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 $W^{(\nu)}_{\phi}(x) = x^{\nu}e^{-\phi(x)} W^{(\nu)}_{pol}(x)$, where $\nu>0$, $W^{(\nu)}_{pol}(x)$ is certain matrix valued polynomial and $\phi$ an entire function. We introduce a pair differential operators $\mathcal{D}$, $\mathcal{D}^{\dagger}$ which are mutually adjoint with respect to the matrix inner product induced by $W^{(\nu)}_{\phi}(x)$. We prove that the Lie algebra generated by $\mathcal{D}$ and $\mathcal{D}^{\dagger}$ is finite dimensional if and only if $\phi$ is a polynomial, giving a partial answer to a problem by M. Ismail. In the case $\phi$ polynomial, we describe the structure of this Lie algebra. The case $\phi(x)=x$, is discussed in detail. We derive difference and differential relations for the MVOPs. We give explicit expressions for the entries of the MVOPs in terms of classical Laguerre and Dual Hahn polynomials.