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Microlocal structure

The ring of microdifferential operators with constant coefficients

$\displaystyle {\mathbf{C}\{\!\{\partial^{-1}_t\}\!\}}\,:=\,\Biggl\{\sum_{i\ge0}...

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$.

Christoph Lossen