I recently learned of a relationship between the representations of the groups $SO(p,q)$ and $SL(2,\mathbb{R})$ which is part of an apparently much larger set of ideas known as Howe Duality. My question is a bit open ended, but can someone point me to a good entry point (review articles, lectures) for learning more about Howe duality and in particular the $SO(p,q)$-$SL(2,\mathbb{R})$ duality? Thanks.
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$\begingroup$ It is related to Capelli identities see wiki and ref therein. Sorry need to run now. $\endgroup$– Alexander ChervovCommented Jan 23, 2012 at 7:08
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$\begingroup$ See also mathoverflow.net/questions/85562/… $\endgroup$– Alexander ChervovCommented Jan 24, 2012 at 5:58
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2 Answers
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A good introduction is "Non-Abelian Harmonic Analysis: Applications of SL(2,R)" by Roger Howe and Eng Chye Tan, especially Chapter III, Section 2.
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$\begingroup$ Jeffrey -- thanks for the reference -- and for your previous answer to what is known about the unitary dual of $O(p,q)$. $\endgroup$ Commented Jan 23, 2012 at 22:03
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$\begingroup$ To complete the reference Jeff makes, this readable softcover book is published in the Universitext series by Springer-Verlag, New York, 1992. $\endgroup$ Commented Jan 29, 2012 at 18:34
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Have you looked in this?
MR0986027 (90h:22015a) Howe, Roger . Remarks on classical invariant theory. Trans. Amer. Math. Soc. 313 (1989), no. 2, 539--570.
answered Jan 23, 2012 at 13:13