Skip to main content
SpringerLink
Log in

Invariant functions in Denjoy–Carleman classes

  • Original Paper
  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Let V be a real finite dimensional representation of a compact Lie group G. It is well known that the algebra \({\mathbb{R}[V]^G}\) of G-invariant polynomials on V is finitely generated, say by σ 1, . . . , σ p . Schwarz (Topology 14:63–68, 1975) proved that each G-invariant C ∞-function f on V has the form f = F(σ 1, . . . , σ p ) for a C ∞-function F on \({\mathbb{R}^p}\). We investigate this representation within the framework of Denjoy–Carleman classes. One can in general not expect that f and F lie in the same Denjoy–Carleman class C M (with M = (M k )). For finite groups G and (more generally) for polar representations V, we show that for each G-invariant f of class C M there is an F of class C N such that f = F(σ 1, . . . , σ p ), if N is strongly regular and satisfies

$$\sup_{k \in \mathbb{N}_{ > 0}}\left(\frac{M_{km}}{N_k}\right)^{\frac{1}{k}} < \infty,$$

where m is an (explicitly known) integer depending only on the representation. In particular, each G-invariant (1 + δ)-Gevrey function f (with δ > 0) has the form f = F(σ 1, . . . , σ p ) for a (1 + δm)-Gevrey function F. Applications to equivariant functions and basic differential forms are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Barbançon G.: Théorème de Newton pour les fonctions de class C r. Ann. Sci. École Norm. Sup. (4) 5, 435–457 (1972)

    MATH  Google Scholar 

  2. Barbançon G.: Invariants de classe C r des groupes finis engendrés par des réflexions et théorème de Chevalley en classe C r. Duke Math. J. 53(3), 563–584 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barbançon G., Raïs M.: Sur le théorème de Hilbert différentiable pour les groupes linéaires finis (d’après E. Noether). Ann. Sci. École Norm. Sup. (4) 16(3), 355–373 (1984)

    Google Scholar 

  4. Bierstone E., Milman P.D.: Resolution of singularities in Denjoy–Carleman classes. Selecta Math. (N.S.) 10(1), 1–28 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bonet J., Braun R.W., Meise R., Taylor B.A.: Whitney’s extension theorem for nonquasianalytic classes of ultradifferentiable functions. Studia Math. 99(2), 155–184 (1991)

    MATH  MathSciNet  Google Scholar 

  6. Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris (1968)

  7. Bronshtein, M.D.: Representation of symmetric functions in Gevrey-Carleman spaces. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 149 (Issled. Linein. Teor. Funktsii.) XV, 116–126, 189 (1986), English transl. in J. Soviet Math. 42(2), 1621–1628 (1988)

  8. Bronshtein, M.D.: On the representation of symmetric functions in Gevrey-Carleman spaces. Izv. Vyssh. Uchebn. Zaved. Mat. (4), 13–18, 85 (1987)

  9. Bruna J.: An extension theorem of Whitney type for non-quasi-analytic classes of functions. J. London Math. Soc. (2) 22(3), 495–505 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carleman T.: Les fonctions quasi-analytiques. Collection Borel, Gauthier-Villars, Paris (1926)

    MATH  Google Scholar 

  11. Chaumat J., Chollet A.-M.: Théorème de Whitney dans des classes ultradifférentiables. C. R. Acad. Sci. Paris Sér. I Math. 315(8), 901–906 (1992)

    MATH  MathSciNet  Google Scholar 

  12. Chaumat J., Chollet A.-M.: Division par un polynôme hyperbolique. Canad. J. Math. 56(6), 1121–1144 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Dadok J., Kac V.: Polar representations. J. Algebra 92(2), 504–524 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  14. Denjoy A.: Sur les fonctions quasi-analytiques de variable réelle. C. R. Acad. Sci. Paris 173, 1320–1322 (1921)

    Google Scholar 

  15. Faà di Bruno C.F.: Note sur une nouvelle formule du calcul différentielle. Quart. J. Math. 1, 359–360 (1855)

    Google Scholar 

  16. Glaeser G.: Fonctions composées différentiables. Ann. of Math. (2) 77, 193–209 (1963)

    Article  MathSciNet  Google Scholar 

  17. Heinrich T., Meise R.: A support theorem for quasianalytic functionals. Math. Nachr. 280(4), 364–387 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hörmander L.: On the division of distributions by polynomials. Ark. Mat. 3, 555–568 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hörmander, L.: The analysis of linear partial differential operators. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256. Springer-Verlag, Berlin (1983), Distribution Theory and Fourier Analysis

  20. Hörmander, L.: Between distributions and hyperfunctions. Astérisque 131, 89–106 (1985). Colloquium in honor of Laurent Schwartz, vol. 1 (Palaiseau, 1983)

  21. Komatsu H.: Ultradistributions. I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973)

    MATH  MathSciNet  Google Scholar 

  22. Komatsu H.: The implicit function theorem for ultradifferentiable mappings. Proc. Japan Acad. Ser. A Math. Sci. 55(3), 69–72 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  23. Komatsu H.: Ultradifferentiability of solutions of ordinary differential equations. Proc. Japan Acad. Ser. A Math. Sci. 56(4), 137–142 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kostov V.P.: On the geometric properties of Vandermonde’s mapping and on the problem of moments. Proc. Roy. Soc. Edinburgh Sect. A 112(3–4), 203–211 (1989)

    MATH  MathSciNet  Google Scholar 

  25. Łojasiewicz S.: Division d’une distribution par une fonction analytique de variables réelles. C. R. Acad. Sci. Paris 246, 683–686 (1958)

    MATH  MathSciNet  Google Scholar 

  26. Łojasiewicz S.: Sur le problème de la division. Studia Math. 18, 87–136 (1959)

    MATH  MathSciNet  Google Scholar 

  27. Mather J.N.: Differentiable invariants. Topology 16(2), 145–155 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  28. Michor P.W.: Basic differential forms for actions of Lie groups. Proc. Amer. Math. Soc. 124(5), 1633–1642 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Michor P.W.: Basic differential forms for actions of Lie groups. II. Proc. Amer. Math. Soc. 125(7), 2175–2177 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Noether E.: Die Funktionalgleichungen der isomorphen Abbildung. Math. Ann. 77(4), 536–545 (1916)

    Article  MATH  MathSciNet  Google Scholar 

  31. Palais R.S.: On the existence of slices for actions of non-compact Lie groups. Ann. of Math. (2) 73, 295–323 (1961)

    Article  MathSciNet  Google Scholar 

  32. Palais R.S., Terng C.-L.: A general theory of canonical forms. Trans. Amer. Math. Soc. 300(2), 771–789 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  33. Palais, R.S., Terng, C.-L.: Critical point theory and submanifold geometry. Lecture Notes in Mathematics, vol. 1353. Springer-Verlag, Berlin (1988)

  34. Petzsche H.-J.: On E. Borel’s theorem. Math. Ann. 282(2), 299–313 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  35. Roumieu C.: Ultra-distributions définies sur R n et sur certaines classes de variétés différentiables. J. Analyse Math. 10, 153–192 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  36. Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)

    MATH  Google Scholar 

  37. Rumberger M.: Finitely differentiable invariants. Math. Z. 229(4), 675–694 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  39. Terng C.-L.: Isoparametric submanifolds and their Coxeter groups. J. Differential Geom. 21(1), 79–107 (1985)

    MATH  MathSciNet  Google Scholar 

  40. Thilliez V.: Sur les fonctions composées ultradifférentiables. J. Math. Pures Appl. (9) 76(6), 499–524 (1997)

    MATH  MathSciNet  Google Scholar 

  41. Thilliez V.: On closed ideals in smooth classes. Math. Nachr. 227, 143–157 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Thilliez V.: On quasianalytic local rings. Expo. Math. 26(1), 1–23 (2008)

    MATH  MathSciNet  Google Scholar 

  43. Weyl H.: The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton (1939)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Armin Rainer

Authors
  1. Armin Rainer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Armin Rainer.

Additional information

Communicated by: P. Michor (Vienna).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rainer, A. Invariant functions in Denjoy–Carleman classes. Ann Glob Anal Geom 35, 249–266 (2009). https://doi.org/10.1007/s10455-008-9135-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-008-9135-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation