Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to half the sum of the positive roots. Let $L_\beta := T \cdot $ the root $SL_2$ subgroup corresponding to the positive root $\beta$.

Quoth Bert: the character of the irrep $V_{n\rho}$ is $T$-isomorphic to the tensor product over all positive roots of the $L_\beta$-irrep with highest weight $n\beta$. (The latter is a $T$-representation by sticking $T$ diagonally into $\prod_{\Delta_+} L_\beta$.) Once someone tells you, it's very easy to prove from the Weyl character formula.

Geometrically, this says the following. Inside ${\mathbb P}^* (V^G_\rho)$, we have a copy of the flag manifold $G/B$ as the orbit of the highest weight vector. (Indeed, this is the smallest embedding by a complete linear series.) Identifying

$V^G_\rho \cong \bigotimes_{\Delta_+} V^{L_\beta}_\beta$

as $T$-representations, we also have in this projective space a Segre-embedded $\prod_{\Delta_+} {\mathbb P}^* (V^{L_\beta}_\beta),$ a product of ${\mathbb P}^1$s. Kostant's observation is that these two subvarieties have the same $T$-equivariant Hilbert series.

"Why" is the flag manifold masquerading as a product of ${\mathbb P}^1$s?

More concretely, we know that the two varieties lie in the same connected component of the Hilbert scheme of this projective space, by Hartshorne's thesis. Can one connect them without breaking the $T$-action? (As far as I know, Hartshorne connectivity doesn't hold in general if you keep track of the multigrading, not just the single grading.)

Since the two varieties are both smooth, (EDIT:) and have different topology, there won't be a flat family over an irreducible base in which one is a general fiber, one the special. Do they have a common degeneration? Hartshorne's thesis just guarantees that we can degenerate, deform, degenerate, deform, ... to get from one to the other, not that they will be on adjacent components of the Hilbert scheme.

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