Okay, I will try this again, last time people didn't like how I asked it :)
So let's say $G$ is a reductive complex algebraic group; it could be $GL_n(\mathbb{C})$ if that makes you happy (and in fact that's my case of interest!)
Let $T$ be the maximal torus of $G$; if $G=GL_n(\mathbb{C})$, then you can take $T$ to be the subgroup of diagonal matrices if you are so inclined :)
I am currently running into the following rather simple question in the project am I working on:
$\textbf{Question:}$ What is the de Rham cohomology of $G/T$.
I simply haven't found a reference and I'm sure it's known, so I was hoping someone here could point me to the result and save me some time :)
One may notice that if $G$ was compact, then I believe that $G/T$ is identified with the full flag variety of the complexification of $G$ (studied by Bott and Samelson).
Also, it's perhaps noteworthy that the orbit of a generic semisimple element in the Lie algebra of $G$ is isomorphic to $G/T$.
Thanks for your help :)