Lie algebras of differential operators for matrix valued Laguerre type polynomials

We study algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) with respect to a weight matrix of the form \(W^{(\nu )}_{\phi }(x) = x^{\nu }e^{-\phi (x)} W^{(\nu )}_\textrm{pol}(x)\) $W_{\varphi }^{(\nu )}(x)=x^{\nu }{e}^{-\varphi (x)}{W}_{\text{pol}}^{(\nu )}(x)$ , where \(\nu >0\) $\nu >0$ , \(W^{(\nu )}_\textrm{pol}(x)\) $W_{\text{pol}}^{(\nu )}(x)$ is a certain matrix valued polynomial and \(\phi \) $\varphi$ is an analytic function. We introduce differential operators \({\mathcal {D}}\) $\mathcal{D}$ , \({\mathcal {D}}^{\dagger }\) ${\mathcal{D}}^{†}$ which are mutually adjoint with respect to the matrix inner product induced by \(W^{(\nu )}_{\phi }(x)\) $W_{\varphi }^{(\nu )}(x)$ . We prove that the Lie algebra generated by \({\mathcal {D}}\) $\mathcal{D}$ and \({\mathcal {D}}^{\dagger }\) ${\mathcal{D}}^{†}$ is finite dimensional if and only if \(\phi \) $\varphi$ is a polynomial. For a polynomial \(\phi \) $\varphi$ , we describe the structure of this Lie algebra. As a byproduct, we give a partial answer to a problem by Ismail about finite dimensional Lie algebras related to scalar Laguerre type polynomials. The case \(\phi (x)=x\) $\varphi (x)=x$ is discussed in detail.