Let $G$ be a reductive, linear algebraic group (variety) over an algebraically closed field $\Bbbk$ of characteristic zero. If $G$ is connected, I know from Humphrey's book that for any Borel subgroup $B\subseteq G,$ there is some opposite Borel subgroup $B^-$ such that $T=B\cap B^-$ is a Torus, and denoting by $U$ and $U^-$ the unipotent radicals of $B$ and $B^-$ respectively, there is an open immersion \begin{align*} \beta: U\times T\times U^- &\longrightarrow G \\ (u,t,v) &\longmapsto utv \end{align*} I am now in a situation where my group is not connected, and I am wondering if the above still holds (to some degree). Searching MO, I found this post which suggests that the above holds for "split reductive" groups. I am not familliar with the term "split reductive", and searching for a definition has yielded very little so far, but wikipedia seems to suggest that this is some issue of separability which I would not have to worry about over an algebraically closed field of characteristic zero.

So, my question is: If $G$ is not connected, can something similar or even the same be said? A reference would be the icing on the cake, I would love to read more about this. Thanks a lot in advance.

dolike the sound of that. Seriously though, why can't I find a single book with the definition in it? $\endgroup$5more comments