Invariant functions in Denjoy–Carleman classes
Let be a real finite dimensional representation of a compact Lie group . It is well-known that the algebra of -invariant polynomials on is finitely generated, say by . Schwarz  proved that each -invariant -function on has the form for a -function on . We investigate this representation within the framework of Denjoy–Carleman classes. One can in general not expect that and lie in the same Denjoy–Carleman class (with ). For finite groups and (more generally) for polar representations we show that for each -invariant of class there is an of class such that , if is strongly regular and satisfies
where is an (explicitly known) integer depending only on the representation. In particular, each -invariant -Gevrey function (with ) has the form for a -Gevrey function . Applications to equivariant functions and basic differential forms are given.
Key words and phrases:Denjoy–Carleman class, invariant functions
2000 Mathematics Subject Classification:26E10, 58C25, 57S15
Let be a real finite dimensional representation of a compact Lie group . By a classical theorem due to Hilbert the algebra of -invariant polynomials on is finitely generated. Choose a system of homogeneous generators of and define . Schwarz  proved a smooth analog of Hilbert’s theorem for orthogonal representations of compact Lie groups : the induced mapping is surjective. Mather  showed that this mapping is even split surjective.
In this paper we treat Schwarz’s theorem in the framework of Denjoy–Carleman classes. These classes of smooth functions play an important role in harmonic analysis and various branches of differential equations (especially Gevrey classes). Let be a non-decreasing sequence of real numbers with . A smooth function in an open subset belongs to the Denjoy–Carleman class if for any compact subset there exist positive constants and such that
for all and . See section 2 for more on Denjoy–Carleman classes. As examples (, see also 3.3) show, one cannot expect in general that a smooth -invariant function on of class has the form for a function of the same class .
For finite groups and (more generally) for polar representations we prove that the representation holds in the context of Denjoy–Carleman classes, where has lower regularity than . More precisely: Let be a subgroup of finite order of . Let and be sequences satisfying some mild conditions which guarantee stability under composition and derivation for and (see 2.1). Assume that is strongly regular (see 2.6) and that
Then for any -invariant function there exists a function such that . In particular: Any -invariant Gevrey function (with ) has the form for a Gevrey function . See theorem 3.4. The result does not depend on the choice of generators , since any two choices differ only by a polynomial diffeomorphism and the involved Denjoy–Carleman classes are stable under composition.
Note that Thilliez  treats a very similar problem: For a compact subset , an analytic mapping on an open neighborhood of , and a function of the form with for an open neighborhood of , the existence of a sequence such that is investigated. This is done by studying the complex setting: Now is compact in , is a -valued holomorphic mapping defined near , and is on and -flat on . However, our results are not covered by Thilliez’, since the minimal number of generators of does in general not coincide with the dimension of the representation space .
We prove the main theorem in section 3. We shall deduce it from an analog theorem (see 3.3) due to Bronshtein [7, 8] which treats the standard representation of the symmetric group in . This method is inspired by Barbançon and Raïs  deploying Weyl’s account  of Noether’s  proof of Hilbert’s theorem.
The rest of the paper is devoted to several applications of this main theorem. In section 4 we treat the presentation in Denjoy–Carleman classes of equivariant functions between representations of a finite group.
In section 5 the main theorem 3.4 is generalized to polar representations, i.e., orthogonal finite dimensional representations of a compact Lie group allowing a linear subspace which meets each orbit orthogonally (see theorem 5.2). The trace of the -action in is the action of the generalized Weyl group which is a finite group. In analogy with a result due to Palais and Terng , which states that restriction induces an isomorphism , we show that each -invariant function on of class has a -invariant extension to of class , where and are sequences with the aforementioned properties. More generally, Michor [28, 29] proved that restriction induces an isomorphism , where is the space of basic -forms on , i.e., -invariant forms that kill each vector tangent to some orbit. Our main theorem 3.4 allows to conclude that each -invariant -form on of class has a basic extension to of class (with and as above).
In  and [28, 29] the isomorphisms and are established in the more general setting of smooth proper Riemannian -manifolds with sections, where there exist closed submanifolds meeting each orbit orthogonally. In section 6 we explain that our analog results in the framework of Denjoy–Carleman classes generalize to real analytic proper Riemannian -manifolds with sections.
2. Denjoy–Carleman classes
2.1. Denjoy–Carleman classes of differentiable functions
We mainly follow  (see also the references therein). We use . For each multi-index , we write , , and .
Let be an increasing sequence () of real numbers with . Let be open. We denote by the set of all such that, for all compact , there exist positive constants and such that
for all and . The set is the Denjoy–Carleman class of functions on . If , for all , then coincides with the ring of real analytic functions on . In general, .
We assume that is logarithmically convex, i.e.,
or, equivalently, is increasing. Considering , we obtain that also is increasing and
Hypothesis (2.1.2) implies that is a ring, for all open subsets , which can easily be derived from (2.1.3) by means of Leibniz’s rule. Note that definition (2.1.1) makes sense also for functions . For -mappings, (2.1.2) guarantees stability under composition (, see also [4, 4.7]).
A further consequence of (2.1.2) is the inverse function theorem for (; for a proof see also [4, 4.10]): Let be a -mapping between open subsets . Let . Suppose that the Jacobian matrix is invertible. Then there are neighborhoods of , of such that is a -diffeomorphism.
Suppose that and satisfy , for all and a constant , or equivalently,
Setting in (2.1.4) yields that if and only if
Since is increasing (by logarithmic convexity), the strict inclusion is equivalent to
We shall also assume that is stable under derivation, which is equivalent to the following condition
Note that the first order partial derivatives of elements in belong to , where denotes the shifted sequence . So the equivalence follows from (2.1.4), by replacing with and with .
2.2. Quasianalytic function classes
Let denote the ring of formal power series in variables (with real or complex coefficients). We denote by the set of elements of for which there exist positive constants and such that
for all . A class is called quasianalytic if, for open connected and all , the Taylor series homomorphism
2.3. Non-quasianalytic function classes
If is a DC-weight sequence which is not quasianalytic, then there are partitions of unity. Namely, there exists a function on which does not vanish in any neighborhood of 0 but which has vanishing Taylor series at 0. Let for and for . From we can construct bump functions as usual.
2.4. Strong non-quasianalytic function classes
2.5. Moderate growth
A DC-weight sequence has moderate growth if
2.6. Strong regularity
2.7. Whitney’s extension theorem
Let be compact. Denote by the Whitney jets on . We say that is a -jet on , or belongs to , if there exist positive constants and such that
for all and and
for all , all with and all , where
2.8. Gevrey functions
Let and put , for . Then is strongly regular. The corresponding class of functions is the Gevrey class .
2.9. More examples
Let and put , for . Then is quasianalytic for and non-quasianalytic (but not strongly) for .
Let and put , for . The corresponding -functions are called -Gevrey regular. Then is strongly non-quasianalytic but not of moderate growth, thus not strongly regular.
2.10. Spaces of -functions
Let be open. For any and compact with smooth boundary, define
It is easy to see that is a Banach space. In the description of , instead of compact with smooth boundary, we may also use open with compact in , like . Or we may work with Whitney jets on compact , like .
The space carries the projective limit topology over compact of the inductive limit over :
One can prove that, for , the canonical injection is a compact mapping (see ). Hence is a Silva space, i.e., an inductive limit of Banach spaces such that the canonical mappings are compact.
2.11. Polynomials are dense in
Let be a DC-weight sequence and let be open. It is proved in [20, 3.2] (see also [17, 3.2]) that the space of entire functions is dense in . Since the polynomials are dense in and the inclusion is continuous, we obtain that the polynomials are dense in . For convenience we give a proof.
Let be a DC-weight sequence and let be open. Then is dense in .
Proof. Let and compact. Let such that , where . Let with , , and compact support . We define for
for all and , and hence
We have for and
By the generalized mean value theorem we have for , , and
Choose such that . Then for , , and
where is a positive constant. For all we have
for a constant independent of . Thus there exist positive constants and independent of , , and such that
We have for
where denotes the Lebesgue measure of . Cauchy’s inequalities imply for each and
where . Choosing we get for
Hence with we obtain for
2.12. Closed ideals
Let be open. Let . Consider the principal ideal generated by .
Assume that is stable under derivation (2.1.5). Let be a linear form on . Then the ideal is closed in . More generally, assume that is a finite product of linear forms . Then is closed in .
Proof. Let . Then , since evaluation at points is continuous. As is stable under derivation, the standard integral formula (after suitable linear coordinate change) implies that for a unique . The same reasoning shows that is closed in , where .
For the general statement it suffices to show: Let be a polynomial and a power of a linear form. If and are relatively prime and both generate closed ideals in , then is closed in . For we find functions with . Since and are relatively prime, we have . By the standard integral formula we obtain as above with . Hence the assertion. ∎
Note that for any hyperbolic polynomial the principal ideal is closed in (e.g. [42, 4.2]). This follows from the fact (due to ) that Weierstrass division holds in for hyperbolic divisors. A polynomial with and , for , is called hyperbolic if, for each , all roots of are real.
Let be a DC-weight sequence, and let be a real analytic manifold. We can define the space of functions of Denjoy–Carleman class on by means of local coordinate systems, since contains the real analytic functions and is stable under composition. Similarly, we may consider the space of -forms of class on .
3. Invariant functions in Denjoy–Carleman classes
Throughout this paper we consider a compact Lie group acting smoothly on a manifold . A function on is said to be -invariant if for all and all . If is a set of functions on , then denotes the subset of -invariant elements in .
3.1. Hilbert’s theorem
(e.g. ) Let be a compact Lie group and let be a real finite dimensional -module. Then, by a theorem due to Hilbert, the algebra of -invariant polynomials on is finitely generated. The generators can be chosen homogeneous and with positive degree.
3.2. Schwarz’s theorem
3.3. Symmetric functions in Denjoy–Carleman classes
In the case that the symmetric group acts in by permuting the coordinates, the statement of Schwarz’s theorem 3.2 is due to Glaeser . In that case is the -th elementary symmetric function, i.e., , and .
The representation of symmetric functions in Denjoy–Carleman (Gevrey) classes was treated by Bronshtein [7, 8]. Since we shall need it later, we present a more general version and we sketch a proof. Let act in by permuting the coordinates. Since , a -invariant function on has the form with .
Assume that and are increasing logarithmically convex sequences with . Then for any function there exists a function such that if and only if
Sketch of proof. We indicate and adapt the main steps in Bronshtein’s proof. The necessity of (3.3.1) is shown by considering the symmetric function (for ) given by
where and . Then with
Since this implies (3.3.1). For one can find a similar example.
Without loss suppose that . Instead of the elementary symmetric polynomials we use the Newton polynomials and put (see remark 3.4(3)). Then we may write where , , and . A direct computation gives