Timeline for What is a "Lefschetz SL2"?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2021 at 1:09 | comment | added | spin | Thanks to everyone for these comments, which are very helpful. | |
| Jun 14, 2021 at 17:43 | comment | added | Will Sawin | There is a natural isomorphism coming from the geometric Satake isomorphism, I believe. But I don't know whether it can be constructed without too much sheaf-theoretic machinery. | |
| Jun 14, 2021 at 14:53 | comment | added | Jason Starr | @pupshaw. You posted just as I was finishing my comment (we both make the same observation). | |
| Jun 14, 2021 at 14:53 | comment | added | Jason Starr | Since the action of the Lie algebra $\mathfrak{sl}_2$ shifts the weights of cohomology classes, it cannot arise as the induced action on cohomology of an action of algebraic groups of $\textbf{SL}_2$ on $G/P$. | |
| Jun 14, 2021 at 14:52 | comment | added | pupshaw | It's generated by multiplication by a class in degree 2, which came from taking c_1 of a line bundle, which in turn comes from your choice of weight, and the adjoint of this operator under the hodge star. Look up "primitive cohomology," those are the highest weight spaces. in particular, it doesn't come from maps of spaces since those preserve degrees. assuming G is connected, the action of elements of G on cohomology should be trivial since they are homotopic to multiplication by the identity. IMO this structure is more just a feature of Hodge theory, it exists on any compact Kähler manifold. | |
| Jun 14, 2021 at 14:44 | history | asked | spin | CC BY-SA 4.0 |