Propagation of singularities for microdifferential equations with multiple selftangential involutory characteristics (Q1116011)

Let M be a real analytic manifold with a complexification X, and let \({\mathcal M}\) be a coherent \({\mathcal E}_ X\) module defined in a neighborhood of \(\rho_ 0\in \dot T^*_ MX\) whose characteristic variety is written in the form (1) Ch(\({\mathcal M})=\{\rho \in T^*X\); \(p_ 1(\rho)\cdot p_ 2(\rho)...p_{\ell}(\rho)=0\}\) by homogeneous holomorphic functions \(p_ 1\), \(p_ 2\) and \(p_{\ell}\) defined in a neighborhood of \(\rho_ 0\in \dot T^*_ MX\). We assume the following conditions: (2) \(p_ 1,...,p_{\ell -1}\) and \(p_{\ell}\) are real valued on \(T^*_ MX\). We set \(S_ j=\{\rho \in T^*_ MX\); \(p_ j=0\}\) (1\(\leq j\leq \ell)\) and assume (3) \(S_ j's\) are non-radical non- singular hypersurfaces and \(\Sigma =\cap_{1\leq j\leq \ell}S_ j\) is a regular involutory submanifold in \(T^*_ MX\) of codimension d. (4) \(S_ i\) and \(S_ j\) are tangent to each other of order \(k_ 0\) on \(\Sigma\). In the above situation, we study propagation of singularities of solutions to \({\mathcal M}\) on the regular involutory submanifold \(\Sigma\).