Elementary symmetric polynomials and martingales for Heckman-Opdam processes

Abstract

We consider the generators $L_k$ of Heckman-Opdam diffusion processes in the compact and non-compact case in $N$ dimensions for root systems of type $A$ and $B$, with a multiplicity function of the form $k=\kappa k_0$ with some fixed value $k_0$ and a varying constant $\kappa \in [0,\infty [$. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the $L_k$ for all $\kappa \in ]0,\infty [$. This leads to martingales associated with the Heckman-Opdam diffusions $(X_{t,1},\dots ,X_{t,N}{)}_{t\ge 0}$. As our results extend to the freezing case $\kappa =\infty$ with a deterministic limit after some renormalization, we find formulas for the expectations $\mathbb{E}({\prod }_{j=1}^N(y-X_{t,j})),$ $y\in \mathbb{C}$.

revised version