Some Conjectures About Invariant Theory and their Applications

It turns out that various algebraic computations can be reduced to the same type of computations: one has to study the series of integrals ∫ K f(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 ∫ K f(k) dk = 0 for any n > 0. Then for any K-finite function g, we have ∫ K f(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.