My question is rather "simple" to ask : what is the dimension of the quotient variety $GL_3/U$, where $U$ is the (closed) group of upper triangular unipotent matrices (= upper triangular matrices with 1's on the diagonal).
Before working on that quotient variety, I've worked on $SL_2/U$ and I could determine the dimension thanks to the isomorphism $SL_2/U \simeq \mathbb{A}^2 \setminus \{(0,0)\}$ (I did use the fact that an open set of an affine variety has the same dimension, is that true?).
Thank you for your help !