Re: complex orthogonal group

[A complimentary Cc of this posting was sent to
Tobias Fritz
<tfritz@xxxxxxxxxxxxxxx>], who wrote in article <Pine.LNX.4.62.0510041655020.5936@xxxxxxxxxxxxxxxxxxx>:
> But if you mean that there is some kind of spectral theorem for
> self-adjoint or normal endomorphisms over arbitrary algebraically closed
> fields, that classifies them up to conjugation by SO_n, that sure sounds
> interesting!

The classification is known for any semisimple group. Although it may
be not easy to find [*], and not easy to state [**]. The case of SO_n is
closer to the category of "standard knowledge", and is actually as
simple as Jordan theorem.

However, I cannot remember how to reduce the classification for SO_n
to some "standard" problem about a pencil of matrices. The question
about so_n IS reduced to the pencil problem, so just look in Thompson
reference I gave
http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V0R-45GMWCX-97-8&_cdi=5653&_user=4420&_orig=search&_coverDate=03%2F31%2F1991&_qd=1&_sk=998529999&view=c&wchp=dGLbVlz-zSkWA&md5=fc63d7e6602266dc8658d479356cb831&ie=/sdarticle.pdf

Now *if* you know that SO_n is exp(so_n) (also true for any semisimple
group over algebraically closed field), this gives the answer for SO_n
too.

I will try to find the general treatement in Stein/Zhelobenko; I
suspect it should cover this topic too. Will report later.

Hope this helps,
Ilya

[*] @article {Dyn52Max,
old = MR14:244d,
AUTHOR = {Dynkin, E. B.},
TITLE = {Maximal subgroups of the classical groups},
JOURNAL = {Trudy Moskov. Mat. Ob\v s\v c.},
VOLUME = {1},
YEAR = {1952},
PAGES = {39--166},
}

[**] I myself "oversimplified" the answer in one of papers of mine.
Fortunately, the editor catched the error, and provided a
reference with a proof of the lemma (in proof of which I used the
"oversimplified" version). A lot thanks for this!

.

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