Table Of ContentLIFTING SMOOTH CURVES OVER INVARIANTS FOR
REPRESENTATIONS OF COMPACT LIE GROUPS
8
9
9
1
Dmitri Alekseevsky
n
Andreas Kriegl
a
J Mark Losik
7 Peter W. Michor
]
G Erwin Schr¨odinger International Institute
D of Mathematical Physics, Wien, Austria
.
h
November 24, 1997
t
a
m
[ Abstract. We show that one can lift locally real analytic curves from the orbit
space of a compact Lie group representation, and that one can lift smooth curves
1
even globally, but under an assumption.
v
9
2
0
1
0 Table of contents
8
9 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
/
h 2. Invariants and their values . . . . . . . . . . . . . . . . . . . . . 2
t
a 3. Local lifting of curves over invariants . . . . . . . . . . . . . . . . . 5
m
4. Global lifting of curves over invariants . . . . . . . . . . . . . . . . 11
:
v
i 1. Introduction
X
r
a In [1] we investigated the following problem: Let
P(t) = xn −σ (t)xn−1 +···+(−1)nσ (t)
1 n
be a polynomial with all roots real, smoothly parameterized by t near 0 in R. Can
we find n smooth functions x (t),...,x (t) of the parameter t defined near 0, which
1 n
are the roots of P(t) for each t? We showed that this is possible under quite general
conditions: real analyticity or no two roots should meet of infinite order.
Thisproblem can be reformulated in the following way: Let the symmetricgroup
S act Rn by permuting the coordinates (the roots), and consider the polynomial
n
1991 Mathematics Subject Classification. 26C10.
Key words and phrases. invariants, representations.
P.W.M. was supported by ‘Fonds zur Fo¨rderung der wissenschaftlichen Forschung, Projekt
P 10037 PHY’.
Typeset by AMS-TEX
1
2 ALEKSEEVSKY, KRIEGL, LOSIK, MICHOR
mapping σ = (σ ,...,σ ) : Rn → Rn whose components are the elementary sym-
1 n
metric polynomials (the coefficients). Given a smooth curve c : R → σ(Rn) ⊂ Rn,
is it possible to find a smooth lift c¯: R → Rn with σ ◦c¯= c?
In this paper we tackle the following generalization of this problem. Consider
an orthogonal representation of a compact Lie group G on real vector space V and
σ = (σ ,...,σ ) : V → Rn, where σ ,...,σ is a system of generators for the
1 n 1 n
algebra R[V]G of invariant polynomials on V. Given an smooth curve c : R → Rm
withvaluesinσ(V),doesthereexistasmoothliftc¯: R → V withc = σ◦c¯? Weshow
that this is true locally under similar conditions as for the case of the symmetric
groupandtheelementarysymmetricfunctions(i.e.thecaseofpolynomials). Global
results we get for smooth curves and for real analytic curves only in the case of
polar representations (in particular for finite groups). In order to glue the local
lifts of a real analytic curve we have to control their non-uniqueness. We try do to
this by looking for orthogonal lifts (which are orthogonal to each orbit thy meet).
Here many interesting unsolved problems arise, we suceed only in the case of polar
representations.
Similar lifting problems have been treated for smooth homotopies in [9], and for
diffeomorphisms (for Coxeter groups only) by [3].
2. Invariants and their values
2.1. The setting. Let Gbea compact Liegroupandρ : G → O(V)anorthogonal
representation in a real finite dimensional Euclidean vector space V with an inner
product h | i. By a classical theorem of Hilbert and Nagata the algebra R[V]G
of invariant polynomials on V is finitely generated. So let σ ,...,σ be a system of
1 n
homogeneous generators of R[V]G of degrees d ≤ ··· ≤ d . Let σ = (σ ,...,σ ) :
1 n 1 n
V → Rn and consider σ(V) ⊆ Rn.
Lemma. Let (y ,...,y ) ∈ σ(V) ⊆ Rn. Then, for each t ∈ R, (td1y ,...,tdny ) ∈
1 n 1 n
σ(V).
Proof. If (y ,...,y ) = σ(v) for v ∈ V, then (td1y ,...,tdny ) = σ(tv). (cid:3)
1 n 1 n
Let t 7→ c(t) ∈ σ(V) be a smooth curve in Rn, defined for t near 0 in R. The
curve c is called (smoothly) liftable if there exists a smooth curve t 7→ c(t) ∈ Rn
such that c = σ ◦c. The curve c is called a smooth lift of c.
Similarly we consider real analytic lifts of real analytic curves.
2.2. Orthogonal lifts. Asmoothliftciscalledanorthogonal liftifitisorthogonal
to each orbit where it meets, i.e. for each t we have c′(t)⊥G.c(t). We suspect that
an orthogonal lift of a curve which is nowhere infinitely flat exists and is unique
up to the action of a constant in G. See 4.2 below where we prove this for polar
representations.
Note that orthogonal lifts may be not unique up to a constant in G if we drop
the assumption that c is not infinitely flat. Namely, consider the action of S1 by
scalar multiplication on C ∼= R2. Then σ(z) = |z|2. Any curve c : R → R which is
non-negative and flat at a discrete set of 0’s can be lifted to R2 along straight lines
to 0 ∈ R2; at 0 we can change the angle arbitrily.
LIFTING SMOOTH CURVES OVER INVARIANTS 3
Note that smooth lifts may exist but not orthogonal lifts, as example [1], 7.4
shows: there the compact group O(2) acts by conjugation on the space V of sym-
metric 2 × 2-matrices, a polar representation. An orthogonal lift would lie in a
section, without loss in the standard section of diagonal matrices, and this would
mean a smooth choice of eigenvalues.
2.3. Removing fixed points. Consider the subspace VG := {v ∈ V : ρ(g)v =
v for all g ∈ G} of G-invariant vectors of V and the G-invariant decomposition
V = VG ⊕(VG)⊥. Obviously we have R[V]G = R[VG]⊗R[(VG)⊥]G.
Let us now order σ ,...,σ in a such way that σ ,...,σ (m = dimVG) are
1 n 1 m
coordinates on VG and σ ,...,σ are generators of R[(VG)⊥]. Put
m+1 n
σ0 = (σ ,...,σ ) : VG → Rm and σ1 = (σ ,...,σ ) : (VG)⊥ → Rn−m.
1 m m+1 n
Since σ0 is an isomorphism of vector spaces and σ = σ0×σ1, the study of σ reduces
to one of σ1: a curve c = (c0,c1) is (orthogonally) liftable over σ = σ0 ×σ1 if and
only if c1 is (orthogonally) liftable over σ1.
So without loss we suppose from now on that VG = {0}. Then there exist no G-
invariant linearforms on V and v 7→ hv|viis a G-invariant homogeneous polynomial
on V of degree 2; so we may also assume that σ (v) = hv|vi and d = 2.
1 1
If y ,...,y are the coordinates on Rn, we have then:
1 n
(1) For x ∈ σ(V), y (x) = 0 implies x = 0.
1
2.4. Description of σ(V). We recall some results of Procesi and Schwarz [4].
Let I := {P ∈ R[Rn] : P ◦ σ = 0} be the ideal of relations of the generators
σ ,...,σ in R[y ,...,y ], and let Z ⊂ Rn be the corresponding real algebraic set
1 n 1 n
Z := {y ∈ Rn : P(y) = 0 for all P ∈ I}. Then σ∗ : R[Z] := R[Rn]/I → R[V]G is an
isomorphism. Since σ is a polynomial map, its image σ(V) ⊂ Z is a semialgebraic
set. Since G is compact, σ is proper and separates orbits of G, thus it induces a
homeomorphism between V/G and σ(V).
Let h | i denote also the G-invariant dual inner product on V∗. The differen-
tials dσ : V → V∗ are G-equivariant, and the polynomials v 7→ hdσ (v)|dσ (v)i are
i i j
in R(V)G and are entries of an n×n symmetric matrix valued polynomial
hdσ (v)|dσ (v)i ... hdσ (v)|dσ (v)i
1 1 1 n
B(v) := ... ... ... .
hdσ (v)|dσ (v)i ... hdσ (v)|dσ (v)i
n 1 n n
There is a unique matrix valued polynomial B˜ on Z such that B = B˜ ◦σ.
For a real symmetric matrix A let A ≥ 0 indicate that A is positive semidefinite.
Theorem. (Procesi-Schwarz [4]) σ(V) = {z ∈ Z : B˜(z) ≥ 0}.
Denote by x (i = 1,...,n) some orthonormal coordinates on V. Then σ =
i 1
Σn x2, σ is a polynomial of degree d in variables x , and dσ = Σn ∂σidx .
i=1 i i i i i j=1∂x j
j
For each 1 ≤ i < ··· < i ≤ n (s ≤ n), consider the matrix Bj1,...,js with entries
1 s i1,...,is
hdσ |dσ i for 1 ≤ p,q ≤ s, a minor of B. We have the following equality:
i j
p q
∂σ ∂σ ∂σ ∂σ ⊤
i1 ... i1 j1 ... j1
∂x1 ∂xn ∂x1 ∂xn
Bj1,...,js = .. .. .. .. .. .. .
i1,...,is . . . . . .
∂σis ... ∂σis ∂σjs ... ∂σjs
∂x1 ∂xn ∂x1 ∂xn
4 ALEKSEEVSKY, KRIEGL, LOSIK, MICHOR
For the determinant ∆j1,...,js = detBj1,...,js we have the following expression (see
i1,...,is i1,...,is
lemma 2.5):
∆j1,...,js = Dk1,...,ksDk1,...,ks,
i1,...,is i1,...,is j1,...,js
1≤k1<X...,<ks≤n
∂σ
where Dk1,...,ks := det ip
i1,...,is (cid:16)(cid:16) xk (cid:17)p,q=1,...,s(cid:17)
q
is the determinant of the minor of dσ(v) determined by the rows with numbers
i ,...,i and the columns with numbers k ,...,k . Clearly, ∆j1,...,js(v) 6= 0 means
1 s 1 s i1,...,is
that the rank of dσ(v) is larger or equal to s. The rank of dσ(v) takes its maximum
at regular v, and it is the codimension of any regular orbit in V.
A simple calculation shows that ∆j1,...,js is a polynomial of degree d + ···+
i1,...,is i1
d +d +···+d −2s on V.
is j1 js
Since ∆j1,...,js is a G-invariant polynomial on V there is a unique polynomial
i1,...,is
∆˜j1,...,js on Z such that ∆j1,...,js = ∆˜j1,...,js ◦σ, by Schwarz’ theorem [8].
i1,...,is i1,...,is i1,...,is
Then we have by Sylvester’s theorem
σ(V) = {z ∈ Z : B˜(z) ≥ 0}
= {z ∈ Z : ∆˜i1,...,ik(z) ≥ 0 for 1 ≤ i < ··· < i ≤ n}.
i1,...,ik 1 k
Note also that ∆ (v) := ∆1(v) = B1(v) = hdσ (v),dσ (v)i = 4σ (v) and conse-
1 1 1 1 1 1
quently ∆˜ (y) = 4y .
1 1
Forfurther use letus alsonotehere how removing offixpointsasin2.3affectsthe
∆˜j1,...,js: If σ0×σ1 : VG×(VG)⊥ → Rk×Rn−k, then we have ∆˜(σ)1,2,...,k,j1,...,js =
i1,...,is 1,2,...,k,i1,...,is
∆˜(σ1)j1,...,js.
i1,...,is
2.5. Lemma. For s ≤ n let A and B be matrices with s rows and n columns.
Then we have
det(A.B⊤) = detA .detB ,
i1,...,is i1,...,is
1≤i1<X···<is≤n
where A is the quadratic minor of A with columns i ,...,i .
i1,...,is 1 s
Proof. Let a and b for i = 1,...,s be the line with number i of A and B, respec-
i i
tively, so a ∈ (Rn)∗ and b⊤ ∈ Rn. Then we have in terms of the standard basis e
i i i
of Rn and their dual basis ei of (Rn)∗:
det(AB⊤) = det(ha ,b⊤i) = ha ∧···∧a ,b⊤ ∧···∧b⊤i
i j 1 s 1 s
= det(ha ,e i)ei1 ∧···∧eis, det(hejq,b⊤i)e ∧···∧e
(cid:28) p iq p j1 js(cid:29)
i1<X···<is j1<X···<js
= det(ha ,e i)det(hb ,e i)
p i p i
q q
1≤i1<X···<is≤n
= detA .detB . (cid:3)
i1,...,is i1,...,is
1≤i1<X···<is≤n
LIFTING SMOOTH CURVES OVER INVARIANTS 5
2.6. Theorem. [7] Let ρ : G → O(V) be an orthogonal, real, finite-dimensional
representation of a compact Lie group G. Let σ ,...,σ ∈ R[V]G be homogeneous
1 n
generators for the algebra R[V]G of invariant polynomials on V. For v ∈ V, let
N := T (G.v)⊥ be the normal space to the orbit at v, and let NGv be the subspace
v v v
of those vectors which are invariant under the isotropy group G .
v
Then gradσ (v),...,gradσ (v) span NGv as a real vector space.
1 n v
3. Local lifting of curves over invariants
3.1. Lemma. Lifting at regular orbits. Suppose that c is a smooth curve in
σ(V) ⊂ Rn such that c(0), corresponds to a regular orbit. Then c is orthogonally
smoothly liftable, locally near 0, and the orthogonal lift is uniquely determined by
its initial value at t = 0. Thus any two orthogonal differentiable lifts differ by the
action of a constant g ∈ G.
If c is real analytic then any orthogonal lift is also real analytic.
Proof. Let v ∈ V be a regular point with σ(v) = c(0). Let S be a normal slice
at v, which can be chosen so small that the isotropy group G acts trivial on S.
v
Then G.S is open in V and equivariantly diffeomorphic to G/G ×S so that S is
v
mapped homeomorphically to an open neighborhood of the orbit through v in the
orbit space V/G ∼= σ(V). Thus σ|S : S → σ(V) ⊂ Rn is injective.
Then also d(σ|S) : T S = T (G.v)⊥ → Rn is injective, by the following argu-
v v v
ment: Choosecoordinatesz onS centered atv, lethbeasmallbumpfunctiononS
i
∼
whichequals1nearv. SinceG/G ×S = G.S equivariantly,hz extendstoasmooth
v i
G-invariant function on V, and by Schwarz’ theorem [8] there exists a smooth func-
tion f ∈ C∞(Rn,R) with hz = f ◦σ. But then z = d(hz )(v) = df (σ(v)).dσ(v),
i i i i i i
so that dσ(v) is injective.
Now (making S smaller if necessary) σ|S : S → Rn is an embedding of a sub-
manifold whose image is open in σ(V), and by the implicit function theorem there
are coordinates on Rn centered at c(0) = σ(v) such that σ|S : (z ,...,z ) →
1 m
(z ,...,z ,0,...,0). Then clearly c is uniquely smoothly liftable near t = 0 to
1 m
c˜: R → S. Via the embedding S ⊂ V we get a smooth lift, which however is not
orthogonal in general (only for polar representations).
Now G.S ∼= G/G ×S −p−r→2 S is a fiber bundle, and orthogonal projection onto
v
the vertical bundle T(G.w) defines a (real analytic) (generalized or Ehresmann)
w
connection on this Sfiber bundle, see [5], section 9. The horizontal lift of the curve
c˜ through any point in G.v is then an orthogonal lift of c, and it is uniquely given
by its initial value c¯(0) ∈ G.v. (cid:3)
3.2. For a smooth function f defined near 0 in R let the multiplicity or order of
flatness m(f) at 0 be the supremum of all integer p such that f(t) = tpg(t) near 0
for a smooth function g. If m(f) < ∞ then f(t) = tm(f)g(t) where now g(0) 6= 0.
Let c be a smooth curve in V such that m(c) ≥ r > 0. Then m(σ ◦c) ≥ rd .
i i
3.3. Multiplicity Lemma. Let c = (c ,...,c ) be a smooth curve in σ(V) ⊆ Rn.
1 n
Then, for integers r, the following conditions are equivalent:
(1) m(c ) ≥ rd for 1 ≤ i ≤ n.
i i
(2) m(∆˜j1,...,js ◦c) ≥ r(d +···+d +d +···+d −2s) for all i ≤ ··· ≤ i
i1,...,is i1 is j1 js 1 s
and i ≤ ··· ≤ j .
j s
6 ALEKSEEVSKY, KRIEGL, LOSIK, MICHOR
(3) m(c ) ≥ 2r.
1
Proof. (1) implies (2): By assumption c (t) = trdib (t) for smooth b . Let us
i i i
choose an arbitrary lift b of b = (b ,...,b ), so that σ ◦ b = b. Then σ(trb(t)) =
1 n
(trd1b (t),...,trdnb (t)) = c(t), and we get
1 n
∆˜j1,...,js(c(t)) = (∆˜j1,...,js ◦σ)(trb(t)) = ∆j1,...,js(trb(t))
i1,...,is i1,...,is i1,...,is
= tr(di1+···+dis+dj1+···+djs−2s)∆j1,...,js(b(t))
i1,...,is
= tr(di1+···+dis+dj1+···+djs−2s)∆˜j1,...,js(σ(b(t)))
i1,...,is
= tr(di1+···+dis+dj1+···+djs−2s)∆˜j1,...,js(b(t))
i1,...,is
so m(∆˜j1,...,js ◦c) ≥ r(d +···+d +d +···+d −2s).
i1,...,is i1 is j1 js
(2) implies (3) since ∆ = y .
1 1
(3)implies(1): Withoutlossletr > 0. By2.3.(1)fromc (0) = 0followsc(0) = 0.
1
Therefore, m(c2),...,m(cn) ≥ 1 and we have ck(t) = tmkck,mk(t) for k = 1,...,n,
where the m are positive integers and c ,c ,...,c are smooth functions,
k 1,2r 2,m2 n,mn
and where we may assume that either m = m(c ) < ∞ or m ≥ rd .
k k k k
Suppose now indirectly that for some k > 1 we have m = m(c ) < rd . Then
k k k
we put
m m
2 n
m := min(r, ,..., ) < r.
d d
2 n
We consider the following continuous curve in Rn for t ≥ 0:
c (t) := (t2r−2mc (t),tm2−md2c (t),...,tmn−mdnc (t)).
(m) 1,2r 2,m2 n,mn
By lemma 2.1 the curve c (t) ∈ σ(V) for t > 0, and since σ(V) is closed in Rn
(m)
by 2.4, also c (0) ∈ σ(V). Since m < r the first coordinate of c (t) vanishes
(m) (m)
at t = 0, thus by 2.3.(1) c (0) = 0. In particular for those k with m = md we
(m) k k
have c (0) = 0 and, therefore, m(c ) > m , a contradiction. (cid:3)
k,m k k
k
3.4. Let r ∈ N and let c be a smooth curve in Rn with values in σ(V) such that
ck(t) = tmkck,m (t) for smooth ck,m , and suppose that mk ≥ rdk. Consider the
k k
smooth curve
c (t) = (c (t)tm1−rd1,...,c (t)tmn−rdn) ∈ Rn.
(r) 1,m1 n,mn
By Lemma 2.1 c (t) lies into σ(V). If c is liftable at 0 and c is its smooth
(r) (r) (r)
(real analytic) lift, t 7→ c¯(t) := trc (t) is a smooth (real analytic) lift of c. If c
(r) (r)
is an orthogonal lift, then also c¯, and conversely, since d (trc (t)) = rtr−1c (t)+
dt (r) (r)
trc ′(t), where c (t)⊥|tr.c (t)|.S(V) ⊃ tr(G.c (t)) = G.(trc (t)). Moreover
(r) (r) (r) (r) (r)
the orthogonal lift c¯is uniquely determined as orthogonal lift of c up to the action
of a constant element in G if and only if c has this property as orthogonal lift of
(r)
c .
(r)
Thus the problem of the existence of an orthogonal smooth lift of c near 0 and
its uniqueness reduces to the one for c .
(r)
LIFTING SMOOTH CURVES OVER INVARIANTS 7
3.5. Let v 6= 0 be a singular point of V. Since VG = 0 we have G 6= G. We
v
consider the action of isotropy group G on the normal plane N := T (G.v)⊥,
v v v
where we translate the vector space structure in such a way, that v = 0 . Since v
N
v
is a fixed point under G , this does not change the representation of G on N . Let
v v v
τ ,...,τ beasystemofhomogeneous generators ofR[N ]Gv andτ = (τ ,...,τ ) :
1 m v 1 m
N → Rm the corresponding map.
v
Since the restriction σ |N ∈ R[N ]Gv there exists a polynomial l ∈ R[Rm] such
s v v s
that σ |N = l (τ ,...,τ ). Puttingl = (l ,...,l ) : Rm → Rn we get σ|N = l◦τ.
s v s 1 m 1 n v
Reduction Lemma. In this situation let c(t) = (c (t),...,c (t)) be a smooth
1 n
curve in σ(V) such that c(0) = σ(v). Then there exists a smooth curve c (t) in
v
τ(N ), defined near t = 0, such that c = l◦c .
v v
N τ Rm
v
w hk cv
y hh
h
l R
'c'
Vu σ Run '*'
w
Moreover c is locally smoothly liftable over σ if and only if c is locally smoothly
v
liftable over τ.
Removing G -fixpoints as in 2.3 then splits
v
τ = τ0 ×τ1 : NGv ×(NGv)⊥ → Rk ×Rm−k
v v
and we have c = (c0,c1) where c0 is smoothly liftable and where c1(0) = 0.
v v v v v
The real analytic analogon of this results is also true.
Proof. Consider a normal slice S at v, which is an open neighborhood of v in N ,
v
and the equivariant representation
q ∼=
G×S −−−−→ G× S −−−−→ G.S
G
v
r
∼=
G/yGv −−−−→ Gy.v,
for the G-invariant open neighborhood G.S of the orbit G.v. Then the restriction
homomorphism R[V]G → R[N ]Gv is a monomorphism.
v
On the other hand, for each f ∈ C∞(S)Gv there exists some f˜∈ C∞(V)G such
˜
that f|S agrees with f near v, by the following argument: (g,y) 7→ f(y) is a smooth
function G×S → R which isG -invariant and thus factorsto aG-invariant function
v
G× S ∼= G.S → R. We may multiply it by a G-invariant smooth function on V
G
v
˜
with support in G.S which equals 1 near the orbit G.v to obtain f.
In particular we obtain τ˜ ∈ C∞(V)G with τ˜|S equals τ near v. By Schwarz’
i i i
theorem there exist h ∈ C∞(Rn,Rm) with τ˜ = h◦σ. Note that τ˜(G.S ) = τ(S ) =
1 1
h(σ(G.S )).
1
Suppose c = (c ,...,c ) is a smooth curve on σ(V) such that c(0) = σ(v).
1 n
Since the image of a slice under σ is an open neighborhood of σ(v) in σ(V) we may
8 ALEKSEEVSKY, KRIEGL, LOSIK, MICHOR
suppose that c(t) is contained in σ(W), where W is a sufficiently small G -invariant
v
neighborhood of v in N .
v
On W we have l◦h◦σ = l◦τ˜ = l◦τ = σ, so l◦h equals the identity on σ(W),
thus c = h ◦ c is a smooth curve such that c (0) = g(v) and l ◦ c = c. Since
v v v
h(σ(G.W)) = τ˜(G.W) = τ(W) the curve c has values in τ(N ).
v v
Any lift c of c over τ into N ⊂ V is also a lift of c over σ.
v v v
Let conversely c¯ be a smooth lift of c over σ. Then c¯ is locally in G.S; we may
∼
project it to a smooth curve in the orbit G.v = G/G ; this we can lift locally to
v
a smooth curve g in G, and then g(t)−1.c¯(t) ∈ S and describes a smooth lift of c
v
over τ. This finishes the proof in the smooth case.
Intherealanalyticcasewehavetoreplacethesmoothmappingh ∈ C∞(Rn,Rm)
from above by a real analytic one. By omitting some of the τ if necessary let us
j
first suppose without loss that the following two equivalent conditions hold:
(a) τ ,...,τ is a minimal system of homogeneous generators of R[N ]Gv, i.e.
1 m v
there does not exist a polynomial relation of the form
τ = P(τ ,...,τ ,τ ,...,τ ).
j 1 j−1 j+1 m
(b) τ ,...,τ induce anR-basisofthevectorspace R[N ]Gv/(R[N ]Gv)2, where
1 m v + v +
we denote by R[N ]Gv the space of allinvariant polynomialson on N which
v + v
vanish at 0 = v.
N
v
Clearly a polynomial relation as in (a) induces a nontrivial linear relation. If con-
versely (b) is wrong then we have a τ = c taα, where the α are multi-
i i |α|≥2 α
indices. Let j be such that a 6= 0Pand thatPτ has minimal degree among them.
j j
Then we get ta = b τ + c′ taα. If on the right hand side we omit
j i>j i i |α|≥2 α
all terms of degree dPifferent to dPegτ all terms involving τ vanish and we have a
j j
relation as forbidden in (a). So (a) and (b) are equivalent statements.
Note that we have
˜
σ|N = l◦τ =: σ(v)+l◦τ,
v
˜
τ = h◦σ|N =: h◦(σ|N −σ(v)), near v = 0 ,
v v N
v
˜ ˜
= h◦l◦τ,
˜ ˜
where l is polynomial and h is smooth. Let us now pass to the infinite Taylor
developments at 0, identify the polynomials with their Taylor series, and get
τ = j∞h˜ ◦˜l◦τ = jNh˜ ◦˜l◦τ,
0 0
since we may truncate the series of h at some order higher than all the degrees
of the τ ’s. Since τ ,...,τ induces a real basis of R[N ]Gv/(R[N ]Gv)2 the last
i 1 m v + v +
equation implies that for the derivative we have
˜ ˜ ˜ ˜
d(h◦l)(0) = dh(0)◦dl(0) = IdRm .
Thus ˜l : Rm → Rn is an immersion at 0 = v and by the real analytic inverse
N
v
functiontheoremthereexistsarealanalyticmappingH fromanopenneighborhood
of 0 in Rn into Rm such that H ◦˜l = IdRm near 0. Using the real analytic mapping
H(σ(v)+ ) insteadof hthe foregoing proof forthe smoothcase can alsobe applied
to the real analytic case. (cid:3)
LIFTING SMOOTH CURVES OVER INVARIANTS 9
3.6. Theorem. Local real analytic lifts. Let ρ : G → O(V) be an orthogonal
representation of a compact Lie group G, and let σ : V → Rn be the mapping whose
components are generators of the algebra of invariant polynomials. Let c : R → Rn
be a real analytic curve with c(R) ⊂ σ(V).
Then there exists a real analytic lift c¯ in V of c, locally near each t ∈ R.
Proof. We may assume that VG = 0, and that all assumptions from 2.3 hold.
We first show that there exist orthogonal lifts of c locally near each point t ∈ R,
0
without loss t = 0, through any v ∈ σ−1(c(0)). We do this by the following
0
algorithm in 3 steps, which always stops, since it either gives an orthogonal local
lift, or it reduces the lifting problem to a situation where the the group is replaced
by an isotropy group of of nonzero vector, which either has smaller dimension, or
at least less connected components (if it is already finite, e.g.).
Step1. If c(0) 6= 0 corresponds toa regularorbit, unique orthogonalreal analytic
lifts exist through all v ∈ σ−1(c(0)), by 3.1.
Step 2. If c(0) 6= 0 corresponds to a singular orbit, we consider the isotropy
representation G → O(N ) with invariant mapping τ : N → Rm, we let l : Rm →
v v v
Rn be a polynomial mapping with σ|N = l◦τ. Then by the reduction lemma 3.5
v
we may lift c to c˜: (R,0) → Rm with l◦c˜= c, and it suffices to lift c˜ to N . Note
v
that now c˜(0) = 0, and that the group G is replaced by the smaller (since VG = 0)
group G , acting on N . By 2.3 we may replace N by (NGv)⊥, or assume that
v v v v
again 0 is the only fixpoint.
Step 3. If c(0) = 0 then c (0) = 0 and c = ∆˜ ◦c ≥ 0, thus m(c ) = 2r for some
1 1 1 1
integer r ≥ 1 or r = ∞. In the latter case c = 0 and by 2.3.(1) c = 0 is constant,
1
which clearly can be lifted.
In the former case by the multiplicity lemma 3.3 we have m(c ) ≥ rd for some
i i
r ≥ 1; so we can write c (t) = trdic (t) and we can consider the curve c =
i i,rdi (r)
(c ,...c ). By lemma 2.1 this curve c takes values in σ(V), and if c¯ is a
1,2r n,rdn (r) (r)
lift of c , then t 7→ (trc¯ (t) is a real analytic lift of c. But now, since m(c ) = 2r,
(r) (r) 1
the first component c (0) of c (0) does not vanish, and we can continue in step
1,2r (r)
1 or step 2. (cid:3)
3.7. Theorem. Local smooth lifts. Let ρ : G → O(V) be an orthogonal
representation of a compact Lie group G, and let σ : V → Rn be the mapping whose
components are generators of the algebra of invariant polynomials. Let c : R → Rn
be a smooth curve with c(R) ⊂ σ(V) such that for some r (not larger than the
codimension of the regular orbits) we have:
(1) ∆˜j1,...,js ◦c = 0 whenever s > r.
i1,...,is
(2) There exists a minor of size r such that ∆˜j1,...,jr ◦ c does not vanish of
i1,...,ir
infinite order at t .
0
Then there exists a smooth lift c¯ in V of c, locally near t ∈ R.
0
Proof. We may assume that VG = 0, and that all assumptions from 2.3 hold.
We first show that we can lift c locally near t ∈ R, without loss t = 0, through
0 0
any v ∈ σ−1(c(0)). We do this by the following algorithm in 3 steps, which always
stops in step 1, since step 2, followed by step 3, reduce the lifting problem to a
situation where the the group is replaced by a smaller isotropy group of a nonzero
vector and smaller s.
10 ALEKSEEVSKY, KRIEGL, LOSIK, MICHOR
Step 1. If c(0) 6= 0 corresponds to a regular orbit, smooth lifts exist through all
v ∈ σ−1(c(0)), by 3.1.
Step 2. If c(0) 6= 0 corresponds to a singular orbit, we consider the isotropy
representation G → O(N ) with invariant mapping τ : N → Rm, we let l : Rm →
v v v
Rn be a polynomial mapping with σ|N = l◦τ. Then by the reduction lemma 3.5
v
we may lift c to c : (R,0) → Rm with l ◦ c = c, and it suffices to lift c to N .
v v v v
Note that now c (0) = 0, and that the group G is replaced by the smaller (since
v
VG = 0) group G , acting on N . Now we have to check that the assumption is
v v
still valid in this new situation. We have
hdσ (v),dσ (v)i = hdσ (v)| ,dσ (v)| i
i j V i N j N N
v v v
∂l ∂l
i j
= (τ(v))dτ (v), (τ(v))dτ (v)
k p
(cid:28) ∂z ∂z (cid:29)
Xk k Xp p Nv
∂l ∂l
i j
= (τ(v)) (τ(v))hdτ (v),dτ (v)i
k p N
∂z ∂z v
Xk k Xp p
so that by matrix theory (use lemma 2.5 twice)
∂l ∂l
(∆˜G)j1,...,js ◦c = det ip ◦c (∆˜Gv)m1,...,ms ◦c det jp ◦c .
i1,...,is (cid:18)∂z v(cid:19)(cid:18) k1,...,ks v(cid:19) (cid:18)∂z v(cid:19)
k1<X···<ks kq mq
m1<···≤ms
Now we remove all fixpoints (including v) in N , as in 2.3. From the formula it
v
follows that then for s > r all (∆˜Gv)1,2,...,m,j1,...,js−m◦c vanish identically, whereas
1,2,...,m,i1,...,is−m v
there is a minor of size r such that (∆˜Gv)1,2,...,m,j1,...,jr−m ◦c does not vanish of
1,2,...,m,i1,...,ir−m v
infinite order at 0. Note that the codimension of a regular orbit in V equals the
codimension of a regular G -orbit in the slice S ⊂ N . By the last remark in 2.4
v v v
we have then
(∆˜(τ1))j1,...,js−k ◦c1 = (∆˜Gv)1,2,...,k,j1,...,js−k ◦c .
i1,...,is−k v 1,2,...,k,i1,...,is−k v
We now reproduced the starting situation for the curve c1 in τ1((NGv)⊥) with
v v
c1(0) = 0, and we reduced r by m = dim(NGv) ≥ 1.
v v
Step 3. If c(0) = 0 then c (0) = 0 and c = ∆˜ ◦ c ≥ 0, thus m(c ) = 2r for
1 1 1 1
some integer r ≥ 1 or r = ∞. In the latter case the multiplicity lemma 3.3 gives
m(∆˜j1,...,js ◦c) = ∞, a contradiction.
i1,...,is
In the former case by the multiplicity lemma 3.3 we have m(c ) ≥ rd for some
i i
r ≥ 1; so we can write c (t) = trdic (t) and we can consider the curve c =
i i,rdi (r)
(c ,...c ). By lemma 2.1 this curve c takes values in σ(V), and if c¯ is a
1,2r n,rdn (r) (r)
lift of c , then t 7→ (trc¯ (t) is a smooth lift of c. But now, since m(c ) = 2r, the
(r) (r) 1
first component c (0) of c (0) does not vanish, and (see the proof of 3.3)
1,2r (r)
∆˜ij11,,......,,ijss ◦c(r) = t−r(di1+···+dis+dj1+···+djs−2s)(∆˜ij11,,......,,ijss ◦c)
is not infinitely flat. So we can continue in step 1 or step 2.
This finishes the proof of the existence of local lifts. (cid:3)
