Why is this universal cover isomorphic to $\text{SL}_2(\mathbb{C})$?

Question

In Lurie's proof of the Borel-Weil theorem http://www.math.harvard.edu/~lurie/papers/bwb.pdf, he states that the universal cover of a Levi factor of $S$, where $C\cup U'B=SB$, is isomorphic to $\text{SL}_2(\mathbb{C})$. Why is this the case? Could someone please give a reference for this? Here $C$ is the Bruhat cell corresponding to a simple root.

Answer 1

There is a lot going on in that sentence. A general reference for the structure theory of reductive algebraic groups would be the book by Springer, or Borel, or any of the online notes circulating. The ones by Herzig and Murnaghan are both useful for this stuff. Specifically, see theorem 180 and the rest of Ch. 6 in these notes. The example to keep in mind that makes reasoning about these things easy is the standard block-upper-triangular parabolic subgroups of $\mathrm{GL}_n$.

Anyway, by construction the Lie algebra of $S$ is isomorphic to $\mathfrak{sl}_2,\mathbb{C})$, so by the theory of Lie groups, $S$ is covered by $\mathrm{SL}(2,\mathbb{C})$. In more detail, if $\alpha=\epsilon_1-\epsilon_2$ is the positive root in question for $\mathrm{GL}_n$, then I think $S$ is can be taken to be generated by the upper-triangular matrices with 1s on the diagonal and and the matrices with 1s on the diagonal, and a single other entry at $(2,1)$, corresponding to $-\alpha$.