$, and it was shown that in {\it each} of such coordinate charts, the Poisson structure $\pi_{\sG/\sQ}$ on $G/Q$ gives rise to a polynomial Poisson algebra that is a {\it symmetric Cauchon-Goodearl-Letzter (CGL) extension (of ${\mathbb{C}}$)}. Primary examples of homogeneous spaces $G/Q$ of $G$ studied in the thesis were the orbits in the double flag variety $(G/B) \times (G/B_-)$ for the diagonal action of $G$, which include the flag variety $G/B$ as the closed orbit, and the symmetric Poisson CGL extensions arising from these examples are certain {\it $T$-mixed products} of the symmetric Poisson CGL extensions defined by generalized Bruhat cells, where $T$ is a maximal torus of $G$.

Symmetric CGL extensions form a particular class of polynomial Poisson algebras that are iterated Poisson Ore extensions and have compatible torus actions. They have been recently introduced and studied by K. Goodearl and M. Yakimov in the context of cluster algebras and quantum groups. The results of the thesis show that important Poisson varieties in Lie theory can be covered by symmetric Poisson CGL extensions.