V-filtration

Since $t^{\alpha-\frac{N}{2\pi i}}$ is invertible, $\psi_\alpha(A):=(i_{*}s(A,\alpha))_0$ defines an inclusion $\psi_{\alpha}:H^n(X^\infty,\mathbf{C})\hookrightarrow \mathcal{G}_0$, satisfying the relations $\partial_t\psi_{\alpha}= \psi_{\alpha-1}\bigl(\alpha-\frac{N}{2\pi i}\bigr)$ and $t\psi_{\alpha}=\psi_{\alpha+1}$, by definition of $s(A,\alpha)$. This implies $(t\partial_t\!-\alpha)\psi_\alpha=\psi_\alpha\bigl(-\frac{N}{2\pi i}\bigr)$, $\exp(-2\pi i t\partial_t)\psi_\alpha=\psi_\alpha M_\lambda$, and the image

$$C_\alpha:=im\psi_\alpha\,=\,Ker\,(t\partial_t-\alpha)^{n+1}$$

of $\psi_\alpha$ is the generalized $\alpha$-eigenspace of $t\partial_t$. Moreover, $T:C_\alpha \to C_{\alpha+1}$ is bijective, and $\partial_t:C_\alpha \to C_{\alpha-1}$ is bijective for $\alpha\neq 0$. The V-filtration $V$ on $\mathcal{G}_0$ is defined by

$$V^\alpha :=\, V^\alpha\mathcal{G}_0 \,:=\, \sum_{\alpha\le\beta}{... ...alpha}\mathcal{G}_0 \,:=\, \sum_{\alpha<\beta}{\mathbf{C}\{t\}} \cdot C_\beta.$$

$V^\alpha$ and $V^{>\alpha}$ are free ${\mathbf{C}\{t\}}$-modules of rank $\mu$ with $V^\alpha\big/V^{>\alpha}\cong C_\alpha$.