Timeline for Richardson Classes and the Bala Carter Theorem
Current License: CC BY-SA 2.5
8 events
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| Dec 7, 2010 at 8:10 | comment | added | Jay Taylor | OK, well thanks for all your help and very useful comments Jim. I'll take a look at the Bala-Carter papers to see if I can sort out my labelling issues. Thanks again. | |
| Dec 7, 2010 at 8:07 | vote | accept | Jay Taylor | ||
| Dec 6, 2010 at 21:58 | comment | added | Jim Humphreys |
P.S. Concerning your first comment, I think you are misreading the complicated page 423 in Carter's book (e.g., how his $\xi_i$ are related to $\lambda$). Concerning Bala-Carter, their original papers may help. But they require pairs: Levi subgroup plus its distinguished parabolic. Class dimensions can be useful in keeping track of partition labels, though in your case the latter by themselves are ambiguous. Too much notation in the whole subject, unfortunately. You need to focus your questions a little more accordingly.
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| Dec 6, 2010 at 19:27 | comment | added | Jim Humphreys | I'll get back into this literature a little later, but at first glance it looks as if your initial concern is about labelling conventions. This gets messy, since one can also look at dual partitions, etc. I don't recall immediately how Bala-Carter connects with partition labels for the classical groups. | |
| Dec 6, 2010 at 19:00 | comment | added | Jay Taylor | So let's just ignore the parabolic coming from the Richardson theory altogether. Can one determine in a nice way specifically the Levi subgroup attached to the class in the Bala-Carter theory? I had the impression that as the partition determines the Jordan normal form of the matrix that this would determine the Levi subgroup. I guess in my mind in the example in $D_4$ the two Jordan blocks of size 4 fit very nicely in some $A_3$ Levi, which will look like $GL_4\times GL_4$ in $SO_8$, (except one $GL_4$ is determined by the other). Using matrices in $SO_{2n}$ would probably make this concrete. | |
| Dec 6, 2010 at 18:51 | comment | added | Jay Taylor | Thanks for your answer Jim. Just a couple of questions. Sticking with the example of $D_4$ and the partition $(4,4)$. Do you really mean to say that the Bala-Carter theory gives a parabolic with Levi complement $A_3$ and the Richardson parabolic is of type $A_2$? What I got the impression that Carter was saying was that the parabolic for which the class is a Richardson class has Levi complement $A_3$. This is mentioned in his book (pg. 423 to be precise) in his description of the Springer correspondence. | |
| Dec 6, 2010 at 18:07 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
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| Dec 6, 2010 at 17:52 | history | answered | Jim Humphreys | CC BY-SA 2.5 |