On the Direct Problem in Differential Galois Theory for the Classical Groups

Let $G$ be a classical group of Lie rank $l$ and let $C$ be an algebraically closed field of characteristic zero. For $l$ differential indeterminates $\boldsymbol{v}=(v_1,\dots,v_l)$ over $C$ we constructed in a previous paper a general Picard-Vessiot extension $\mathcal{E}$ of the differential field $C\langle \boldsymbol{s}(\boldsymbol{v})\rangle$ having differential Galois group $G(C)$. Here $\boldsymbol{s}(\boldsymbol{v})=(s_1(\boldsymbol{v}),\dots,s_l(\boldsymbol{v}))$ are certain differential polynomials in $C\{\boldsymbol{v} \}$ which are differentially algebraically independent over $C$. The linear differential equation defining $\mathcal{E}$ is defined by the normal form matrix $A_{G}( \boldsymbol{s}(\boldsymbol{v}))$ lying in the Lie algebra of $G$. In the first part of this paper we analyze the structure of $\mathcal{E}$ induced by the action of the standard parabolic subgroups of $G(C)$ on $\mathcal{E}$. In the second part we consider specializations $A_{G}(\boldsymbol{s}(\boldsymbol{v})) \to A_{G}(\overline{\boldsymbol{s}})$ with $\overline{\boldsymbol{s}} \in C(z)^l$ of the normal form matrix for $G$ of type $A_l$, $B_l$, $C_l$ or $\mathrm{G}_2$ (here $l=2$). We show how one can combine the results of the first part with known algorithms for the computation of the differential Galois group and its Lie algebra to determine the differential Galois group of certain specialized equations $\partial(\boldsymbol{y}) = A_{G}(\overline{\boldsymbol{s}})\boldsymbol{y}$ over $C(z)$ with $C$ a computable algebraically closed field of characteristic zero.