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ISOMETRIC ACTIONS OF LIE GROUPS AND INVARIANTS

Peter W. Michor Erwin Schr"odinger Institute for Mathematical Physics,

Boltzmanngasse 9, A-1090 Wien, Austria

and Institut f"ur Mathematik,

Universit"at Wien, Strudlhofgasse 4, A-1090 Wien, Austria.

July 31, 1997

This are the notes of a lecture course held by P. Michor at the University of Vienna in the academic year 1992/1993, written by Konstanze Rietsch. The lecture course was held again in the academic year 1996/97 and the notes were corrected and enlarged by P. Michor.

Table of contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Polynomial Invariant Theory . . . . . . . . . . . . . . . . . . . . 5 3. C1-Invariant Theory of Compact Lie Groups . . . . . . . . . . . . . 11 4. Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . 34 5. Proper Actions . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6. Riemannian G-manifolds . . . . . . . . . . . . . . . . . . . . . . 48 7. Riemannian Submersions . . . . . . . . . . . . . . . . . . . . . . 57 8. Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 9. The Generalized Weyl Group of a Section . . . . . . . . . . . . . . . 77 10. Basic Differential Forms . . . . . . . . . . . . . . . . . . . . . . 84 11. Basic versus equivariant cohomology . . . . . . . . . . . . . . . . 91

1991 Mathematics Subject Classification. 53-02, 53C40, 22E46, 57S15. Key words and phrases. invariants, representations, section of isometric Lie group actions, slices.

Supported by `Fonds zur F"orderung der wissenschaftlichen Forschung, Projekt P 10037 PHY'.

Typeset by AMS-TEX

1 1. Introduction Let S(n) denote the space of symmetric n \Theta n matrices with entries in R and O(n) the orthogonal group. Consider the action:

` : O(n) \Theta S(n) \Gamma ! S(n) (A; B) 7! ABA\Gamma 1 = ABA t If \Sigma is the space of all real diagonal matrices and Sn the symmetric group on n letters, then we have the following

1.1. Theorem.

(1) \Sigma meets every O(n)-orbit. (2) If B 2 \Sigma , then `(O(n); B) " \Sigma , the intersection of the O(n)-orbit through

B with \Sigma , equals the Sn-orbit through B, where Sn acts on B 2 \Sigma by permuting the eigenvalues. (3) \Sigma intersects each orbit orthogonally in terms of the inner product hA; Bi =

tr(AB t) = tr(AB) on S(n).

(4) R [S(n)]O(n), the space of all O(n)-invariant polynomials in S(n) is isomorphic to R [\Sigma ]Sn, the symmetric polynomials in \Sigma (by restriction). (5) The space C1 (S(n))O(n) of O(n)-invariant C1-functions is isomorphic to

C1 (\Sigma )Sn, the space of all symmetric C1-functions in \Sigma (again by restriction), and these again are isomorphic to the C1-functions in the elementary symmetric polynomials. (6) The space of all O(n)-invariant horizontal p-forms on S(n), that is the