  We will define the Alexander duality for strongly stable ideals. More
precisely, for a strongly stable ideal $I \subset \Bbbk[x_1, \ldots, x_n]$ with
${\rm deg}(\mathsf{m}) \le d$ for all $\mathsf{m} \in G(I)$, its dual $I^*
\subset \Bbbk[y_1, \ldots, y_d]$ is a strongly stable ideal with ${\rm
deg}(\mathsf{m}) \le n$ for all $\mathsf{m} \in G(I^*)$. This duality has been
constructed by Fl$\o$ystad et al. in a different manner, so we emphasis
applications here. For example, we will describe the Hilbert serieses of the
local cohomologies $H_\mathfrak{m}^i(S/I)$ using the irreducible decomposition
of $I$ (through the Betti numbers of $I^*$).
