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.