Root Parametrized Differential Equations for the classical groups

Abstract

Let $C \langle t_1, \dots t_l\rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound criterion for the differential Galois group $G(C)$ of a matrix parameter differential equation $\partial(\boldsymbol{y})=A(\boldsymbol{t})\boldsymbol{y}$ over $C \langle t_1, \dots t_l\rangle$ and we prove that every connected linear algebraic group is the Galois group of a linear parameter differential equation over $C\langle t_1 \rangle$. As a second application we compute explicit and nice linear parameter differential equations over $C\langle t_1, \dots, t_l \rangle$ for the groups $\mathrm{SL}_{l+1}(C)$, $\mathrm{SP}_{2l}(C)$, $\mathrm{SO}_{2l+1}(C)$, $\mathrm{SO}_{2l}(C)$, i.e. for the classical groups of type $A_l$, $B_l$, $C_l$, $D_l$, and for $\mathrm{G}_2$ (here $l=2$).