Skip to main content Accessibility help
×
Home
Hostname: page-component-544b6db54f-4nk8m Total loading time: 0.189 Render date: 2021-10-24T15:33:36.279Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Lifting Quasianalytic Mappings over Invariants

Published online by Cambridge University Press:  20 November 2018

Armin Rainer*
Affiliation:
Fakultät für Mathematik, Universität Wien, A-1090 Wien, Austria email: armin.rainer@univie.ac.at
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\rho :\,G\,\to \,\text{GL}\left( V \right)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$ , and let ${{\sigma }_{1}},\ldots ,{{\sigma }_{n}}$ be a system of generators of the algebra of invariant polynomials $\mathbb{C}{{\left[ V \right]}^{G}}$ . We study the problem of lifting mappings $f:\,{{\mathbb{R}}^{q}}\,\supseteq \,U\,\to \,\sigma \left( V \right)\,\subseteq \,{{\mathbb{C}}^{n}}$ over the mapping of invariants $\sigma \,=\,\left( {{\sigma }_{1}},\ldots ,{{\sigma }_{n}} \right):\,V\,\to \,\sigma \left( V \right)$ . Note that $\sigma \left( V \right)$ can be identified with the categorical quotient $V//G$ and its points correspond bijectively to the closed orbits in $V$ . We prove that if $f$ belongs to a quasianalytic subclass $C\subseteq {{C}^{\infty }}$ satisfying some mild closedness properties that guarantee resolution of singularities in $C$ , e.g., the real analytic class, then $f$ admits a lift of the same class $C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\text{SB}{{\text{V}}_{\text{loc}}}$ , i.e., special functions of bounded variation. If $\rho $ is a real representation of a compact Lie group, we obtain stronger versions.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Alekseevsky, D., Kriegl, A. Losik, M. and Michor, P. W., Lifting smooth curves over invariants for representations of compact Lie groups. Transform. Groups 5(2000), no. 2, 103110. http://dx.doi.org/10.1007/BF01236464 Google Scholar
[2] Ambrosio, L., Fusco, N. and Pallara, D. Functions of bounded variation and free discontinuity problems. The Clarendon Press, Oxford University Press, New York, 2000.Google Scholar
[3] Bierstone, E., Lifting isotopies from orbit spaces. Topology 14(1975), no. 3, 245252. http://dx.doi.org/10.1016/0040-9383(75)90005-1 Google Scholar
[4] Bierstone, E. and Milman, P. D., Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 542.Google Scholar
[5] Bierstone, E. and Milman, P. D., Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(1997), no. 2, 207302. http://dx.doi.org/10.1007/s002220050141 Google Scholar
[6] Bierstone, E. and Milman, P. D., Resolution of singularities in Denjoy-Carleman classes. Selecta Math. 10(2004), no. 1, 128. http://dx.doi.org/10.1007/s00029-004-0327-0 Google Scholar
[7] Dadok, J. and Kac, V. Polar representations. J. Algebra 92(1985), no. 2, 504524. http://dx.doi.org/10.1016/0021-8693(85)90136-X Google Scholar
[8] De Giorgi, E. and L.Ambrosio, New functionals in the calculus of variations. (Italian) Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82(1988), no. 2, 199210 .Google Scholar
[9] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I. II. Ann. of Math. (2) 79 (1964), 109203; 205–326.Google Scholar
[10] Kriegl, A., Losik, M. Michor, P. W., and Rainer, A. Lifting smooth curves over invariants for representations of compact Lie groups. II. J. Lie Theory 15(2005), no. 1, 227234.Google Scholar
[11] Kriegl, A., Lifting smooth curves over invariants for representations of compact Lie groups. III. J. Lie Theory 16(2006), no. 3, 579600.Google Scholar
[12] Kriegl, A., Addendum to: “Lifting smooth curves over invariants for representations of compact Lie groups. III” [J. Lie Theory 16 (2006), no. 3, 579–600]. J. Lie Theory 22(2012), no. 1, 245249.Google Scholar
[13] Kriegl, A., Lifting mappings over invariants of finite groups. Acta Math. Univ. Comenian. 77(2008), no. 1, 93122.Google Scholar
[14] Kriegl, A. and Michor, P. W., The convenient setting of global analysis. Mathematical Surveys and Monographs 53. American Mathematical Society, Providence, RI, 1997,CrossRefGoogle Scholar
[15] Kriegl, A., Michor, P. W., and Rainer, A. The convenient setting for non-quasianalytic Denjoy-Carleman differentiable mappings. J. Funct. Anal. 256(2009), no. 11, 35103544. http://dx.doi.org/10.1016/j.jfa.2009.03.003 Google Scholar
[16] Losik, M., Lifts of diffeomorphisms of orbit spaces for representations of compact Lie groups. Geom. Dedicata 88(2001), no. 1-3, 2136. http://dx.doi.org/10.1023/A:1013180828701 Google Scholar
[17] Losik, M., Michor, P. W., and Rainer, A. A generalization of Puiseux's theorem and lifting curves over invariants. Rev. Mat. Complut., Published online 22 February, 2011. http://dx.doi.org/10.1007/s13163-011-0062-y CrossRefGoogle Scholar
[18] Luna, D., Slices étales. In: Sur les groupes algébriques. Bull. Soc. Math. France Mém. 33. Société Mathématique de France, Paris, 1973, pp. 81105.Google Scholar
[19] Montgomery, D. and Yang, C. T., The existence of a slice. Ann. of Math. 65(1957), 108116. http://dx.doi.org/10.2307/1969667 Google Scholar
[20] Palais, R. S., The classification of G-spaces. Mem. Amer. Math. Soc. No. 36, 1960.Google Scholar
[21] Procesi, C. and Schwarz, G. Inequalities defining orbit spaces. Invent. Math. 81(1985), no. 3, 539554. http://dx.doi.org/10.1007/BF01388587 Google Scholar
[22] Rainer, A., Quasianalytic multiparameter perturbation of polynomials and normal matrices. Trans. Amer. Math. Soc. 363(2011), no. 9, 49454977. http://dx.doi.org/10.1090/s0002-9947-2011-05311-0 Google Scholar
[23] Schwarz, G. W., Lifting smooth homotopies of orbit spaces. Inst. Hautes Études Sci. Publ. Math. (1980), no. 51, 37135.Google Scholar
[24] Thilliez, V., On quasianalytic local rings. Expo. Math. 26(2008), no. 1, 123.Google Scholar
[25] Vinberg, È. B. and Popov, V. L., Invariant theory. In: Algebraic Geometry 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 137314, 315.Google Scholar
You have Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Lifting Quasianalytic Mappings over Invariants
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Lifting Quasianalytic Mappings over Invariants
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Lifting Quasianalytic Mappings over Invariants
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *