Séminaires et Congrès - 2 - pages 263-279 < Séminaires et Congrès < 2

Algèbre non commutative, groupes quantiques et invariants Septième contact Franco-Belge, Reims, Juin 1995
J. Alev, G. Cauchon (Éd.)
Séminaires et Congrès 2 (1997), 304 pages

Some Conjectures About Invariant Theory and their Applications
Olivier MATHIEU
Séminaires et Congrès 2 (1997), 263-279
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Résumé :
It turns out that various algebraic computations can be reduced to the same type of computations: one has to study the series of integrals $\int _{K}f^{n}(k) g(k)\,dk$, where f,g are complex valued K-finite functions on a compact Lie group K. So it is tempting to state a general conjecture about the behavior of such integrals, and to investigate the consequences of the conjecture.

MAIN CONJECTURE : Let K be a compact connected Lie group and let f be a complex-valued K-finite function on K such that $\int _{K} f^{n}(k)\,dk=0$ for any n> 0. Then for any K-finite function g, we have $\int _{K} f^{n}(k)g(k)\,dk=0$ for n large enough.

Especially, we prove that the main conjecture implies the jacobian conjecture. Another very optimistic conjecture is proposed, and its connection to isospectrality problems is explained.

Class. math. : 14E07

ISBN : 2-85629-052-3
Publié avec le concours de : Centre National de la Recherche Scientifique