Timeline for Global Affine Flag Variety and Affine Flag Variety
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 16, 2015 at 23:02 | vote | accept | Qiao | ||
| Jun 16, 2015 at 23:02 | vote | accept | Qiao | ||
| Jun 16, 2015 at 23:02 | |||||
| Mar 6, 2015 at 0:07 | answer | added | Timo Richarz | timeline score: 3 | |
| Feb 3, 2015 at 18:12 | answer | added | Qiao | timeline score: 3 | |
| Jan 27, 2015 at 19:47 | comment | added | Allen Knutson | The family is (topologically) trivial; the subfamily is not. For a smaller example, imagine the curve $xz-t\ y^2$ in $\mathbb P^2$ with coordinates $x,y,z$ and parameter $t$. This is a nontrivial subfamily of the (algebraically!) trivial family over $Spec\ k[t]$ with constant fiber $\mathbb P^2$. When you say a family is trivial (in whatever sense), you mean you can correspond each fiber to some fixed fiber; I'm just saying that in the Gaitsgory example, that correspondence can be made $T$-equivariant. Therefore the (discrete set of) fixed points must correspond. | |
| Jan 26, 2015 at 17:40 | comment | added | Qiao | Thanks! Is there any references for your comment? Why the family being topologically trivial imply that we could correspond the $T-$fixed points? In the example in my question, a copy of $\mathbb{P}^1$ degenerates to two copies of $\mathbb{P}^1$ glued at a point. | |
| Jan 26, 2015 at 0:57 | comment | added | Allen Knutson | $K$ is the maximal compact in $G$, e.g. $U(n)$ inside $GL(n)$. I'm being careless notationally using $T$ as the maximal (compact) torus in $K$, rather than (complex) in $G$. Because the family is topologically trivial, you can correspond the $T$-fixed points in the general and special fibers. Then $X$ and $X_{\epsilon\to 0}$ must induce the same elements of equivariant $K$-homology, i.e. must be writable as the same combination of $T$-fixed points with $frac(K_T)$-coefficients. | |
| Jan 24, 2015 at 16:35 | comment | added | Qiao | Thanks! I know $Fl$ is topologically trivial, but I am not sure how to make use of that at the moment. As for your comment, what is $K$? Is $T$ the maximal torus in $G$? | |
| Jan 24, 2015 at 3:02 | comment | added | Allen Knutson | One thing that's handy about it is that the family is topologically trivial, at least once one fattens $Fl$ to $LK/T$ and $Gr$ to $LK/K \cong \Omega K$, since $LK \cong K \times \Omega K$ gives $LK/T \cong K/T \times \Omega K$. | |
| Jan 22, 2015 at 22:29 | history | edited | Qiao | CC BY-SA 3.0 |
deleted 24 characters in body
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| Jan 22, 2015 at 22:17 | history | asked | Qiao | CC BY-SA 3.0 |