ag.algebraic geometry - Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?

Your first question is about two objects becoming isomorphic after quantization, and you're asking "Why?"

Here, the relevant quantum object is the spin representation of $mathfrak g$, which is a representation of $mathfrak gltimes mathit{Cliff}(mathfrak g)$,
where $mathit{Cliff}(mathfrak g)$ is the Clifford algebra of (the underlying vector space of) $mathfrak g$, with respect to some invariant inner product; a $mathbb Z/2$-graded algebra.

Let $S$ be the unique (up to grading reversal) irreducible $mathbb Z/2$-graded representation of $mathit{Cliff}(mathfrak g)$. It has a graded-commuting action of
$$C:=begin{cases}mathbb C&text{ if }quad dim(mathfrak g) text{ is even} \ mathit{Cliff}(1)&text{ if }quad dim(mathfrak g) text{ is odd.} end{cases}$$
Let $V$ be the category of modules of the above algebra, so that $Sin V$.
To make things a bit more canonical, one can use the graded Morita equivalence between $C$ and $mathit{Cliff}(mathfrak h)$ to identify $V$ with the category of $mathit{Cliff}(mathfrak h)$-modules.

Let $alpha$ denote the adjoint action of $G$ on $mathit{Cliff}(mathfrak g)$. For any element $gin G$, we can pre-compose the action of $mathit{Cliff}(mathfrak g)$ on $S$ by $alpha_g$ to get a new, isomorphic $mathit{Cliff}(mathfrak g)$-module in $V$ [here, I'm using that $G$ is connected].
The map that sends $g$ to such an isomorphism is unique up to scalar, and so we get a projective representation of $G$ on $S$.

If $G$ is simply connected, this lifts to an honest action of $G$ on $S$, and so we get an action of $Gltimesmathit{Cliff}(mathfrak g)$ on the object $Sin V$.

All in all, (the underlying vector space of) $S$ has actions of $G$, of $mathit{Cliff}(mathfrak g)$, and of $mathit{Cliff}(mathfrak h)$.
Now, we can "cancel" two $mathit{Cliff}(mathfrak h)$ actions to get a vector space with actions of $G$ and of $mathit{Cliff}(mathfrak gominus h)$.
That's the irreducible $G$-rep with highest weight $rho$.

This vector space has two descriptions:
(1) The irreducible $G$-rep with highest weight $rho$.
(2) The irreducible $mathit{Cliff}(mathfrak gominus h)$-module.

Note however that this operation of "canceling" the two $mathit{Cliff}(mathfrak h)$ actions is a bit unnatural.
For example, the vector space (1) is purely even, while (2) is genuinely $mathbb Z/2$-graded. You probably see that same weirdness on the classical side of the problem when you try to identify the flag variety with a product of $mathbb P^1$s.

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