Timeline for An example in symplectic geometry
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28, 2021 at 22:24 | comment | added | Maria | I apologize if my comments are very messy and too long, but please feel free to ask me to clarify any thing you don't understand in what I've wrought. | |
| Jan 28, 2021 at 22:18 | comment | added | Maria | The last time he asked me to prove that the set $ S_M $ contains 5 elements: $\lbrace T, T_e, T_1 ,T_2, T_3 \rbrace $ and then to prove that $\mu(T_1) = \lbrace \Delta_1, \Delta_2 , \Delta_3 \rbrace $ and $\mu(T_2) = \lbrace \Delta'_1, \Delta'_2 , \Delta'_3 \rbrace $ and $\mu(T_3) = \lbrace \Delta''_1, \Delta''_2 , \Delta''_3 \rbrace $ where $\Delta_i , \Delta'_i \Delta''_i$ are the faces and the diagonals in the hexagon . | |
| Jan 28, 2021 at 22:06 | comment | added | Maria | it's a different perspective to find the set $\Sigma$ . | |
| Jan 28, 2021 at 17:54 | comment | added | LSpice | @Maria, I have not had a chance to think through your latest comments. Is there a new question, or is it only more information / a different perspective? | |
| Jan 21, 2021 at 21:44 | comment | added | Maria | And realizing this, i start now to work with the second description of the set rather then the first one in the post ( wich i denote $\Sigma$ ) since it seems much easier ! Please let me know if you've understand what I've written in my comments ! | |
| Jan 21, 2021 at 21:26 | comment | added | Maria | Then For each sub torus T' in T s.t $T' / S_M$ is not finite we take each connected component Z of $M^{T'}$, it's a symplectic submanifold of M provided with the restricted symplectic 2-form of M and the restricted moment map $\mu_{|Z}$. Applying the Atiyah-Guillemin-Sternberg convexity theorem to each connected component Z, we see that $ \mu(Z)$ is a convex polytope equal to $ Conv \lbrace \mu(F)$ , F is connected component of $Z^T \rbrace$ | |
| Jan 21, 2021 at 21:10 | comment | added | Maria | thank you for your answer! Regarding my question, i didn't have the opportunity yet to ask the professor who wrote the notes that I'm trying to understand, but from rereading them I think the set $\Sigma$ in my question is the same as the set $ \lbrace\bigcup\limits_{S_M \subset T'} \mu(M^{T'}) , $ $T'\subset T$ and $T'/S_M$ is infinite $ \rbrace $. Where $S_M := \bigcap\limits_{m \in M} Stab(m)$ is the intersection of stabilizers in M, and there are five type of them. Please see this math.stackexchange.com/questions/3936484/… | |
| Jan 21, 2021 at 19:28 | comment | added | LSpice | That's a good question. It's clear that each $\operatorname{Ad}^*(Ln)X^*$ with $n \in \operatorname N_G(T)$ is connected, and that it depends only on $\operatorname N_L(T)n$; but I don't see at the moment why two such connected sets can't intersect. I will think more about it. \\ In the meantime, did you clarify that the statement you were trying to prove was the correct one? | |
| Jan 20, 2021 at 17:42 | vote | accept | Maria | ||
| Jan 20, 2021 at 17:42 | |||||
| Jan 20, 2021 at 17:39 | comment | added | Maria | Hi @LSpice, Could you please explain to me why the connected components of $M^{T'}$ = $\Ad^*(L\cdot\Norm_G(T))X^*$ are indexed by $\Norm_L(T)\backslash\Norm_G(T)$, $\Ad^*(L)X^*$, $\Ad^*(L s_\beta)X^*$ and $\Ad^*(L s_\beta s_\alpha)X^*$ ? | |
| Nov 14, 2020 at 14:09 | comment | added | Maria | Oh Thanks @LSpice, these are a worthwhile references | |
| Nov 14, 2020 at 3:06 | comment | added | LSpice | Most of this is just stuff I've picked up in the course of learning the general structure theory of reductive groups, rather than coadjoint orbits and symplectic geometry specifically. Nonetheless, Kirillov is the master of the (coadjoint) orbit method; probably Kirillov - Lectures on the orbit method is a place to start. There's a paper by Kottwitz called "Harmonic analysis on reductive $p$-adic groups and Lie algebras" (MSN) that I seem to remember discusses these polytopes, too. | |
| Nov 14, 2020 at 2:47 | comment | added | Maria | Thank you so much for your answer and for your time @LSpice ! I'll be looking for the rest of it ! But to be honest, my understanding of this was very little, since I'm a beginner in symplectic geometry! Could you please recommend some good references about properties of convex hulls of coadjoint orbits and their relation with weyl group (As your answer seems to be based on)? | |
| Nov 14, 2020 at 0:24 | comment | added | LSpice | Wait a minute, I now think that the components are actually the edges between $\mu(X^*)$ and $\mu(s_\alpha X^*)$ ($[AB]$), between $\mu(s_\beta X^*)$ and $\mu(s_\alpha s_\beta X^*$ ($[EF]$), and between $\mu(s_\beta s_\alpha X^*)$ and $\mu(s_\alpha s_\beta s_\alpha X^*)$ ($[CD]$), and I don’t see how to get the diagonals. I will keep thinking. | |
| Nov 13, 2020 at 23:08 | history | answered | LSpice | CC BY-SA 4.0 |