We study Bessel processes on Weyl chambers of types A and B on $\mathbb R^N$. Using elementary symmetric functions, we present several space-time-harmonic functions and thus martingales for these processes $(X_t)_{t\ge0}$ which are independent from one parameter of these processes. As a consequence, $p(y):=\mathbb E(\prod_{i=1}^N (y-X_t^i))$ can be expressed via classical orthogonal polynomials. Such formulas on characteristic polynomials admit interpretations in random matrix theory where they are partially known by Diaconis, Forrester, and Gamburd.