isometric actions of lie groups and invariants [jnl article] - p. michor

[...]... 1997 P Michor, 3.19 30 3 Invariant theory of compact Lie groups, 3.20 ~ ~! 3.20 Lemma Consider : S ; S ,! G:S as in the proof of 3.15, and de ne ~! := jG:S : S ; Rk The i form a minimal system of generators for R Rk and we denote the germ of i (resp i ) by i (resp i ) Then there are germs of 1 smooth functions Bi 2 C0 (Rk ) such that j = Bj ( 1 : : : k ): Proof of lemma 3.20 Since is a Gp -invariant... representations of compact Lie groups with more than one xed point It turns out that it is July 31, 1997 P Michor, 3.21 3 Invariant theory of compact Lie groups, 3.22 33 3.22 Corollary Let G be a compact Lie group with an orthogonal representation on Rn and = ( 1 : : : k ) the corresponding Hilbert generators, homogeneous and of positive degree Then C 1 (Rk f0g) = C 1 (Rn f0g)G: Proof Schwarz' theorem implies... ;1 ;R Rk(;1, ;and itSnhas) theon B((;1 ;1) (S )) This H is in C 1 0 desired property: B H jR (Sn;1) = h jR (Sn;1) : July 31, 1997 P Michor, 3.14 3 Invariant theory of compact Lie groups, 3.15 25 3.15 The main part of the proof of Schwarz' theorem will be carried out by induction To be able to state the induction hypothesis, we make the following de nition: For two compact Lie groups G and G0 we will... forward in the proof of Schwarz' theorem Proof of the Corollary In (a) as well as in (b) the inclusion \ " is clear So let us just concern ourselves with the surjectivity of (id jSn;1 ) in each case (a) is a consequence of the identity C 1 (R July 31, 1997 Rk f0g Rk ) = C 1 (Rk C 1 (R f0g)) = C 1 (Rk ) ^ C 1 (R f0g) P Michor, 3.16 26 3 Invariant theory of compact Lie groups, 3.17 and the resulting... be chosen homogeneous and of positive degree: Since the action of G is linear, the degree of a polynomial p 2 R V ] is invariant under G Therefore, if we split each Hilbert polynomial up into its homogeneous parts, we get a new set of Hilbert polynomials Let us denote these by i and the corresponding degrees by di > 0 July 31, 1997 P Michor, 3.7 3 Invariant theory of compact Lie groups, 3.8 17 3.8 Corollary... smooth extension f~ of f Rk on R Rk but it need not be at at zero So consider a submanifold chart ( U ) of Graph around 0 and de ne fU : U ; R ! Rk ( pr2 f ; ! Rk ; id) Graph ; R: ; ;;; ! ! ~ Then fU is a smooth extension of f on U and is at at zero Now f and fU patched together with a suitable partition of unity give a function f 2 C 1 (R Rk 0) such that i f = f End of the Proof of 3.7 Recall from... polynomial relation of the type j = P( 1 July 31, 1997 : : : j;1 j+1 : : : k ): P Michor, 3.17 3 Invariant theory of compact Lie groups, 3.17 27 Remark If an algebra is nitely generated, then it automatically possesses a minimal system of generators We only have to take an arbitrary nite set of generators and recursively drop any elements which can be expressed as polynomials in the others Proof of 3.15 Let... But this map is just for := (idU ), and we are done July 31, 1997 P Michor, 3.10 18 3 Invariant theory of compact Lie groups, 3.11 3.11 We shall use the following notation: For a manifold M and a closed submanifold K M let C 1 (M K ) := ff 2 C 1 (M ) : f is at along K g: Lemma For the proof of theorem 3.7 it su ces to show that C 1 (V 0)G ; C 1 (Rk 0) is surjective Proof Consider the following diagram:... behavior of H" for " ! 0 Our strategy will be as follows: (1) Show that L is 1-regular Referring back to 3.5, the topology on E 1 (L) is then generated by the family of seminorms fj : jL : m 2 N 0 g m (2) Show that J 1 H" is a Cauchy sequence for " ! 0 in terms of the family of seminorms fj : jL : m 2 N 0 g m H" is smooth on 0 1) July 31, 1997 Rk P Michor, 3.14 22 3 Invariant theory of compact Lie groups, ... subspace of all Whitney jets m de ned as follows For each a 2 K there is a map Ta : J m (K ) ; R Rn ] given (in ! multi-index notation) by X (x ; a)k k TamF (x) = F (a) j k j m k! which assigns to each m-jet its would-be Taylor polynomial of degree m With it we can de ne as the remainder term (an m-jet again): m Ra F := F ; J m (TamF ): If F is the set of partial derivatives restricted to K of some C m -function