210409B Li Yu

April 9, 2021

Title: Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra

Speaker: Li Yu (Junior talk, 4:30pm)


Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $bar H$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $fh$ of $H$, we define an analogous compactification $bar fh$ of $fh$, to be referred to as the wonderful compactification of $fh$. We establish a bijection between the set of irreducible components of the boundary $bar fh - fh$ of $fh$ and the set of maximal closed root subsystems of the root system for $(G, H)$ of rank $r - 1$, where $r$ is the dimension of $fh$. An algorithm based on Borel-de Siebenthal theory that constructs all such root subsystems is given. We prove that each irreducible component of $bar fh - fh$ is isomorphic to the wonderful compactification of a Lie subalgebra of $fh$ and is of dimension $r - 1$. In particular, the boundary $bar fh - fh$ is equidimensional. We
describe a large subset of the regular locus of $bar fh$. As a consequence, we prove that $bar fh$ is a normal variety.

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