Timeline for Kodaira dimension of co-adjoint orbit
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 5, 2014 at 13:30 | vote | accept | CommunityBot | ||
| Nov 3, 2014 at 14:35 | comment | added | user21574 | Ohhh, thanks a lot, Now I undrestand that why Hirzebruch said that the first chern class of flag variaty is $2ρ$ | |
| Nov 3, 2014 at 14:30 | comment | added | Allen Knutson | A $G$-equivariant line bundle $\mathcal L$ on $G/B$ is uniquely determined by the $T$-weight on the line $\mathcal L|_{w_0 B/B}$ over the top point $w_0 B/B \in G/B$; if it's $\lambda$ (and dominant) then the space of sections is the irrep with high weight $\lambda$. Now, the tangent space $T_{B/B} G/B$ at the basepoint is $\mathfrak g/\mathfrak b$, with weights $\Delta_-$. Add those up and you get $-2\rho$. Now dualize (because we want the cotangent space, for $K_X$) and hit with $w_0$, obtaining $-2\rho$ again. In particular, when you add up all the negative roots, the result isn't dominant. | |
| Nov 2, 2014 at 14:17 | comment | added | user21574 | Allen Knutson@, I need more detail, to undrestand | |
| Nov 2, 2014 at 10:49 | comment | added | Allen Knutson | Indeed, $K_X = -2\rho$ so not dominant, under the usual indexing of line bundles by weights. | |
| Oct 30, 2014 at 17:23 | comment | added | user21574 | Thanks YangMils@, you could you of also en.wikipedia.org/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem , | |
| Oct 30, 2014 at 15:36 | history | answered | YangMills | CC BY-SA 3.0 |