Some Conjectures About Invariant Theory and their Applications
Some Conjectures About Invariant Theory and their Applications
Séminaires et Congrès | 1997
- Consulter un extrait
- Année : 1997
- Tome : 2
- Format : Papier
- Langue de l'ouvrage :
Anglais - Class. Math. : 14E07
- Pages : 263-279
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.