By R. Herb, R. Lipsman, J. Rosenberg
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Extra info for Lie Group Representations I
Therefore it is a homeomorphism, and G:x is an embedded submanifold of M. 5. Lemma. Let (M ) be a Riemannian manifold and ` : G M ;! e. g:x = x for all x 2 M ) g = e), such that `(G) Isom(M ) is closed in the compact open topology. Then ` is proper. Proof. Let gn 2 G and xn x y 2 M such that gn :xn ! y and xn ! x then we have to show that gn has a convergent subsequence which is the same as proving that fgn : n 2 N g is relatively compact, since `(G) Isom(M ) is closed. Let us choose a compact neighborhood K of x in M .
Now we can use Glaeser's lemma. Take i and 0 2 S~. Then there is a smooth function i 2 C 1 (Rk ) such that T01 i = T 1(0) i T01 : Since both i and are polynomials, we can disregard the T01. T 1(0) i is a power series in (t ; (0)). If we take 'i 2 R t] to be the power series in t with the same coe cients, then the above formula turns into (**) i = 'i ( ; (0)): Since i is homogeneous of degree > 0, 'i has no constant term. So we can write it as 'i = Li + higher order terms Li 2 R t] 1 In particular, if we insert (*) into (**) this implies (***) July 31, 1997 i ; Li (A1 ( ) : : : Ak ( )) 2 (R t]G+p )2 : P.
July 31, 1997 P. 13 38 4. 14 (3) If G:x is a principal orbit and Gx compact, then Gy = Gx for all y 2 S if the slice S at x is chosen small enough. In other words, all orbits near G:x are principal as well. (4) If two Gx -orbits Gx :s1 Gx :s2 in S have the same orbit type as Gx -orbits in S , then G:s1 and G:s2 have the same orbit type as G-orbits in M . (5) S=Gx = G:S=G is an open neighborhood of G:x in the orbit space M=G. Proof. 12(1). 12(2). (3) By (2) we have Gy Gx , so Gy is compact as well.