LIFTS OF DIFFEOMORPHISMS OF THE ORBIT

SPACE FOR A FINITE REFLECTION GROUP

M. Losik

Abstract. For a finite reflection group W acting on a real vector space V it is proved that any diffeomorphism of the orbit space V/W is liftable to a diffeomorphism of V. The analogous statement is proved for the induced action of W on the complexification V ⊗ C of V and biholomorphic maps of V ⊗ C onto itself preserving the discriminant locus.

0. Introduction

Let W be a finite reflection group acting on n-dimensional real vector space V. By the Chevalley theorem (see [3]) the algebra R[V]^{W} of W -invariant polynomials is a free algebra with n homogeneous algebraically independent generators. Let p^{1}, …, p^{n} be such generators and p = (p^{1}, …, p^{n}) : V → R^{n} the corresponding map. The image P (V) is a closed subset of R^{n} homeomorphic to the orbit space V/W. The bijective map f : p(V) → p(V) is called a diffeomorphism of the orbit space V/W if it can be extended to a C^{∞}-diffeomorphism of a neghborhood of p(V) into R^{n}. It is easily seen that this definition is independent of the choice of generators p^{i}.

The main aim of this paper is to prove that each diffeomorphism f of V/W can be lifted to a diffeomorphism of V and it is analytic whenever f is analytic (Main Theorem 3.1). We prove also the local version of this result and extend it to a smooth family of diffeomorphisms of V/W, in particular, to a Lie group of diffeomorphisms of V/W. Besides, it is proved that a smooth vector field on p(V), tangent to the discriminant locus of p, is liftable to V.

Furthermore, we consider the corresponding problems for the extended action of W on the complexification V_{C} of V. In this case the map p induces the biholomorphic map of the orbit space V_{C}/W onto C^{n} and a diffeomorphism of V_{C}/W is a biholomorphic map V_{C}/W → V_{C}/W which preserves the discriminant locus of p. It is proved that each diffeomorphism of V_{C}/W can be lifted to a biholomorphic map of V_{C} onto itself and this statement is true for a smooth or holomorphic family of diffeomorphisms, in particular, for a Lie group of diffeomorphisms.

Finally, the above results imply that the structure of Frobenius manifold on V/W and V_{C}/W defined by Dubrovin [4] has no nontrivial global and local automorphisms both in the real and in the complex case.

Section 1 contains the summary of known facts on finite reflection groups.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20F55.

Key words and phrases. finite reflection group, orbit space, lift.

Supported by by Max-Planck-Institut für Mathematik in Bonn.

In Section 2 we prove the existence of lifts under the map p for some class of smooth curves, containing all analytic curves, in V/W . This result is one of the principal tools in the proof of Main theorem 3.1.

In Section 3 we prove Main Theorem 3.1 and its natural consequences Theorems 3.6, 3.7, 3.8, and Corollary 3.9.

In Section 4 we prove the existence of lifts for some holomorphic curves in p(V_{C}) to V_{C} and prove the version of Main Theorem 3.1 in the complex case (Theorem 4.1).

In Section 5 we prove that each global or local automorphism of the Frobenius manifold V/W or V_{C}/W is trivial.

Throughout the paper a smooth map means C^{∞}-map.

1. Preliminaries

In this section we formulate some standard notions and facts of the theory of finite reflection groups (see, for example, [2] and [6]), which will be used later.

Let V be a real n-dimensional vector space V endowed with a positive definite inner product ⟨x, y⟩. A reflection of V is a linear operator s which sends some nonzero vector α ∈ V to −α and fixes pointwise the hyperplane H_{α} orthogonal to α. We may write s = s_{α}.

A finite set Φ of nonzero vectors of V such that

(1) Φ ∩ Rα = {α, −α} for all α ∈ Φ;

(2) s_{α}(Φ) = Φ for all α ∈ Φ is called a root system in V and the vectors belonging to Φ are called roots. A finite reflection group W is a finite subgroup of O(V) generated by reflections s_{α} (α ∈ Φ) and Φ is called the root system of W. It is known that the reflections s_{α} (α ∈ Φ) exhaust all reflections in W.

A subset Δ of Φ called a simple system if Δ is a basis for the R-span of Φ in V such that each α ∈ Φ is a linear combination of Δ with all all nonzero coefficients of the same sign. The vectors of Δ are called simple roots. If Δ is a simple system and w ∈ W, wΔ is also a simple system. This action of W on the set of simple systems is simply transitive.

The subset Π of Φ consisting of roots which are linear combinations of Δ with non-negative coefficients is called a positive system and the roots belonging to Π are called positive roots.

1.1. Theorem. For any finite reflection group W a simple system Δ exists and W is generated by the reflections s_{α} (α ∈ Δ).

For each simple system Δ and α ∈ Δ define the open subspace

A_{α} := {λ ∈ V |⟨λ, α⟩ > 0},

the open cone

C := ∩_{α∈Δ}A_{α},

called a chamber, and a closed cone C̄ which is the closure of C. The hyperplanes H_{α} (α ∈ Δ) are called walls of the chamber C. It is known that C̄ is a fundamental domain for the action of W on V, i.e. each λ ∈ V is conjugate under W to one and only one point of C̄.

For any w ∈ W wC is the chamber for the simple system wΔ with the corresponding fundamental domain wC̄ and the walls wH_{α}. The simple transitive action of W on the set of simple systems induces the simple transitive action of W on the set of chambers. The chambers are characterized topopogically as the connected components of V \ ∪_{α∈Π}H_{α}.

1.2. Theorem. If U is any subset of V, the subgroup of W fixing U pointwise is generated by those reflections which it contains.

There is the following theorem due to Chevalley (see [2], [3]).

1.3. Theorem. Let R = R[V]^{W} be a subalgebra of R[V] consisting of W -invariant polynomials. Then R is generated, as an R-algebra, by n homogeneous, algebraically independent polynomials p^{1}, …, p^{n} of positive degree.

Let p^{1}, …, p^{n} be the set of generators of the algebra R of degrees d_{1}, …, d_{n}, respectively. In the sequel we suppose that d_{1} ≤ ⋯ ≤ d_{n}. It is known that the degrees d_{1}, …, d_{n} are invariants of the group W. Consider the map p = (p_{1}, …, p_{n}) : V → R^{n}, and the matrix valued function G on V with entries g^{ij} := ⟨dp^{i}, dp^{j}⟩^{∗}, where ⟨x, y⟩^{∗} means the dual inner product. Since g^{ij} ∈ R there is a unique polynomial g̃^{ij} ∈ R[R^{n}] such that g^{ij} = g̃^{ij}(p_{1}, …, p_{n}). Denote by G̃ the matrix valued function on R^{n} with the entries g̃^{ij} and by M̃_{i_{1}...i_{k}} (i_{1} < …, < i_{k}) its minor formed by the rows and the columns of G̃ with the numbers i_{1}, …, i_{k}, by M the determinant of the matrix G = (g^{ij}), and by M̃ the determinant of the matrix G̃.

Let x^{i} (i = 1, …, n) be some orthogonal coordinates in V,

Jm := (\frac{∂p^{i}}{∂x^{j}} )

the jacobian matrix of p, and J = det(Jm). It is easily seen that G = Jm(Jm)^{t}, where (Jm)^{t} denote the transpose matrix of Jm, and then M = J^{2}.

A polynomial f ∈ R[V] is called anti-invariant if for all w ∈ W f ∘ w = det wf. For each root α define a linear form l_{α} as follows: l_{α}(x) = ⟨α, x⟩ (x ∈ V).

1.4. Theorem. (a) For each positive system Π

J = c ∏_{α∈Π} l_{α},

where c is a nonzero constant.

(b) A polynomial f is anti-invariant iff it is a product of J and a W -invariant polynomial. In particular, J is an anti-invariant polynomial.

A point x ∈ V is called regular if it belongs to some chamber C and critical if it belongs to a wall of some chamber. By Theorem 1.4. the set V_{reg} of regular points coincides with the set {x ∈ V |J(x) ≠ 0}. The set of critical values of the map p, i.e. D := p(V \ V_{reg}) ⊂ R^{n}, is called the discriminant locus. Recall the following theorem due to Shephard and Todd [13].

1.5. Theorem. Let V be an n-dimensional real vector space and W a finite subgroup of GL(V) acting naturally on the algebra R[V]. Suppose the algebra R[V]^{W} of W -invariant polynomials is generated by n algebraically independent homogeneous polynomials p^{i} (deg p^{i} = d_{i}). Then W is generated by the reflections it contains and |W | = d_{1} ⋯ d_{n}, where |W | is the order of the group W.

Theorem 1.5 means that the finite reflection group W is determined uniquely by its ring of W -invariant polynomials.

Formulate now the following partial case of the Procesi-Schwarz theorem [8] for compact Lie groups.

1.6. Theorem. The map p : V → R^{n} is proper. The image p(V) of the map p is a semialgebraic subset defined by the inequalities

M̃_{i_{1}...i_{k}} ≥ 0 (k = 1, …, n; i_{1} < ⋯ < i_{k})

and it is homeomorphic to the orbit space V/W equipped with the quotient topology.

By Theorem 1.6 the discriminant locus D is the intersection of p(V) with the set of zeros of the polynomial M̃. Remark that the choice of generators p^{i} does not affect the structure of the semialgebraic subset p(V) because the elements of any system of generators of R are polynomials in p^{i}.

In the sequel we use the following results of Luna, Schwarz, and Solomon.

Let k = R, C. Suppose a group G acts linearly on k^{n} and this action is completely reducible. Then by the classical Hilbert theorem the algebra k[k^{n}]^{G} of G-invariant polynomials on k^{n} has a finite system p^{1}, …, p^{m} of generators. Put p = (p^{1}, …, p^{m}) : k^{n} → k^{m}. Let U be an open subset of k^{n} and 𝔄(U) the algebra of analytic functions on U.

1.7. Theorem. (see [7])

(1) For each G-invariant function f ∈ 𝔄(R^{n}) there exists an open subset U containing p(k^{n}) and g ∈ 𝔄(U) such that f = g ∘ p;

(2) For each G-invariant function f ∈ 𝔄(C^{n}) there exists a function g ∈ 𝔄(C^{m}) such that f = g ∘ p.

1.8. Theorem. (see [12]) Let G be a compact Lie group acting orthogonally on R^{n} and f a smooth G-invariant function on R^{n}. Then there exists a smooth function g on R^{m} such that f = g ∘ p.

1.9. Theorem. (see [14]) Consider the action of a finite reflection group W on a real vector space V or the induced action on the complexification V ⊗C of V. Then each W -invariant differential form on V (or on V ⊗C) with polynomial coefficients is the pull-back via the map p of some differential form with polynomial coefficients on R^{n} (or on V ⊗ C/W = C^{n}).

2. Lifts of curves in the orbit space of a finite reflection group

Consider a smooth (analytic) curve c : R → R^{n} such that c(R) ⊂ p(V). A smooth (analytic) curve c̃: R → V is called a lift of c if p ∘ c̃ = c. The aim of this Section is to prove the existence of lifts under the map p for some class of smooth curves, containing all analytic curves, in p(V). Our proof is a generalization to an arbitrary finite reflection group the solution of this problem given in [1] for A_{n}.

Let Φ be a root system in V and W the corresponding finite reflection group. Denote by V_{1} the R-span of Φ in V. Obviously W acts trivially on the orthogonal complement V_{2} of V_{1} in V and Φ is a root system in V_{1} for the restriction of the action of W to V_{1}.

2.1. Lemma. R[V]^{W} = R[V_{1}]^{W} ⊗ R[V_{2}].

The proof is obvious.

By Lemma 2.1 one can suppose that the set of homogeneous generators p^{1}, …, p^{n} of R[V]^{W} is the union of the set of homogeneous generators of the algebra R[V_{1}]^{W} and the basis of the dual space V_{2}^{∗} of V_{2}. Hence, d_{1} = ⋯ = d_{m} = 1 and d_{m+1} = 2, where m = dim V_{2}, and p(V) = R^{m} ⊕ p_{1}(V_{1}), where p_{1} := (p^{m+1}, …, p^{n}) : V_{1} → R^{n−m}.

Consider a smooth (analytic) curve c(t) = (c_{1}(t), …, c_{n}(t)) in p(V). It is clear that it has a lift in V iff the curve c′(t) = (c_{m+1}(t), …, c_{n}(t) has a lift in V_{1}.

Thus, the lifting problem is reduced to the case when the R-span of Φ coincides with V and d_{1} = 2. Moreover, under this assumption one can choose p^{1}(x) = ⟨x, x⟩. Then we have

(2.1)  For a = (a_{1}, …, a_{n}) ∈ p(V), p^{1}(a) = 0 implies a = 0.

First we study the existence of a local lift, i.e. we consider the smooth (analytic) curve t → c(t) ∈ p(V) defined near t_{0} ∈ R and look for its lift in V. Without loss of generality one can suppose that t_{0} = 0.

We consider the following three cases: (1) c(0) ∈ p(V) \ D; (2) c(0) = 0; (3) c(0) ∈ D and c(0) ≠ 0.

2.2. Lemma. Lifting outside of discriminant locus. The restriction of p to each chamber C is an analytic diffeomorphism of C onto p(V) \ D. Then each smooth (analytic) curve c(t) such that c(0) ∈ p(V) \ D has a unique lift c̃(t) near 0 such that c̃(0) ∈ C.

Proof. The restriction of p to C is a bijective map of C to p(V)\D and the jacobian J is nonzero on C. This implies the statement of the lemma. □

For any real-valued smooth function f defined near 0 in R the multiplicity m(f) at 0 is a supremum of all integers p such that f(t) = t^{p}g(t) near 0 for a smooth function g. If m(f) < ∞ then f(t) = t^{m(f)}g(t), where g(0) ≠ 0. It is easily checked that m(f) ≥ p iff f and its derivatives f′, …, f^{(p−1)} vanish at 0. A smooth nonzero function f defined near t_{0} ∈ R is called flat at t_{0} if f and all its derivatives at t_{0} vanish. For a smooth curve c(t) in V defined near 0 we write m(c) ≥ r if for each linear form ω on V m(ω ∘ c) ≥ r.

Suppose c(t) is a smooth curve in V defined near 0 and m(c) ≥ r. Since deg p^{i} = d_{i}, m(p^{i} ∘ c) ≥ rd_{i}.

Let t → c(t) ∈ p(V) be a smooth curve in R^{n} defined near 0. Denote by M̃_{i_{1}...i_{k},j_{1}...j_{k}} (i_{1} < … i_{k}, j_{1} < ⋯ < j_{k}) the minor of the matrix M̃ formed by the rows with the numbers i_{1}, …, i_{k} and by the columns with the numbers j_{1}, …, j_{k}.

2.3. Multiplicity Lemma. Suppose the R-span of Φ coincides with V and p^{1}(x) = ⟨x, x⟩. Let t → c(t) ∈ p(V) be a smooth curve in R^{n} defined near 0 and c^{i} = p^{i} ∘ c. Then the following conditions are equivalent:

(1) m(c^{i}) ≥ rd_{i} for 1 ≤ i ≤ n;

(2) m(M̃_{i_{1}...i_{k},j_{1}...j_{k}} ∘c) ≥ r(d_{i_{1}} +⋯+d_{i_{k}} +d_{j_{1}} +⋯+d_{j_{k}} −2k) for all 1 ≤ k ≤ n, i_{1} < ⋯ < i_{k}, and j_{1} < ⋯ < j_{k};

(3) m(c^{1}) ≥ 2r.

Proof. (1) implies (2): By assumption c^{i}(t) = t^{rd_{i}}c′_{i}(t) for smooth c′_{i}. Consider p^{i} as a graded indeteminate of degree d_{i}. Then deg g̃^{ij} = d_{i}+d_{j} −2 and it can be easily checked that all summands of the minor M̃_{i_{1}...i_{k},j_{1}...j_{k}} have degree r(d_{i_{1}} +⋯+d_{i_{k}} + j_{1} +⋯+j_{k} −2k). Then m(M̃_{i_{1}...i_{k},j_{1}...j_{k}} ∘c) ≥ r(d_{i_{1}} +⋯+d_{i_{k}} +d_{j_{1}} +⋯+d_{j_{k}} −2k).

(2) implies (3) since M̃_{1,1} = 4p^{1}.

(3) implies (1): If r = 0 we are done. Let r > 0. By (2.1) c^{1}(0) = 0 implies c(0) = 0. Therefore m(c^{2}), …, m(c^{n}) ≥ 1 and we have c^{i}(t) = t^{m_{i}}c_{m_{i}}^{i}(t) for i = 1, …, n, where m_{1}, …, m_{n} are positive integers, c_{m_{i}}^{i} are smooth functions, and m_{1} = 2r. We may assume that for each i either m_{i} = m(c^{i}) < ∞ or m_{i} ≥ rd_{i}. Suppose now that for some i > 1 we have m_{i} = m(c^{i}) < rd_{i}. Then we put

m := min(r, \frac{m_{2}}{d_{2}} , …, \frac{m_{n}}{d_{n}}) < r.

Consider the following continuous curve in R^{n} for t ≥ 0:

c_{(m)}(t) := (t^{2r−2m}c_{2r}^{1}(t), t^{m_{2}−md_{2}}c_{m_{2}}^{2}(t), …, t^{m_{n}−md_{n}}c_{m_{n}}^{n}(t)).

Since for each x ∈ V and t ∈ R p^{i}(tx) = t^{d_{i}}p^{i}(x), c_{(m)}(t) ∈ p(V) for each t > 0. By Theorem 1.6 p(V) is a closed set in R^{n}, then c_{(m)}(0) ∈ p(V). Since m < r, c_{2r}^{1}(0) = 0 and then by (2.1) c_{(m)}(0) = 0. In particular, for all i such that m_{i} = md_{i} c_{m_{i}}^{i}(0) = 0. Then m(c^{i}) > m_{i}. This contradiction concludes the proof. □

Let one of the conditions (1), (2), and (3) of Multiplicity Lemma 2.3 be satisfied for the curve c(t). Then c^{i}(t) = t^{rd_{i}}c_{i,rd_{i}}(t) (i = 1, …, n), where c_{i,rd_{i}} is a smooth function. Consider the smooth curve

(2.2)  c_{(r)}(t) := (c_{1,2r}(t), c_{2,rd_{2}}(t), …, c_{n,rd_{n}}(t)).

It is clear that c_{(r)}(t) ∈ p(V) and c̃(t) = t^{r}c̃_{(r)}(t) is a lift of c(t) near 0 iff c̃_{(r)}(t) is a lift of c_{(r)}(t) near 0.

Let Π be a positive system, x ∈ V \ V_{reg}, and W_{x} the isotropy group of x. By Theorem 1.2 W_{x} is a finite reflection group generated by the set of reflections {s_{α}|α ∈ Π, s_{α}(x) = x}. Suppose Φ_{x} be a root system of W_{x} and Π_{x} is the corresponding positive system. Put

V_{x} := ∩_{α∈Π_{x}}H_{α}.

Then by Lemma 2.1

R[V]^{W_{x}} = R[V′_{x}]^{W_{x}} ⊗ R[V_{x}],

where V′_{x} is the orthogonal complement of the subspace V_{x} in V.

Denote by q^{1}, …, q^{n} a set of algebraically independent homogeneous generators of the algebra R[V]^{W_{x}} of W_{x}-invariant polynomials on V, where q^{1}, …, q^{m} is the basis of the dual space of V_{x}, and put

q := (q^{1}, …, q^{n}) : V → R^{n}.

Since R[V]^{W} ⊂ R[V]^{W_{x}}, for each i = 1, …, n there is a unique polynomial P^{i} ∈ R[R^{n}] such that p^{i} = P^{i}(q^{1}, …, q^{n}). Put

P = P_{x} := (P^{1}, …, P^{n}) : R^{n} → R^{n}.

2.4. Reduction Lemma. ^{1}

(1) The map P = P_{x} : R^{n} → R^{n} is a local analytic diffeomorphism in a neighborhood of q(x) ∈ R^{n};

(2) If c(t) is a smooth (analytic) curve near 0 such that c(0) = p(x), it has a unique smooth (analytic) lift c̃(t) to R^{n} under the map P such that c̃(0) = q(x);

(3) The set of lifts of the curve c near 0 is the union of sets of lifts for the curves P_{x}^{−1} ∘ c near 0 (x ∈ p^{−1}(c(0))).

Proof. Let e_{i} (i = 1, …, n) be an orthonormal basis of V and x^{i} the corresponding coordinates. Denote by J(p), J(q), and J(P) the jacobians of the maps p, q, and P with respect to these coordinates, respectively. It is sufficient to prove that J(P)(q(x)) ≠ 0.

Since p = q ∘ P we have

(2.3)  J(p) = J(q)(J(P) ∘ q).

By definition J(p), J(q), and J(P) are polynomials in x^{1}, …, x^{n}. Hence, by Theorem 1.4

J(P) ∘ q = \frac{J(p)}{J(q)}.

Denote by Φ_{x} the root system of W_{x} and put Π_{x} := Π ∩ Φ_{x}. Therefore,

J(p) = c ∏_{α∈Π} l_{α} and J(q) = c′ ∏_{α∈Π_{x}} l_{α},

where c, c′ are nonzero constants. Then

J(P) = \frac{c}{c′} ∏_{α∈Π\Π_{x}} l_{α}.

Since for each α ∈ Π \ Π_{x} l_{α}(x) ≠ 0, J(P)(q(x)) ≠ 0. This proves (1).

(2) and (3) follow from (1) immediately. □

Remark. If x is a critical point distinct from 0 the order |W_{x}| of the group W_{x} is strictly less than the order |W | of the group W. Therefore, in this case Reduction Lemma 2.4 reduces problem of the existence of local lift for W to the same problem for the finite reflection group W_{x} of strictly less order.

Now we consider the class of the smooth curves c(t) defined near t_{0} satisfying the following condition (A):

There exists an integer k ≤ n such that

(A_{1}) rank G̃ ∘ c ≤ k in some neighborhood of t_{0};

(A_{2}) There is a nonzero minor M̃_{i_{1}...i_{k},j_{1}...j_{k}} ∘ c which is not flat at t_{0}.

Note that the condition (A) is satisfied for any analytic curve c.

It is easily seen that the condition (A) is preserved under the reduction of the lifting problem by Lemma 2.1 and when we replace the curve c by the curve c_{(r)} using Multiplicity Lemma 2.3.

1This lemma was proved with D.V.Alekseevsky.

Lemma 2.5. The condition (A) is preserved under the reduction of the lifting problem by Reduction Lemma 2.4 to the group W_{x}.

Proof. Without loss of generality we may assume that t_{0} = 0. Let Jm(P) be the jacobian matrix of the map P and G̃_{x} = (g̃_{x}^{ij}) the matrix constructed from the set of generators q^{1}, …, q^{n}. Then G̃ = Jm(P)G̃_{x}(Jm(P))^{t}. Since J(P)(x) ≠ 0, rank G̃_{x} ∘ c ≤ k near 0, i.e. the condition (A_{1}) is satisfied for the group W_{x}.

Denote by M̃_{x;i_{1}...i_{k},j_{1}...j_{k}} the minors of the matrix G̃_{x} formed by the rows with numbers i_{1}, …, i_{k} and by the columns with numbers j_{1}, …, j_{k}. Then there is a minor M̃_{x;i_{1}...i_{k},j_{1}...j_{k}} which is not flat at 0. Actually, assume that, for all i_{1} < ⋯ < i_{k} and j_{1} < ⋯ < j_{k}, the minors M̃_{x;i_{1}...i_{k},j_{1}...j_{k}} ∘ c either vanish identically near 0 or are flat at 0. It is known (see, for example, [5]) that each p-minor of the product of two matrices is equal to a sum of products of p-minors for factors. Therefore, all minors of the matrix G̃ ∘ c of order k either vanish or are flat at 0. This contradiction proves the condition (A_{2}) for the group W_{x}. □

2.6.Theorem. Suppose the R-span of Φ coincides with V and p_{1}(x) = ⟨x, x⟩. Let c be a smooth (analytic) curve defined near t_{0} which lies in p(V), x ∈ p^{−1}(c(t_{0})) and satisfies the condition (A). There is a unique smooth (analytic) lift c̃ of c near t_{0} such that c̃(t_{0}) = x. All lifts of c near t_{0} differ by transformations belonging to W.

Proof. Without loss of generality we may assume that t_{0} = 0. We point out the following algorithm in 3 steps to get the lift c̃.

Step 1.If c(0) ∈ p(V) \ D, there is a unique local lift through each x ∈ p^{−1}(c(0)) by Lemma 2.2.

Step 2. If c(0) = p(x), where c(0) ∈ D and c(0) ≠ 0, the existence of a lift for c near 0 by Reduction Lemma 2.4 reduces to the existence of local lift for the group W_{x} such that |W_{x}| < |W | and for the curve uniquely determined by c. By Lemma 2.5 for this reduction the condition (A) remains true and we reduce the lifting problem to the case c(0) = 0.

Step 3. Let c(0) = 0. Reduce the lifting problem to the case when the R-span of Φ coincides with V and p^{1}(x) = ⟨x, x⟩ by Lemma 2.1. For this reduction the condition (A) remains true.

Consider the curve c_{(r)}(t) defined by (2.2). Since the condition (A) is true for the curve c_{(r)} we reduce the lifting problem to the case c(0) ≠ 0. Then we can proceed in step 1 or 2.

By Lemmas 2.1, 2.2, 2.3, and 2.4, the lifts obtained by the above algorithm give all lifts of c near 0 and they differ by transformations belonging to W. This concludes the proof. □

2.7. Theorem. Let c(t) = (c^{i}(t)) (t ∈ R) be a smooth (analytic) curve lying into p(V) and satisfying the condition (A) for each t ∈ R. Then for each x ∈ p^{−1}(0) there is a smooth (analytic) lift c̃(t) (t ∈ R) in V such that c̃(0) = x. All lifts of c(t) differ by transformations belonging to W.

Proof. By Theorem 2.6 one can cover R by a family U_{s} of open intervals such that there is a lift c̃_{s}(t) of c for each of these intervals. We may suppose that for U_{s} containing 0 c̃_{s}(0) = x and for each two overlapping U_{s_{1}} and U_{s_{2}} there is a point t_{0} ∈ U_{s_{1}} ∩ U_{s_{2}} such that c̃_{s_{1}}(t_{0}) = c̃_{s_{2}}(t_{0}). Then we can glue the lift c̃ from this family of lifts. This follows from the following statement:

Let c̃_{1} and c̃_{2} be two lifts of the curve c on an interval (a, b) of R and c̃_{1}(t_{0}) = c̃_{1}(t_{0}) for some t_{0} ∈ (a, b). Then c̃_{1} = c̃_{2}.

Actually, by Theorem 2.6 the set {t|c̃_{1}(t) = c̃_{2}(t)} is open and closed simultaneously and then it equals (a, b). The uniqueness of the lift c̃(t) follows from the same statement. □

Note that by Theorem 2.7 any analytic curve c has a global analytic lift c̃.

3. Proof of Main Theorem

By Theorem 1.6 p(V) is a closed subset of R^{n} and its interior is equal to p(V) \ D. A smooth (analytic) diffeomorphism of V/W = p(V) is a bijective map f : p(V) → p(V) which can be extended to a smooth (analytic) diffeomorphism of some neighborhood of p(V) into V. The aim of this Section is to prove the following

3.1. Main Theorem. Let f be a smooth (analytic) diffeomorphism of V/W and (C, C′) a pair of chambers. Then there is a unique smooth (analytic) diffeomorphism F : V → V such that F (C) = C′ and the following diagramm

V ⟶^{F} V
p↓  ^{p}↓
V/W ⟶^{f} V/W

is commutative.

Remarks. 1. Since D is the boundary of p(V) each diffeomorphism f of V/W induces the diffeomorphism of the interior p(V) \ D onto itself and then p(D) = D.

2. Let F be a diffeomorphism of V of Main Theorem 3.1 and w_{1}, w_{2} ∈ W. Then the statement of Main Theorem 3.1 is true for the diffeomorphism w_{1} ∘ F ∘ w_{2} of V and the pair of chambers (w_{2}^{−1}(C), w_{1}(C′)). Hence, it is sufficient to prove Main Theorem 3.1 for C′ = C.

3. By Lemma 2.1 it is sufficient to prove Theorem 3.1 in the case when the R-span of Φ is equal to V. Further we suppose that this condition is satisfied and p^{1}(x) = ⟨x, x⟩ (x ∈ V).

Since for each x ∈ V the tangent space T_{x} to V at x is canonically isomorphic to V the polynomial p^{i} define on T_{x} a unique symmetric d_{i}-form, i.e. the tensor P_{i}(x) of type (0, d_{i}) on T_{x}. Then the map x → P_{i}(x) is an analytic symmetric tensor field of type (0, d_{i}) on V. By Theorem 1.5 the tensor fields P_{i} (i = 1, …, n) define on V, as an affine space, the structure of W -module up to a choice of 0. In particular, g(V) := P_{1} is the Riemannian metric of the Euclidean space V.

The restriction of map p to a chamber C is a diffeomorphism of C onto p(V)\D. Then for a pair of chambers (C, C′) there is a unique diffeomorphism F_{C,C′} : C → C′ such that p ∘ F_{C,C′} = f ∘ p on C. Consider the pull-back F_{C,C′}^{∗}P_{i} of the restriction of P_{i} to C′. Since the polynomial p^{i} is W -invariant, F_{C,C′}^{∗}P_{i} does not depend on C′. Denote by P_{i,0}^{∗} the symmetric tensor field of type (0, d_{i}) on V_{reg} such that its restriction to each chamber C equals F_{C,C′}^{∗}P_{i}. In particular, g_{0}^{∗}(V) := P_{1,0}^{∗} is a flat Riemannian metric on V_{reg}. By construction the tensor field P_{i,0}^{∗} is W -invariant.

The proof of Theorem 3.1 is based on the following 3.2. Theorem. There is a unique smooth W -invariant tensor field P_{i}^{∗} of type (0, d_{i}) (i = 1, …, n) on V such that the restriction of P_{i}^{∗} to V_{reg} coincides with P_{i,0}^{∗}. The tensor field P_{i}^{∗} is analytic whenever the diffeomorphism f is analytic.

Let us fix some orthonormal basis of the Euclidean space V and the corresponding coordinates x^{j}. Then each tensor field P_{i} is defined by its components P_{i,j_{1}...j_{m}}, where m = d_{i}, with respect to the coordinates x^{j}. By definition P_{1,jk} = δ_{jk}. In just the same way we define the tensor fields on R^{n} by their components with respect to the natural coordinates of R^{n}. Denote by P^{i} the tensor field on V of type (m, 0) corresponding to P_{i} under the canonical isomorphism of the Euclidean space V with its dual space V^{∗}.

First we prove the following three lemmas.

Let T V = V ⊕ V be the tangent bundle of V with the induced action of W and π : T V → V the natural projection. We identify a function ϕ on V with the corresponding function ϕ ∘ π on T V.

3.3.Lemma. Suppose ϕ is a smooth (analytic) W -invariant real-valued function on T V which vanishes on the restriction of the tangent bundle T V to V \ V_{reg}. Then M divides ϕ in the ring 𝔉(T V) of smooth functions on T V, i.e. there is a smooth real-valued function ψ on T V such that ϕ = ψM. If ϕ is analytic, ψ is analytic as well.

Proof. Let Δ be a simple system and Π the corresponding positive system for W. Since ϕ vanishes on each hyperplane π^{−1}(H_{α}) (α ∈ Π) of T V = V ⊕ V, l_{α} divides ϕ in the ring 𝔉(T V). Then, by Theorem 1.4 (a) there is a smooth function η on T V such that ϕ = ηJ and by Theorem 1.4 (b) the function η is anti-invariant, i.e. for each w ∈ W η ∘ w^{∗} = det(w)η, where w^{∗} is a natural extension of w to T V.

Let α ∈ Φ. Then η ∘ s_{α}^{∗} = −η. This forces that η vanishes on the hyperplane π^{−1}(H_{α}) and, therefore, J divides η in the ring 𝔉(T V).

Finally, by Theorem 1.4 M = J^{2} divides the function ϕ in the ring 𝔉(T V).

Since M is a polynomial, the function ψ is analytic whenever the function ϕ is analytic. This concludes the proof. □

3.4. Lemma. The function h = M̃ ∘ f ∘ p equals kM, where k is a W -invariant smooth positive function on V which is analytic whenever the diffeomorphism f is analytic.

Proof. Since h is a W -invariant non-negative smooth function on V which vanishes on V \ V_{reg}, by Lemma 3.3 h = kM, where k is a smooth non-negative function on V.

The function M = M̃ ∘ p is W -invariant, therefore k is also W -invariant. By Theorem 1.8 k = k̃ ∘ p, where k̃ is a smooth function on R^{n}. It is clear that k̃ is positive on p(V) \ D.

Replace in the above reasoning f by f^{−1}. Then one has M̃ ∘ f^{−1} ∘ p = k_{1}M, where k_{1} = k̃_{1} ∘ p and k̃_{1} is a smooth function on R^{n} which is positive on p(V) \ D. Hence, k̃(k̃_{1} ∘ f) = 1 on p(V) \ D and, therefore, on p(V). Thus, k = k̃ ∘ p is a positive function on V.

For the analytic diffeomorphism f the proof is the same but we use Theorem 1.7 instead of Theorem 1.8. □

3.5. Lemma. Let b = (b_{jk}) be the symmetric tensor field of type (0, 2) on R^{n} such that the components b_{jk} are equal to the entries of the adjoint G̃^{∗} of the matrix G̃. Then the pull-back p^{∗}b of b equals Mg(V).

Proof. The statement of the lemma follows from the evident identity

G̃^{−1} = M̃^{−1}G̃^{∗}.

□

Proof of Theorem 3.2. Let b be the tensor of Lemma 3.5. Denote by b^{∗} the pullback of b via the map f ∘ p and put u(τ) = b^{∗}(τ, τ) for each tangent vector τ at x ∈ V. It is clear that b^{∗} is a smooth (analytic) W -invariant function on T V.

Suppose x ∈ V \V_{reg}. Consider the straight line a(t) = x+τt in V near 0, where we identify the tangent vector τ with the corresponding vector of V, and its image c := f ∘ p ∘ a under the map f ∘ p in p(V). By definition the tangent vector \frac{d}{dt}a(0) of a(t) at 0 equals τ and then by the definition of pull-back

u(τ) = b (\frac{d}{dt}c(0), \frac{d}{dt}c(0)) .

Since the curve p ∘ a(t) is analitic it satisfies the conditions of Theorem 2.6. It is clear that these conditions are invariant under the diffeomorphism f. Then the curve c(t) satisfies these conditions also and there is a lift c̃(t) of c(t) near 0. This implies the equality

u(τ) = (p^{∗}b) (\frac{d}{dt}c̃(0), \frac{d}{dt}c̃(0)) .

Since c(0) ∈ D and by Lemma 3.5 p^{∗}b = Mg(V), u(τ) = 0. By Lemma 3.3 there is a smooth function v on T V such that u = kMv, where k is the function on V defined by Lemma 3.4.

The restriction of v to the tangent space T_{x} is a quadratic form on T_{x} and, hence, v defines the W -invariant smooth symmetric tensor field g^{∗}(V) of type (0, 2) on V. By Lemma 3.4 g^{∗}(V) coincides with g_{0}^{∗}(V) on V_{reg}. It is easily checked that g^{∗}(V) is non-degenerate on V .

Since the Riemannian metric g^{∗}(V) is flat there is a local isometry I of (V, g^{∗}(V)) to (V, g(V)) such that I(0) = 0. We may suppose that I coincides with the isometry F_{C,C} : C → C on the intersection of the domain of I and C. Then on this intersection the pull-back I^{∗}P_{i} coincides with P_{i}^{∗}. By definition P_{i} is a W -invariant parallel tensor field on the Riemannian manifold (V, g(V)). Then I^{∗}P_{i} is the W - invariant parallel tensor field on the flat Riemannian manifold (V, g^{∗}(V)). Denote by P_{i}^{∗} the parallel extension of I^{∗}P_{i} to the whole of V. By construction the tensor field P_{i}^{∗} is W -invariant and its restriction to V_{reg} equals P_{0,i}^{∗}.

It remains to note that all tensor fields of the above proof are analytic whenever the diffeomorphism f is analytic. □

Proof of Main Theorem 3.1. Since the Riemannian metric g^{∗}(V) is flat it defines the Euclidean structure in some neighborhood U of 0. Therefore, the parallel tensor field P_{i}^{∗} defines a polynomial p^{∗,i} in U such that p^{∗,i}(0) = 0 and then by Theorem 1.5 U is isomorphic to a neighborhood U′ of 0 of W -module V.

Fix such an isomorphism I : U → U′. By definition, for each i p^{∗,i} = p^{i} ∘ I on U, i.e. p ∘ I = f ∘ p on U. Moreover, I establishes a one-to-one correspondence C → C′ between chambers such that I(C ∩ U) = C′ ∩ U′ and, hence, I coincides with F_{C,C′} on C ∩ U. Therefore, one can glue the smooth diffeomorphism F_{0} of U ∪ V_{reg} onto U′ ∪ V_{reg} from I and the maps F_{C,C′}.

We prove now that F_{0} has a smooth extension F to the whole of V. Since U ∪V_{reg} is a dense subset of V this extension is unique, whenever it exists, and F ∘p = p∘f. Suppose V_{0} is the maximal open subset of V for which such an extension, denoted again by F, exists. Since F is an isometry of V_{0} into the Euclidean manifold (V, g(V)) it is a diffeomorphism. We claim that V_{0} = V and F (V_{0}) = V.

Suppose V_{0} ≠ V and x ∉ F (V_{0}). The curve a(t) = p(tx) (t ∈ R) is an analytic curve in p(V) and, therefore, it satisfies the conditions of Theorem 2.7. Since these conditions are preserved under the diffeomorphism f^{−1} the curve f^{−1} ∘ a has a global smooth lift ã such that ã(0) = 0. By Theorem 2.6 we may assume that ã coincides with the curve t → F_{0}^{−1}(tx) near 0 since it is a lift of the curve f^{−1} ∘ a near 0.

Consider the set of intervals (0, t) such that ã((0, t)) ⊂ V_{0} and let (0, t_{0}) be the maximal of them. Consider a local isometry I_{1} of some neighborhood U_{1} of the point ã(t_{0}) in the Riemannian manifold (V, g^{∗}(V)) into the Euclidean space (V, g(V)) such that I_{1}(ã(t)) = tx for all t ∈ (0, t_{0}). Then I_{1}(ã(t_{0})) = t_{0}x and we may assume that for all i I_{1}^{∗}P_{i} = P_{i}^{∗} on U_{1}. But this means that I_{1} coincides with F on V_{0} ∩ U_{1} and, therefore, F_{0} has a smooth extension to V_{0} ∪ U_{1} ≠ V_{0}. This implies that either V_{0} = V or F (V_{0}) = V.

Suppose V = V_{0}. Then F ∘ ã is a global lift of the curve p ∘ a which coincides with a near 0 by construction. Therefore, F ∘ã = a and, in particular, F (ã(1)) = x. This means that x ∈ F (V). This contradiction implies F (V) = V.

Suppose F (V_{0}) = V. Then V_{0} is a complete open Riemannian submanifold of (V, g^{∗}(V)). But it is possible only if V_{0} = V. This concludes the proof in the smooth case.

It remains to note that if f is analytic diffeomorphism, the diffeomorphism F is analytic as well. □

Now we indicate the local version of Theorem 3.1.

3.6. Theorem. Let f be a diffeomorphism of a connected open neighborhood U of y ∈ p(V) ⊂ R^{n} into R^{n} such that f(U ∩p(V)) ⊂ p(V) and f(D∩U) ⊂ D. Then for each x ∈ V such that p(x) = y, there is a neighborhood Ũ of x and a diffeomorphism F of Ũ into V such that p ∘ F = f ∘ p on Ũ. If (C, C′) is a pair of chambers, the diffeomorphism F is determined uniquely by the condition F (U ∩ C) ⊂ C′.

Proof. Note that Lemma 3.3 is true for functions ϕ and ψ defined in an open subset of V. Therefore, the local version of Lemma 3.4 is also true. Then the proof of Theorem 3.6 coincides with first part of the proof of Main Theorem 3.1. □

3.7. Theorem. Let Y be a smooth (analytic) manifold and f_{y} (y ∈ Y) a smooth (analytic) family of smooth (analytic) diffeomorphisms of V/W. Then for each pair of chambers (C, C′) there is a unique smooth (analytic) family F_{y} of smooth (analytic) diffeomorphisms of V such that for each y ∈ Y F_{y}(C) = C′ and p ∘ F_{y} = f_{y} ∘ p. In particular, let Y be a Lie group and f_{y} an action of Y on p(V) by diffeomorphisms of V/W. Then for C = C′ the above family F_{y} gives an action of the Lie group Y on V independent of C.

Proof. It is sufficient to consider the case C′ = C. Note that Lemma 3.3 is true for a smooth family of functions ϕ_{y} (y ∈ Y) satisfying the conditions of this lemma and the corresponding family ψ_{y} is smooth. Denote by g^{∗}(V)_{y} the smooth family of Riemannian metrics on V corresponding to the family of diffeomorphisms f_{y} by Theorem 3.2.

From the proof of Theorem 3.1 it is easily seen that for C = C′ the exponential map for the Riemannian manifold (V, g^{∗}(V)) at 0 exp_{0} : V = T_{0}V → V is equal to F^{−1}. Therefore, for the smooth family g^{∗}(V)_{y} the corresponding family F_{y} of diffeomorphisms of V is smooth as well.

If Y is a Lie group acting on V/W, it is easily to check that the corresponding family F_{y} defines an action of the Lie group Y on V.

In the analytic case the proof is the same. □

Now we prove the following analog of the Zakalyukin theorem [15] on the lift of a vector field, tangent to the discriminant locus, for the real case.

A function or a vector field on V/W = p(V) is called smooth (analytic) if it has a smooth (analytic) extension to a neighborhood of p(V).

3.8. Theorem. Let X be a smooth (analytic) vector field on p(V), tangent to the discriminant locus. Then there is a unique smooth (analytic) vector field X̃ on V such that p_{∗}X̃ = X. The map X → X̃ is an isomorphism of the Lie algebra of smooth (analytic) vector fields on p(V) onto the Lie algebra of smooth (analytic) W -invariant vector fields on V.

Proof. Since X defines a local flow in a neghborhood of each x ∈ p(V) and this flow preserves the discriminant locus, the proof of the existence and uniqueness of X̃ is analogous to one of Theorem 3.7 for a Lie group of transformations of p(V) using Theorem 3.6 instead of Theorem 3.1.

Let X̃ be a smooth (analytic) W -invariant vector field on V. Since X̃ is invariant under each reflection s_{α} (α ∈ Φ) it is tangent to the hyperplane H_{α}. This implies that the vector field X = p_{∗}X̃ is tangent to the discriminant locus. This concludes the proof. □

3.9 Corollary. Let 𝔉(p(V)) be the ring of smooth (analytic) functions on p(V). The Lie algebra of smooth (analytic) vector fields on p(V), tangent to the discriminant locus, is a free 𝔉(p(V))-module with n generators

∑_{j=1}^{n} g̃^{ij}∂_{j} (i = 1, …, n).

Proof. It is evident that the inner product on V defines a one-to-one correspondence between the space of W -invariant smooth (analytic) vector fields and the space of W -invariant smooth (analytic) 1-forms on V. By Theorem 1.8 each smooth W - invariant function g on V can be represented as h(p^{i}), where h is a smooth function on R^{n}. By Theorem 1.7 the same assertion is true for an analytic W -invariant function g and a function h which is analytic in some neighborhood of p(V). Using these theorems one can easily extend the proof of Theorem 1.9 (see [14]) about the W -invariant polynomial forms to smooth (analytic) forms. Therefore, each smooth (analytic)W -invariant form is a sum of forms g(p^{j})dp^{i}, where g is a smooth function in R^{n} (an analytic function in some neighborhood of p(V)). Hence, each smooth (analytic) W -invariant vector field on V is a sum of vector fields g(p^{j}) grad p^{i}. It remains to note that p_{∗} grad p^{i} = g̃^{ij}∂_{j}. This concludes the proof. □

4. Main Theorem in the complex case

Consider the natural action of W on the complexification V_{C} = V ⊗ C of V and the corresponding subalgebra R_{C} of C[V_{C}] of W -invariant polynomials. Then p^{1}, …, p^{n} are again homogeneous generators of the algebra R_{C} and

p_{C} = (p^{1}, …, p^{n}) : V_{C} → C^{n}

is a bijective map. The complexification ⟨x, y⟩_{C} of the inner product ⟨x, y⟩ of V is a complex orthogonal Euclidean metric on V_{C}.

In just the same way as for the real case we define the set V_{C,reg} of regular points of V_{C}, the discriminant locus D_{C}, the matrix valued function G_{C} = (g_{C}^{ij}) on V_{C}, the matrix valued function G̃_{C} = (g̃_{C}^{ij}) on V_{C}/W = C^{n}, the determinant M_{C} of the matrix G, and the determinant M_{C} of the matrix G. For the sake of simplicity we usually drop the subscript C in p_{C} and ⟨x, y⟩_{C} till the end of this Section.

Let f : C^{n} → C^{n} be a biholomorphic map. A pair (x, x′) of points of V_{C} is called f-compatible if f ∘ p(x) = p(x′). In this Section we prove the following analog of Main Theorem 3.1.

4.1. Theorem. Let f : C^{n} → C^{n} be a biholomorphic map such that f(D) = D. Then, for each pair (x, x′) of regular f-compatible points of V_{C}, there exists a unique biholomorphic map F : V_{C} → V_{C} such that F (x) = x′ and p ∘ F = f ∘ p.

Since Lemma 2.1 is true for the algebra R_{C} it is sufficient to consider the case when C-span of Φ coincides with V_{C}, d_{1} = 2, and p^{1}(x) = ⟨x, x⟩.

To prove Theorem 4.1 we need to lift some holomorphic curves in V_{C}/W to V_{C} but, unfortunately, only the part of results of Section 2 remains true in this case. More precisely, the map p is biholomorphic in a neighborhood of every point x ∈ V_{C} and Reduction Lemma 2.4 is true for the biholomorphic map P : C^{n} → C^{n}. Thus we have a holomorphic lift of a holomorphic curve lying outside of discriminant, all these lifts differ by a transformation belonging to W, and the problem of local lifting of a holomorphic curve, passing through x, such that x ∈ D and x ≠ 0 reduces to the case when x = 0. Theorems 2.6 and 2.7 are not true in the complex case in general and we need to find the appropriate substitute.

As in Section 3 we consider the holomorphic tensor fields P_{i} of type (0, d_{i}) (i = 1, …, n) on V_{C} which are uniquely defined by the polynomials p^{i}. Since the map p is biholomorphic in a neighborhood of every regular point of V_{C}, for each pair (x, x′) of f-compatible points of V_{reg} the maps p and f define a biholomorphic map F_{x,x′} of a neighborhood U_{x} of x onto a neighborhood of x′ such that p∘F_{x,x′} = f ∘p in U_{x}. Hence, there are unique W -invariant holomorphic tensor fields P_{i,0}^{∗} of type (0, d_{i}) on V_{C,reg} such that for each pair (x, x′) of regular f-compatible points of V_{C} (f ∘ p)_{∗}(P_{i,0}^{∗}(x)) = p_{∗}(P_{i}(x′)).

First we recall some facts we need further.

Denote by 𝔉(V)^{W} the ring of W -invariant holomorphic functions on V, by 𝔛(V)^{W} the Lie algebra of W -invariant holomorphic vector fields on V, by 𝔉(V_{C}/W) the ring of holomorphic functions on C^{n} = V_{C}/W, and by 𝔛(V_{C}/W) the Lie algebra of holomorphic vector fields on V_{C}/W, tangent to the discriminant locus D.

There is the following theorem due to Zakalyukin (see [15]).

4.2. Theorem. The map p induces the isomorphism of the Lie algebra 𝔛(V)^{W} onto the Lie algebra 𝔛(V_{C}/W). The Lie algebra 𝔛(V)^{W} is a free 𝔉(V)^{W} -module with the generators grad p^{i} (i = 1, …, n) and the Lie algebra 𝔛(V_{C}/W) is a free 𝔉(V_{C}/W)-module with the generators

g^{i} = p_{∗} grad p^{i} = ∑_{j=1}^{n} g̃^{ij}∂_{j} (i = 1, …, n),

where ∂_{i} = \frac{∂}{∂z^{i}} and z^{i} are the natural coordinates on C^{n}.

This theorem implies, in particular, that 0 is a unique common zero point of all g^{i} and therefore f(0) = 0.

Let x ∈ V_{C}. Put

V_{x} := ∩_{α∈Π_{x}}H_{α},

where Π_{x} is a positive system for the isotropy group W_{x} of x. Then V_{x} is equal to the C-span of the system of vectors grad p^{i}(x) (i = 1, …, n) and by Theorem 4.2 the restriction of the map p_{∗} : T_{x}V_{C} → T_{p(x)}C^{n} to V_{x} ⊂ V_{C} = T_{x}V_{C} is a monomorphism.

4.3. Theorem. There is a unique holomorphic W -invariant tensor field P_{i}^{∗} of type (0, d_{i}) (i = 1, …, n) on V such that the restriction of P_{i}^{∗} to V_{reg} coincides with P_{i,0}^{∗}. In particular, g^{∗}(V_{C}) := P_{1}^{∗} is a holomorphic flat Riemannian metric on V_{C}.

Proof. Note that in the complex case Lemma 3.3 is true for holomorphic functions ϕ and ψ, Lemma 3.4 is true for a nonzero holomorphic function k, and Lemma 3.5 is true for a holomorphic tensor field b_{ij}. As in the proof of Theorem 3.2 put b^{∗} := (f ∘ p)^{∗}b and b^{∗}(τ) := b^{∗}(τ, τ) for each tangent vector τ at x ∈ V_{C}. It is clear that b^{∗}(τ) is a W -invariant holomorphic function on T V_{C}.

Let x = 0. By definition b vanishes at 0, then b^{∗}(τ) = 0 for each tangent vector τ at 0.

Let x ∈ V_{C} \ V_{C,reg}, x ≠ 0, and let (x, x′) be a pair of f-compatible points of V_{C}. Consider the tangent spaces T_{x}V_{x} and T_{x′}V_{x′} of the subspaces V_{x} and V_{x′} at points x and x′, respectively. By Theorem 4.2 (f ∘ p)_{∗}(T_{x}V_{x}) = p_{∗}(T_{x′}V_{x′}).

Note that Reduction lemma 2.4 is true in the complex case. Apply this lemma to the point x and denote by V′′ the orthogonal complement of V_{x}. Since V_{x} is the set of all fixed points of W_{x} the action of W_{x} is trivial on V_{x}, V′′ is an invariant subspace of this action, and W_{x} has no fixed nonzero points in V′′. Fix an orthonormal basis (e_{i}) in V_{C} such that e_{1}, …, e_{m} is a basis of the subspace V_{x} and denote by z^{i} the corresponding coordinates in V_{C}. By Lemma 3.1 for the complex case choose the homogeneous generators q^{1}, …, q^{n} of C[V_{C}]^{W_{x}} so that q^{1}, …, q^{m} is the dual basis of V_{x}^{∗} and q^{m+1}, …, q^{n} are homogeneous generators of the algebra of W_{x}-invariant polynomials on V′′. Then

\frac{∂q^{i}}{∂z^{j}} = δ_{j}^{i} (i, j = 1, …, m)

and the other partial derivatives \frac{∂q^{i}}{∂z^{j}} vanish on the subspace P_{x}. Since p = P ∘ q, p_{∗}(τ) = p_{∗}(τ′), where τ = τ′ + τ′′, τ ∈ P_{x}, and τ′′ ∈ P′′. Therefore, by Theorem 4.2 (f ∘ p)_{∗}(τ) lies in the C-span of the vectors g^{i}(p(x′)) and there is a tangent vector τ_{1} at x′ such that p_{∗}(τ_{1}) = τ. Hence, by Lemma 3.5 for the complex case b^{∗}(τ) = M(x′)g(V)(τ_{1}, τ_{1}) = 0.

The rest of the proof coincides with one of Theorem 3.2. □

Now we prove the following local version of Theorem 4.1.

4.4. Theorem. Let f be a biholomorphic map of a connected open neighborhood U of y ∈ C^{n} onto some neighborhood of y′ ∈ C^{n} such that f(y) = y′ and f(U ∩ D) ⊂ D. Then, for x, x′ ∈ V_{C} such that p(x) = y and p(x′) = y′ there exists a neighborhood Ũ of x ∈ V_{C} and a biholomorphic map F of Ũ onto some neighborhood of x′ ∈ V_{C} such that p∘F = f∘p on Ũ and F (x) = x′. The map F is uniquely defined by the condition F (x_{1}) = x′_{1}, where x_{1}, x′_{1} ∈ V_{C,reg}, p(x_{1}) ∈ U, and f ∘ p(x_{1}) = p(x′_{1}).

Proof. If the biholomorphic map f of Theorem 4.1 is defined only in a neighborhood U of y ∈ C^{n} the above constructions and results, including Theorem 4.3, remain true for some neighborhood of each x ∈ V_{C} such that p(x) = y. Then there exists a biholomorphic isometry F of a neighborhood Ũ of x in the Riemannian manifold (V_{C}, g^{∗}(V_{C})) onto some neighborhood of x′ in the Riemannian manifold (V_{C}, g(V_{C})) such that F (x) = x′. We may choose this isometry so that F^{∗}P_{i} = P_{i}^{∗} on Ũ and then p ∘ F = f ∘ p on Ũ. The last statement of the proof is obvious. □

Now we can prove the following substitute of Theorem 2.6.

4.5. Lemma. Let c(t) be a holomorphic curve in V_{C} defined near 0 such that c(0) = 0. Then the curve c_{1} = f ∘ p ∘ c has a holomorphic lift c̃_{1} near 0. All such lifts differ by tranformations belonging to W.

Proof. If c(t) ≡ 0 we are done.

Let c(t) ≢ 0. By Theorem 4.4 c̃_{1} = F ∘ c gives the lift of the curve c_{1}.

Suppose r is a maximal integer such that c̃_{1}(t) = t^{r}ã_{(r)}(t), where ã_{(r)}(t) is a holomorphic curve in V_{C}. Since c(0) = 0, r > 0. Let c_{1} = (c_{1}^{i}). Then m(c_{1}^{i}) ≥ rd_{i}, i.e. there are holomorphic functions a_{(r)}^{i}(t) (i = 1, …, n) such that c_{1}^{i}(t) = t^{rd_{i}}a_{(r)}(t). It is clear that the existence of the holomorphic lift for the curve c_{1} near 0 is equivalent to one for the curve a_{(r)} = (a_{(r)}^{i}) and ã_{(r)}(t) is a holomorphic lift of a_{(r)} near 0.

By the choice of r ã_{(r)}(0) ≠ 0. If ã_{(r)}(0) ∈ V_{C,reg}, all lifts of the curve a_{(r)} differ by transformations belonging to W. Otherwise, apply the version of Reduction Lemma for the comlex case to the curve a_{(r)}. Then, for each x ∈ p^{−1}(a_{(r)}(0)), we get the corresponding lifting problem for the group W_{x} such that |W_{x}| < |W | and some holomorphic curve a′_{(r)} such that a′_{(r)}(0) = 0. Moreover, we know that there is w ∈ W such that w ∘ c̃_{1} is a holomorphic lift of a′_{(r)} near 0. Therefore, the last statement of the lemma is proved by induction on the order |W | of W. □

4.6. Theorem. Let the map f satisfy the conditions of Theorem 4.1 and let c(t) (t ∈ C) be a holomorphic curve in V_{C}. Then there exists a global holomorphic lift c̃(t) of the curve f ∘ p ∘ c and all such lifts differ by transformations belonging to W.

Proof. The proof follows from the versions of Lemmas 2.2 and 2.5 for the complex case and Lemma 4.5. Actually, by the above lemmas there are local holomorphic lifts of f ∘ p ∘ c near each t ∈ C and one can choose these lifts so that they are glued to a global lift. Since local lifts are defined up to transformations belonging to W the same is true for a global lift. □

Proof of Theorem 4.1. First we prove that for each x ∈ V_{C} there is a unique maximal geodesic curve c(t) (t ∈ C) in the Riemannian space (V_{C}, g^{∗}(V_{C})) such that c(0) = 0 and c(1) = x.

Let (x, x′) be a pair of f-compatible points of V_{C} and F a biholomorphic map defined by Theorem 4.4. Consider the complex straight line a(t) = tx′ (t ∈ C) and the holomorphic lift ã of the curve f^{−1} ∘ p ∘ a defined by Theorem 4.6. We may suppose that ã coincides with the curve F^{−1} ∘ a near 0 and this condition determines ã uniquely. By definition c̃ is a geodesic curve of the Riemannian space (V_{C}, g^{∗}(V)) near 0 and by the uniqueness of holomorphic extension c is a geodesic curve of this Riemannian manifold in the whole of its domain. Obviously there is w ∈ W such that w ∘ ã(1) = x, i.e. replacing F by F ∘ w^{−1} we may suppose that ã(1) = x. Therefore, c = ã is a maximal geodesic curve which is unique by construction.

Let us fix some biholomorphic map F defined by Theorem 4.4. For each x ∈ V_{C} denote by c̃ the lift of the curve f ∘ p ∘ c, where c is the defined above for x ∈ V_{C} geodesic curve of (V_{C}, g^{∗}(V_{C})), which coincides with F ∘ c near 0. The curve c̃ is a determined uniquely complex straight line of the Euclidean space (V_{C}, g(V_{C})). Let F̃ be a map x ↦ c̃(1). By construction F̃ is a well defined bijective map of V_{C} onto itself which coincides with F in the domain of F. By the definitions of geodesics and the tensor fields P_{i}^{∗}, p ∘ F̃ = f ∘ p. Then, for each x ∈ V_{C,reg}, F̃ coincides with F_{x,F̃(x)} near x and, therefore, is biholomorphic near x.

Let x ∈ V_{C} \ V_{reg}. Consider the local isometry I of a neighborhood of x onto a neighborhood of F_{1}(x). Since the set V_{C,reg} is connected we may suppose that I coincides with F̃ at regular points. But this means that F̃ is biholomorphic in a neighborhood of x. Since F̃ is a bijective map of V_{C} onto itself which is biholomorphic near each x ∈ V_{C}, it is biholomorphic.

The above construction implies that for each pair (x, x′) of regular f-compatible points of V_{C} there is a biholomorphic map F̃ such that F̃(x) = x′ and this map coincides with F_{x,x′} near x. Hence, by the uniqueness of holomorphic extension the map F̃ is unique. This concludes the proof. □

4.7. Theorem. Let Y be a connected smooth (real analytic, complex) manifold, y_{0} ∈ Y, and f_{y} (y ∈ Y) a smooth (analytic, complex analytic) family of biholomorphic maps of C^{n} onto itself such that for each y ∈ Y f_{y}(D) = D. Then for each pair (x, x′) of regular f_{y_{0}}-compatible points of V_{C} there is a unique smooth (real analytic, complex analytic) family F_{y} of biholomorphic maps of V_{C} onto itself such that F_{y_{0}}(x) = x′ and p ∘ F_{y} = f_{y} ∘ p. In particular, let Y be a Lie group and f_{y} an action of G on V_{C} by biholomorphic maps of C^{n} onto itself. Then, for y_{0} equal to the identity element of Y and x′ = x, the above family F_{y} is a Lie group of diffeomorphisms of V independent of x.

Proof. Let T be a fundamental domain of the action of W on V_{C} containing x′. There is a continuous map h : Y → T such that h(y_{0}) = x′ and for each y ∈ Y (x, h(y)) is a pair of f_{y}-compatible points of V_{C}. By Theorem 4.1 there is a unique biholomorphic map F_{y} such that p ∘ F_{y} = f_{y} ∘ p and F_{y}(x) = h(y). It is easily checked that the family F_{g} satisfies the statements of the theorem. □

5. Automorphisms of some Frobenius manifolds

For an irreducible finite reflection group W acting on a real vector space V Dubrovin constructed the natural structure of Frobenius manifold on V_{C}/W = C^{n} (see [4]). It is determined by two complex orthogonal flat Riemannian metrics on V_{C}/W. One of them is given by the tensor field g̃^{ij}, defined in Section 4, and it is non-degenerate outside of the discriminant locus D. Another metric η, which gives a flat structure on V_{C}/W, was defined by Saito, Sekuguchi, and Yano (see [11]) by direct calculations for all irreducible finite reflection groups except E_{7} and E_{8}. The general construction of η for an arbitrary irreducible finite reflection group is given in [9] and [10]. The metric η posesses flat coordinates which are given by an appropriate choice of generators p^{i} of the algebra R[V]^{W}.

By Dubrovin [4] the above two metrics form the so-called flat pencil of metrics which defines a unique structure of Frobenius manifold on V_{C}/W. Note that the components of the above metrics are polynomials with real coefficients and, therefore, there is the corresponding structure on V/W = p(V) ⊂ R^{n}. Further we consider both V_{C}/W and V/W with the above structures and call them Frobenius manifolds. An automorphism of the Frobenius manifold V/W (V_{C}/W) is a smooth diffeomorphism of V/W (a biholomorphic map of V_{C}/W onto itself) which preserves the above flat pencil of metrics.

5.1. The Frobenius manifolds V_{C}/W and V/W have no nontrivial automorphisms.

Proof. Let f be an automorphism of the Frobenius manifold V/W. Consider the pull-back of the metrics on V/W, forming a flat pencil, via the map p. The pullback of the first of them is the Euclidean metric g(V) and it defines the structure of Euclidean space on V. The pull-back of the second one is a flat metric on V which is non-degenerate on V_{reg}. The flat coordinates of this metric are the compositions of the flat coordinates of the metric η with p and then they give some system of generators of R[V]^{W}. Therefore, by Theorem 1.5 the above pull-backs of metrics determine uniquely the structure of W -module on V.

Consider the lift F of the automorphism f given by Theorem 3.1. The definition of the automorphism f implies that F is an automorphism of the W -module V and, hence, f is the identity map of V/W.

Note that each automorphism f of the Frobenius manifold V_{C}/W preserves the set of degenerate points of the metric g̃^{ij}, i.e. the discriminant locus D. Then the statement of the theorem for V_{C}/W follows from Theorem 4.1. □

Note that using Theorems 3.6 and 4.4, i.e. the local versions of Theorems 3.1 and 4.1, one can prove that the Frobenius manifolds V/W and V_{C}/W have no nontrivial local automorphisms.

References

1. D.V. Alekseevsky, A. Kriegl, M. Losik, P.W. Michor, Choosing roots of polynomials smoothly, to appear in Israel J. Math..

2. N. Bourbaki, Groupes et algèbres de Lie, vol. Ch. 4–6, Masson, Paris, 1981.

3. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782.

4. B. Dubrovin, Geometry of 2D topological field theories, In: Springer LMN 1620 (1996), 120– 348.

5. F.R.Gantmacher, The theory of matrices, vol. 1, Chelsea publishing company, New Jork, 1990.

6. J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.

7. D. Luna, Fonctions différentiables invariantes sous l’operation d’une groupe réductif, Ann. Inst. Fourier, Grenoble 26 (1976), no. 1, 33–49.

8. C. Prochesi, G.W. Schwarz, Inequalities defining orbit spaces, Invent. Math. 81 (1985), 539– 554.

9. K. Saito, On a linear structure of a quotient variety by a finite reflection group, Preprint RIMS-288 (1979).

10. K. Saito, On a linear structure of the quotient variety by a finite reflection group, Publ. RIMS, Kyoto Univ. 29 (1993), 535–579.

11. K. Saito, T. Yano, J. Sekuguchi, On a certain generator system of a ring of invariants of a finite reflection group, Comm. in Algebra 8(4) (1980), 373-408.

12. G.W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68.

13. G.C. Shephard, J.A. Todd, Unitary groups generated by reflections, Canad. J. Math. 6 (1954), 274–304.

14. L. Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57–64.

15. V.M. Zakalyukin Perestroikas of wave fronts that depend on a parameter, Funktsional. Anal.i Prilozhen. 10 (1976), no. 2, 69–70 (Russian); English transl. in Funct. Anal. Appl. 10 (1976), 139–140.

Saratov University, Astrakhanskaya 83, 410026 Saratov, Russia