# tangent bundle of the toric variety of the wonderful compactification.

Let G be a adjoint group over $k$,algebraically closed of caracteristic zero. Let $\overline{G}$ be its wonderful compactification.

I denote by $\overline{T}$ the closure of the torus $T$ in $\overline{G}$.

How can we describe the cotangent bundle $\Omega_{\overline{T}}$?

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I'm not sure what you would like in an answer. Do you want some moduli-theoretic interpretation? In which case even $\overline T$ is tricky to understand. – Allen Knutson Nov 30 '12 at 18:45
We know that the closure is a smooth toric variety with $T$ as an open dense subset. I would like a description of the vector bundle $\Omega_{\overline{T}}^{1}$ relating with the Lie algebra of $T$. – prochet Nov 30 '12 at 19:02
I believe we also know that the fan of this toric variety is essentially the Weyl chamber decomposition. There is an analog of the Euler sequence for toric varieties, namely the proposition in Section 4.3, p 87 of Fulton's "Introduction to Toric Varieties". It should be possible to describe the cotangent bundle using this. – Jason Starr Nov 30 '12 at 21:15
How can we indentify the sheaves $\mathcal{O}_{D_{i}}$ then? – prochet Dec 1 '12 at 5:08
@prochet: "How can we indentify the sheaves ..." These are the boundary divisors. They are in one-to-one correspondence with the positive simple roots of $G$. – Jason Starr Dec 1 '12 at 20:20