# SO(p,q) and Howe Duality

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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It is related to Capelli identities see wiki and ref therein. Sorry need to run now. –  Alexander Chervov Jan 23 '12 at 7:08
–  Alexander Chervov Jan 24 '12 at 5:58

## 2 Answers

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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Jeffrey -- thanks for the reference -- and for your previous answer to what is known about the unitary dual of $O(p,q)$. –  Mark Mueller Jan 23 '12 at 22:03
To complete the reference Jeff makes, this readable softcover book is published in the Universitext series by Springer-Verlag, New York, 1992. –  Jim Humphreys Jan 29 '12 at 18:34

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.

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Thanks -- looks helpful. –  Mark Mueller Jan 23 '12 at 22:04