부산대학교 도서관

Suppose that one has a connected, complex, semisimple, linear, algebraic group $G$ with a fixed Borel subgroup $B\subseteq G$. Each unipotent element $u\in G$ then determines a Springer fiber $(G/B)_u\subseteq G/B$ as follows: $$ (G/B)_u\coloneq \{gB\in G/B: g^{-1}ug\in B\}. $$ This projective variety enjoys a number of remarkable properties, both representation-theoretic and topological. \par In the interest of generalizing aspects of the above setup, one might replace $B$ with a different closed subgroup $K\subseteq G$ and consider $$ (G/K)_u\coloneq \{gK\in G/K: g^{-1}ug\in K\}\subseteq G/K. $$ In this paper the authors do precisely this when $u$ is regular, $G=\roman{SL}_n(\Bbb{C})$, and $K$ is one of $\roman{SO}_n(\Bbb{C})$ and $\roman{Sp}_n(\Bbb{C})$. Recognizing that $(G/K)_u$ is no longer projective in these cases, the authors consider the closure $X^{u}$ of $(G/K)_u$ in a wonderful compactification of $G/K$. They subsequently use a Białynicki-Birula cell decomposition of $X^u$ to determine its Poincaré polynomial. Other results include a description of the partial order on the Białynicki-Birula cells when $K=\roman{SO}_n(\Bbb{C})$.