Microlocal structure

The ring of microdifferential operators with constant coefficients

$${\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}\,:=\,\Biggl\{\sum_{i\ge0}... ...}_t]]\;\Bigg\vert\; \sum_{i\ge0}\frac{a_i}{i!}t^i\in{\mathbf{C}\{t\}}\Biggr\}$$

is a discrete valuation ring and ${\mathbf{C}\{t\}}$ is a free ${\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}$-module of rank $1$. For $\alpha>-1$, $\partial_t t: C_\alpha \to C_\alpha$ is bijective and hence $t^{\partial_t t\vert _{C_\alpha}}$ is a ${\mathbf{C}\{t\}}$-automorphism of ${\mathbf{C}\{t\}} \cdot C_\alpha$ mapping the trivial ${\mathbf{C}\{t\}}[\partial_t t]$-structure to that of ${\mathbf{C}\{t\}} \cdot C_\alpha$. Hence, ${\mathbf{C}\{t\}} \cdot C_\alpha$ is a free ${\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}$-module of rank $\dim_\mathbf{C}C_\alpha$.

In particular, $V^\alpha$, resp. $V^{>\alpha}$, is a free ${\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}$-module of rank $\mu$ for $\alpha>-1$, resp. $\alpha\ge-1$. Hence, $\mathcal{G}_0$ is a $\mu$-dimensional vector space over the quotient field $\mathbf{C}(\partial^{-1}_t):={\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}[\partial_t]$. Since $\partial^{-1}_t\mathcal{H}''_0\subset\mathcal{H}''_0$ and $\mathcal{H}''_0\subset V^{>-1}$, $\mathcal{H}''_0$ is a free ${\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}$-module of rank $\mu$.