In Lie algebra course, I just learned about the fundamental affine space of $\mathfrak{sl}(n, \mathbb{C})$, which is defined as the following way: space $X_{n}$ is the subspace of $$ \mathbb{C}^{n}\times \wedge^{2}\mathbb{C}^{n}\times\cdots \times \wedge^{n-1}\mathbb{C}^{n} $$ where the elements are $n$ tuples of primitive forms $(a_{1,1}, a_{2,1}\wedge a_{2, 2}, a_{3, 1}\wedge a_{3, 2}\wedge a_{3, 3}, \dots)$ such that $a_{i, 1}\wedge\cdots \wedge a_{i, i} = 0$ or $\mathrm{Span}\{a_{i-1, 1}, \dots, a_{i-1, i-1}\}\subset\mathrm{Span}\{a_{i, 1}, \dots, a_{i, i}\}$. One can check that this is an affine variety. Interesting point is the following:
Theorem. Let $\mathcal{O}(X_{n})$ be a space of functions on $X_{n}$. (Coordinate ring of $X_{n}$) Then $$ \mathcal{O}(X_{n})\simeq \bigoplus_{\lambda\in \Lambda^{+}}V_{\lambda} $$ where $V_{\lambda}$ is the irreducible representation of $\mathfrak{sl}(n, \mathbb{C})$ that has highest weight $\lambda\in \Lambda^{+}$.
Professor said that this is Russian kind of viewpoint, and that's the reason why I can't find this in any English Lie algebra textbook or even in google. There are some other interesting part: let $X_{n}^{0}\subset X_{n}$ be a subspace of elements where $a_{i, 1}\wedge\cdots\wedge a_{i, i}\neq 0$ for all $i$. Then $X_{n}^{0}$ is $G = \mathrm{SL}(n, \mathbb{C})$-homogeneous space. In fact, we have $X_{n}^{0}\simeq G/N$ where $N = [B, B]$, and $B$ is a Borel subgroup of $G$.
Is there anyone know any possible English references that I can find these things for general Lie algebra? Also, I want to know how these things can be used for $p$-adic Lie groups, such as $\mathrm{SL}(n, \mathbb{Q}_{p})$. Thanks in advance.
Edit. I also heard that this might be equivalent to Borel-Weil-Bott theorem.