# root system generalizations of Sekiguchi-Debiard (aka Laplace-Beltrami) operators

For the root system $A_n$, taking the limit $q = t^\alpha$ and $t \to 1$, and letting $Y = (t-1) X -1$ one obtains from the Macdonald operator the so-called Sekiguchi-Debiard operator: $$D_\alpha(X) = a_\delta(x)^{-1} \sum_{w \in S_n} \epsilon(w) x^{w \delta} \prod_{j=1}^n (X + (w \delta)_j + \alpha x_j \partial_j).$$

The eigenfunctions of this family of operators indexed by powers of $X$ are called symmetric Jack polynomials with parameter $\alpha$. It is well-known that Heckman-Opdam polynomials generalize Jack symmetric polynomials to other root systems. My question is, are there generalizations of the Laplace-Beltrami operators to root systems as well? The operator is defined in $A_{n+1}$ up to affine transformation as $$D_\alpha^2 f = \frac{\alpha}{2} \sum_{i=1}^n (x_i \partial_i)^2 + \sum_{i < j} \frac{x_i^2 \partial_i - x_j^2 \partial_j}{x_i - x_j}.$$

Applying this operator to power sum polynomial $p_\lambda$ has a natural random walk interpretation, see the paper "a probabilistic interpretation of Macdonald polynomials" for detail.

Edit: I figured out that the notation $\partial_\alpha$ with respect to a weight $\alpha$ is indeed the same as $x_i \partial_i$ since the latter is differentiation with respect to maximal torus element, i.e., $\partial_\alpha e^{k \alpha} = k e^{k \alpha} = \partial_{x_i} x_i^k$ if $e^\alpha = x_i$. Thus Heckman-Opdam operators do provide the Laplace Beltrami operators in the other root systems.

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