The wonderful compactication and the universal centralizer

February 26, 2018 PIRSA:18020085 Scientific Series Mathematical physics

DOI10.48660/18020085

Balibanu, A. (2018). The wonderful compactication and the universal centralizer. Perimeter Institute. https://pirsa.org/18020085

Balibanu, Ana. The wonderful compactication and the universal centralizer. Perimeter Institute, Feb. 26, 2018, https://pirsa.org/18020085

@misc{ pirsa_PIRSA:18020085,
  doi = {10.48660/18020085},
  url = {https://pirsa.org/18020085},
  author = {Balibanu, Ana},
  keywords = {Mathematical physics},
  language = {en},
  title = {The wonderful compactication and the universal centralizer},
  publisher = {Perimeter Institute},
  year = {2018},
  month = {feb},
  note = {PIRSA:18020085 see, \url{https://pirsa.org}}
}

Abstract

Let $G$ be a complex semisimple algebraic group of adjoint type and $\overline{G}$ the wonderful compacti
cation. We show that the closure in \overline{G} of the centralizer $G^e$ of a regular nilpotent $e \in Lie(G)$ is isomorphic to the Peterson variety. We generalize this result to show that for any regular $x \in Lie(G)$ , the closure of the centralizer $G^x$ in $\overline{G}$ is isomorphic to the closure of a general $G^x$ -orbit in the flag variety. We consider the family of all such centralizer closures, which is a partial compactication of the universal centralizer. We show that it has a natural log-symplectic Poisson structure that extends the usual symplectic structure on the universal centralizer.