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I am interested in understanding Lemma A.2 in the paper "Moduli spaces of principal F-bundles" by varshavsky which you can find here. It uses so called "Plücker" coordinates of the flag variety for which I would like a reference.

Let me explain the statement : Let $G$ be a split reductive group over a field $k$ which in my case I want to be of characteristic 0 even though Varshavsky works over $\mathbf{F}_q$ but I don't think it is a problem. Fix $T$ a split maximal torus of $G$ and $B$ a borel of $G$ containing $T$. For each dominant weight $\lambda \in X^*(T)$ of $G$ with respect to $B$ we define $V_\lambda$ to be the irreducible representation of $G$ with highest weight $\lambda$.

Let $G^{der}$ be the derived group of $G$, $G^{sc}$ be its simlply connected cover and let $G^{ad}$ be the adjoint quotient of $G$.

Define a quasi-fundamental weight of $G$ to be the smallest positive multiple of a fundamental weight of $G^{sc}$, which belongs to $X^*(T^{ad}) \subset X^*(T)$. Let $(\omega_i)_{i \in I}$ be the set of quasi-fundamental weights of $G$ and for $i \in I$ let $V_i := V_{\omega_i}$

Then varshavsky says that we have a closed enbeding : $$ G/B \hookrightarrow \prod_{i \in I} \mathbf{P}(V_i) $$ and gives the following description of it's image. If $S$ is a $k$-scheme, a point of $\prod_{i \in I} \mathbf{P}(V_i)$ is an $I$-tuple $(\mathscr{L}_i)_{i \in I}$ of line sub-bundles of $V_i \otimes_k \mathscr{O}_S$. It is in the image of the above inclusion if and only if for each tuple of non-negative integers $(k_i)_{i \in I}$ the line subbundle $\otimes_{i \in I} L_i^{\otimes k_i} \subset \otimes_{i \in I} V_i^{\otimes k_i} \otimes_k \mathscr{O}_S$ is contained $V_{\sum_{i \in I} k_i \omega_i} \otimes_k \mathscr{O_S}$.

Does anyone have a reference for this (or a proof) ? The case of $\text{GL}_n$ is very well known but I can't find a reference in the kind of generality that I am working in.

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    $\begingroup$ You are ask several questions: (1) why is $V_i$ nonzero, (2) why is the morphism $f:G/B \to\prod_i \mathbb{P}(V_i)$ an embedding, (3) why is the image of $f$ defined by the relations above? Let me address only (2), assuming (1). First, the cone of effective curve classes on $G/B$ is spanned by the "Demazure curves", i.e., the fibers of the $\mathbb{P}^1$-bundles $G/B\to G/P_i$ used by Demazure in his "simple proof" of Borel-Weil-Bott. The fundamental weights span the dual cone, so the fibers of $f$ contain no curves, i.e., they are finite. Now use homogeneity and simple connectedness. $\endgroup$
    – Jason Starr
    Commented Jul 1, 2018 at 15:17
  • $\begingroup$ Dear Jason. I think $(1)$ is clear no ? I mean $V_i$ is the irreducible representation associated to a non trivial dominant weight. In any case thank you very much for your answer about $(2)$ (it will take some time for me to parse since I don't know some of the notions you are using). $\endgroup$
    – cccp
    Commented Jul 1, 2018 at 15:35

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