Timeline for Characterization of restricted weights of representations of real semisimple Lie groups
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2016 at 21:59 | history | edited | Ilia Smilga | CC BY-SA 3.0 |
clarified a reference (I initially forgot to include chapter number)
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| Mar 7, 2016 at 23:56 | vote | accept | Ilia Smilga | ||
| Mar 7, 2016 at 18:48 | answer | added | Jim Humphreys | timeline score: 2 | |
| Mar 7, 2016 at 2:17 | comment | added | Ilia Smilga | Yes, I mean a real Lie algebra, thank you. Sorry about that! | |
| Mar 7, 2016 at 2:16 | history | edited | Ilia Smilga | CC BY-SA 3.0 |
Added that $G$ must be real.
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| Mar 6, 2016 at 22:51 | comment | added | nfdc23 | Also, Corollary 5.8 in Borel-Tits paper on reductive groups in IHES 27 is the same integrality result for connected semisimple groups (over any field), in characteristic 0 equivalent to your question in terms of Lie algebras. Unfortunately there is a gap in that proof (the error is that the appeal to 5.7 there is insufficient if the weight $a$ is divisible). I don't know if Thm. 21.6 in Borel's LAG textbook (mentioned in my previous comment) has the same error (as I didn't learn this material from Borel's book, so I never read that part), so I don't know a simple reference. Sorry! | |
| Mar 6, 2016 at 22:41 | comment | added | nfdc23 | Here is a wider context (maybe of no interest to you). A semisimple $\mathfrak{g}$ over a field $k$ of characteristic 0 is ${\rm{Lie}}(G)$ for a Zariski-connected semisimple linear algebraic group $G$ over $k$ (Cor. 7.9 Ch. II of Borel's book "Linear Algebraic Groups"), and we can arrange that linear representations of $\mathfrak{g}$ and $G$ coincide. Maximal split $k$-tori $S$ of $G$ are $G(k)$-conjugate (Thm. 20.9(ii) in LAG), and for $k=\mathbf{R}$ their Lie algebras are the maximally non-compact Cartan subalgebras of $\mathfrak{g}$. Then Thm. 21.6 of LAG is the result (in vast generality). | |
| Mar 6, 2016 at 19:39 | history | asked | Ilia Smilga | CC BY-SA 3.0 |