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Let $G$ be a reductive group over ${\mathbb C}$ and let $G[[t]]$ denote the corresponding group over the formal power series ring ${\mathbb C}[[t]]$. This is a group scheme, so one can speak about its …

Let $V$ be the basic (3-dimensional) representation of $GL(3)$. Then $SL(3)/U$ is the set of all pairs $x\in V, y\in V^*$ where $x$ and $y$ are non-zero and $(x,y)=0$. The quotient $GL(3)/U$ is non- …

I think this is true for any affine variety $X$ over $F$: by Noether normalization lemma it can be represented as a finite cover of an affine space, for which the statement is clearly true (then take …

It is probably hard to give a complete description. A lot of partial information about this is contained in this paper of Kostant https://arxiv.org/abs/1201.4494 (in particular there is a description …

Let $G$ be a semi-simple group over complex number; for simplicity let us assume that it is simply laced. Let $X$ be the orbit of the highest root line in the adjoint representation of $G$. This is a …

Let $X$ denote the flag variety of a semi-simple group $G$ (in characteristic 0) and let $T_X$ denote its tangent bundle. I would like to ask the following question(s): 1) Is it true that for any $n …

Let $G$ be a split semi-simple simply connected group over a global field $F$ and let $\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well known that $\omega …

As you said, the main thing is to construct the central extension. The story is relatively straightforward for groups of type $A$ and gets more complicated in the general case. First, let $\mathcal …