Laguerre functions on symmetric cones and recursion relations in the real case (Q849611)

The Laguerre polynomials over \(\mathbb R\) are defined by the formula \[ L^\nu_m(x)=\sum_{k=0}^m \frac{\Gamma(m+\nu)}{\Gamma (k+\nu)} \left( \begin{matrix} m\\ k\end{matrix}\right) (-x)^k , \] and the Laguerre functions are defined by \[ \ell_m^\nu(x)=e^{-x}L^\nu_m(2x). \] The following differential recursion relations hold: \[ \left( xD^2+\nu D-x\right) \ell_m^\nu (x) =-(2m+\nu)\ell_m^\nu (x), \] \[ \left( xD^2+(2x+\nu)D +(x+\nu)\right) \ell_m^\nu (x) =-2m(\nu+m-1)\ell_{m-1}^\nu (x), \] \[ \left( xD^2-(2x-\nu)D +(x-\nu)\right) \ell_m^\nu (x) =-2\ell_{m+1}^\nu (x). \] The article under review generalizes these formulas via the representation theory of \(\text{Sp}(n,\mathbb R)\) to Laguerre functions defined on the cone of positive definite real symmetric matrices, where \[ \text{Sp}(n,\mathbb R)=\left\{ g\in \text{SL}(2n,\mathbb R) : gjg^t=j\right\}, \] \[ j=\left( \begin{matrix} 0& I\\ -I & 0 \end{matrix}\right)\in \text{SL}(2n,\mathbb R) \quad \text{ and } I \text{ is an } n\times n \text{ identity matrix}. \] Let \(J=\text{Sym}(n,\mathbb R)\) be the Jordan algebra of symmetric real \(n\times n\) matrices, \(\Omega\) be an open selfdual cone of positive definite matrices, and \(T(\Omega)=\Omega +iJ\) be a tube domain. The authors discuss the action of the group \(\text{Sp}(n,\mathbb R)\) on \(T(\Omega)\) and derive differential recursion relations for the Laguerre functions on the cone \(\Omega\) of positive definite real matrices. The highest weight representations of the group \(\text{Sp}(n,\mathbb R)\) play an important role in this. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain.