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Timeline for Wonderful compactification

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Jul 27, 2016 at 2:56 vote accept Ramin
Jul 23, 2016 at 19:53 comment added Friedrich Knop @Ramin: $BwB$ is dense in $P$. That's it. Your question can be seen in the context of spherical varieties for which Borel group orbits have been studied to some extent. In your case, the group acting is $G\times G$ and $B\times B$ is its Borel subgroup.
Jul 23, 2016 at 18:14 comment added Ramin @FriedrichKnop Friedrich and Jason, thank you both for comments. Friedrich, what is a reference for the statement about the closure of P?
Jul 23, 2016 at 1:04 comment added Jason Starr @FriedrichKnop. I agree with you. In my answer below, I was just using the closure of $M$ in $\widehat{G}$ as the definition of $\overline{M}$.
Jul 22, 2016 at 18:36 comment added Friedrich Knop Let $w$ be the longest Weyl group element of $M$. Then the closure of $P$ is the closure of the $B\times B$-orbit $BwB$ in $X$. The orbit closures of arbitrary $B\times B$-orbits have been studied thoroughly in a series of papers by Springer. In particular, they are all normal with rational singularieties.
Jul 22, 2016 at 18:33 comment added Friedrich Knop There is a problem: Since $M$ is not semisimple, let alone of adjoint type, it doesn't have a wonderful embedding.
Jul 22, 2016 at 13:37 answer added Jason Starr timeline score: 6
Jul 22, 2016 at 12:15 comment added Jason Starr For an adjoint semisimple group $G$, for a Borel $B$, $M$ is a maximal torus in $T$ in $G$, and the closure $\overline{T}$ is fixed (setwise, not pointwise) by the restriction of conjugation to a Weyl group $W\subset N_G(T)$. The GIT quotient of $\widehat{G}$ by conjugation equals the finite quotient $\overline{T}/W$. The maximal open of $\widehat{G}$ on which the GIT quotient is defined (the "semistable locus") was studied by Xuhua He and myself. The maximal open of $\overline{B}$ on which the projection $B\to \overline{T}$ is defined should follow from that . . .
Jul 22, 2016 at 12:08 comment added Jason Starr For $G=\textbf{PGL}_2$ and $P=B$ a Borel subgroup, something seems wrong. The wonderful compactification of $G$ is $\mathbb{P}\text{Mat}_{2\times 2}$, the $\mathbb{P}^3$ of $2\times 2$ matrices. I believe that $M$ is a maximal torus in $G$, isomorphic to $\mathbb{G}_m$, and the closure $\overline{M}$ is $\mathbb{P}^1$. Yet the closure $\overline{B}$ is a $\mathbb{P}^2$ in $\mathbb{P}\text{Mat}_{2\times 2}$. There is no nonconstant morphism from $\mathbb{P}^2$ to $\mathbb{P}^1$. This all seems familiar . . .
Jul 21, 2016 at 22:57 comment added Ramin @JasonStarr: that's certainly true. At any rate, the issue is not the reductive part as the closure of M in X is Y. It is not clear to me what the closure of U is. One might be tempted to say that it is isomorphic to G/P, but that's not true--as this guy is not bi-equivariant for U.
Jul 21, 2016 at 21:10 comment added Jason Starr . . . if I remember correctly.
Jul 21, 2016 at 21:08 comment added Jason Starr The closure in $X$ of a maximal torus $T_G$ is the toric variety $\overline{T}_G$ whose fan is the Weyl fan of $G$. So for the maximal torus $T_M$ of the Levi factor the closure of $T_M$ inside $X$ equals the closure of $T_M$ in $\overline{T}_G$. Does that toric variety admit a morphism to the toric variety $\overline{T}_M$ of $M$? That should be something about Weyl fans . . .
Jul 21, 2016 at 20:56 history edited Ramin
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Jul 21, 2016 at 20:43 history asked Ramin CC BY-SA 3.0