$\displaystyle\simeq\bigoplus_{\underline{k}\vdash
l\colon\ell(\underline{k})=r}\mathcal{H}^{l}_{a}(\operatorname{IC}(N_{S_{1}}\times\ldots\times
N_{S_{r}},\Lambda^{k_{1}}_{S_{1}}\boxtimes\ldots\boxtimes\Lambda^{k_{r}}_{S_{r}}))$
$\displaystyle\simeq\bigoplus_{\underline{k}\vdash
l\colon\ell(\underline{k})=r}\>\>\bigoplus_{\underline{l}\vdash
l\colon\ell(\underline{l})=r}\mathcal{H}^{l_{1}}_{a}(\operatorname{IC}(N_{S_{1}},\Lambda^{k_{1}}_{S_{1}}))\otimes\ldots\otimes\mathcal{H}^{l_{r}}_{a}(\operatorname{IC}(N_{S_{r}},\Lambda^{k_{r}}_{S_{r}})),$

with $l=k$ (i.e. the two summation indexes both running over the same set of
partition of $l$ of length $r$). For each pair {$\underline{k}$,
$\underline{l}$} of partitions of $l$ there are two options: either
$l_{i}=k_{i}$ for all $i\in[r]$, or there exists $i_{0}\in[r]$ such that
$l_{i_{0}}>k_{i_{0}}$. In the latter case,
$\mathcal{H}^{l_{i_{0}}}_{a}(\operatorname{IC}(N_{S_{i_{0}}},\Lambda^{k_{i_{0}}}_{S_{i_{0}}}))=0$
by 9.6. Thus we obtain

$\displaystyle\mathcal{H}^{l}_{a}(\operatorname{IC}(N_{\underline{S}},\Lambda^{l}_{\underline{S}}))\simeq\bigoplus_{\underline{l}\vdash
l\colon\ell(\underline{l})=r}\mathcal{H}^{l_{1}}_{a}(\operatorname{IC}(N_{S_{1}},\Lambda^{l_{1}}_{S_{1}}))\otimes\ldots\otimes\mathcal{H}^{l_{r}}_{a}(\operatorname{IC}(N_{S_{r}},\Lambda^{l_{r}}_{S_{r}})).$

We conclude that

$\displaystyle\mathcal{H}^{2c(n,g)}_{a}(\mathscr{S}(\mathscr{L}_{\underline{n}}(d)|_{N_{\underline{S}}})$
$\displaystyle\simeq\bigoplus_{l\geq
0}\mathcal{H}^{l}_{a}(\operatorname{IC}(N_{\underline{S}},\Lambda^{l}_{\underline{S}}))\otimes\mathbb{Q}^{r(\underline{n},d)}$
$\displaystyle\simeq\bigg{(}\bigotimes^{r}_{i=1}\bigoplus_{l_{i}\geq
0}\mathcal{H}^{l_{i}}_{a}(\operatorname{IC}(N_{S_{i}},\Lambda^{l_{i}}_{S_{i}}))\bigg{)}\otimes\mathbb{Q}^{r(\underline{n},d)}$
$\displaystyle\simeq\bigg{(}\bigotimes^{r}_{i=1}\mathcal{H}_{a}^{2c(n_{i},g)}(\mathscr{S}_{\\{n_{i}\\}}|_{N_{n_{i}/\underline{n}_{i}}})\bigg{)}\otimes\mathbb{Q}^{r(\underline{n},d)}.$

∎

##  10\. Proof of Theorem 1.5 and Theorem 1.6

###### Proof of Theorem 1.5.

By Theorem 9.1 it suffices to check that
$(R^{2c(n,g)}\chi(n,0)_{*}\mathbb{Q}_{{M}(n,0)})_{a}$ has no proper support at
the general point $a\in S_{\underline{m}}$ for any $\underline{m}$. We prove
it by induction on the length of $\underline{m}$. If $\underline{m}$ has
length $1$, then $a\in A^{\mathrm{reg}}$ avoids any potential proper support
of the decomposition theorem, so there is nothing to prove.

Suppose now that $\underline{m}$ is a permutation of length $s>1$, and no
summands of
$R^{2c(n,g)}\chi(n,0)_{*}\mathbb{Q}_{{M}(n,0)}|_{A^{\mathrm{red}}}$ is
supported on $S_{\underline{n}}\cap A^{\mathrm{red}}$, for any
$n\in\mathbb{Z}_{>0}$ and for any partition $\underline{n}$ of $n$ such that
$\underline{m}<\underline{n}<\\{n\\}$. We show that
$(R^{2c(n,g)}\chi(n,0)_{*}\mathbb{Q}_{{M}(n,0)})_{a}$ has no summands at $a\in
S_{\underline{m}}$. Otherwise, there exists a non-zero $\mathbb{Q}$-vector
space $\mathcal{F}_{a}$ such that

$(R^{2c(n,g)}\chi(n,0)_{*}\mathbb{Q}_{{M}(n,0)})_{a}\simeq\mathcal{H}^{2c(n,g)}_{a}(\mathscr{S}_{\\{n\\}})\oplus\mathcal{F}_{a}.$

Let $e$ be an integer with $\gcd(e,n)=1$. 9.7 gives

$\displaystyle(R^{2c(n,g)}$
$\displaystyle\chi(n,e)_{*}\mathbb{Q}_{{M}(n,e)})_{a}\simeq\bigoplus_{\underline{n}\geq\underline{m}}\mathcal{H}^{2c(n,g)}_{a}(\mathscr{S}(\mathscr{L}_{\underline{n}}(e)))$
$\displaystyle\simeq\bigoplus_{\underline{S}\vdash[s]}\bigg{(}\bigotimes^{\ell(\underline{S})}_{i=1}\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}}|_{N_{S_{i}}})\bigg{)}\otimes\mathbb{Q}^{r(\ell(\underline{S}),e)}$
$\displaystyle\simeq\mathcal{H}^{2c(n,g)}_{a}(\mathscr{S}_{\\{n\\}})\oplus\bigoplus_{\begin{subarray}{c}\underline{S}\vdash[s]\\\
\underline{S}\neq\\{[s]\\}\end{subarray}}\bigg{(}\bigotimes^{\ell(\underline{S})}_{i=1}\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}}|_{N_{S_{i}}})\bigg{)}\otimes\mathbb{Q}^{r(\ell(\underline{S}),e)},$

where we set $\sigma_{i}\coloneqq\sum_{j\in S_{i}}m_{j}$. By induction
hypothesis we have that for any $\underline{S}\vdash[s]$

$\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}}|_{N_{S_{i}}})\simeq\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}})\simeq(R^{2c(\sigma_{i},g)}\chi(\sigma_{i},0)_{*}\mathbb{Q}_{{M}(\sigma_{i},0)})_{a}.$

Recall that the vertices of the dual graph $\Gamma[\underline{m}]$ are
labelled by $[s]$. Let $\Gamma(S_{i})$ be the maximal subgraph of
$\Gamma[\underline{m}]$ on the vertex set $S_{i}$. Theorem 7.8 gives

$\dim\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}}|_{N_{S_{i}}})=C(Z_{\Gamma(S_{i})}).$

Now set
$\omega_{\underline{m}}(e)=(m_{1}e/n,\ldots,m_{s}e/n)\in\mathbb{Q}^{s}$ with
$\underline{m}=\\{m_{1},\ldots,m_{s}\\}$, and write

$\displaystyle C(Z_{\Gamma[\underline{m}]}+\omega_{\underline{m}}(e))$
$\displaystyle=\dim(R^{2c(n,g)}\chi(n,e)_{*}\mathbb{Q}_{{M}(n,e)})_{a}$
$\displaystyle=\dim\mathcal{H}^{2c(n,g)}_{a}(\mathscr{S}_{\\{n\\}})+\sum_{\begin{subarray}{c}\underline{S}\vdash[s]\\\
\underline{S}\neq\\{[s]\\}\end{subarray}}r(\ell(\underline{S}),e)\cdot\prod^{|\underline{S}|}_{i=1}\dim\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}}|_{N_{S_{i}}})$
$\displaystyle=\dim\mathcal{H}^{2c(n,g)}_{a}(\mathscr{S}_{\\{n\\}})+\sum_{\begin{subarray}{c}\underline{S}\vdash[s]\\\
\underline{S}\neq\\{[s]\\}\end{subarray}}r(\ell(\underline{S}),e)\cdot\prod^{|\underline{S}|}_{i=1}C(Z_{\Gamma(S_{i})}).$

Note that $r(\ell(\underline{S}),e)\geq(\ell(\underline{S})-1)!$ by the Hodge-
to-singular correspondence [28, Thm 1.1 and Prop. 4.11.(8)]. Together with the
combinatorial identity of 5.4.(i)

$C(Z_{\Gamma[\underline{m}]}+\omega_{\underline{m}}(e))=\sum_{\underline{S}\vdash[s]}(\lvert\underline{S}\rvert-1)!\prod_{j=1}^{\lvert\underline{S}\rvert}C(Z_{\Gamma(S_{j})}),$

we conclude that $r(\ell(\underline{S}),1)$ actually equals
$(\ell(\underline{S})-1)!$,333Alternatively, the equality
$r(\ell(\underline{S}),1)=(\ell(\underline{S})-1)!$ follows via the explicit
computation of the Cattani–Kaplan–Schmid complex for a versal family of
$C_{a}$ in [12, Cor. 6.20]. and so

$\dim\mathcal{H}^{2c(n,g)}_{a}(\mathscr{S}_{\\{n\\}})=C(Z_{\Gamma[\underline{m}]})=\dim(R^{2c(n,g)}\chi(n,0)_{*}\mathbb{Q}_{{M}(n,0)})_{a},$

which implies $\mathcal{F}_{a}=0$, i.e. $S_{\underline{m}}$ is not a support
of $R^{2c(n,g)}\chi(n,0)_{*}\mathbb{Q}_{{M}(n,0)}$. ∎

###### Proof of Theorem 1.6.

Let $a$ be a general point in $S_{\underline{m}}$ for any partition
$\underline{m}=\\{m_{1},\ldots,m_{s}\\}$. 9.7 gives

$(R^{2c(n,g)}\chi(n,d)_{*}\mathbb{Q}_{{M}(n,d)})_{a}\simeq\bigoplus_{\underline{S}=\\{S_{i}\\}\vdash\underline{m}}\mathscr{L}_{\\{|S_{i}|\\}}(d)_{a}\otimes\bigg{(}\bigotimes^{\ell(\underline{S})}_{i=1}\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}}|_{N_{S_{i}}})\bigg{)}.$
(29)

The LHS of (29) is an $\operatorname{Aut}(\underline{m})$-representation,
while the summands of the RHS are
$\operatorname{Aut}(\underline{m})$-representations induced by the injection
$\operatorname{Stab}(\underline{S})\hookrightarrow\operatorname{Aut}(\underline{m})$.
For convenience, write $\omega$ for the vector $\omega_{\underline{m}}(d)$
defined in (2). By Theorem 7.8 and Theorem 6.4, the following
$\operatorname{Aut}(\underline{m})$-representations are isomorphic

$\displaystyle(R^{2c(n,g)}$
$\displaystyle\chi(n,d)_{*}\mathbb{Q}_{{M}(n,d)})_{a}\simeq\mathcal{C}(Z_{\Gamma[{\underline{m}}]}+\omega_{\underline{m}}(d))$
$\displaystyle\simeq$
$\displaystyle\,\mathcal{C}(Z_{\Gamma[{\underline{m}}]})\oplus\bigoplus_{\underline{S}\in\Pi_{\omega}(\underline{m})/\operatorname{Aut}(\Gamma[\underline{m}])}\operatorname{Ind}_{\operatorname{Stab}(\underline{S})}^{\operatorname{Aut}(\Gamma[\underline{m}])}\big{(}\operatorname{sgn}\otimes\widetilde{H}_{\ell(\underline{S})-3}(\Delta(\overline{\Pi}_{\omega,\geq\underline{S}}))\otimes\mathcal{C}(Z_{\Gamma_{\underline{S}}})\big{)},$
$\displaystyle\simeq$
$\displaystyle\,(R^{2c(n,g)}\chi(n,0)_{*}\mathbb{Q}_{{M}(n,0)})_{a}\oplus$
$\displaystyle\oplus\bigoplus_{\underline{S}\in\overline{\Pi}_{\omega}(\underline{m})/\operatorname{Aut}(\Gamma[\underline{m}])}\operatorname{Ind}_{\operatorname{Stab}(\underline{S})}^{\operatorname{Aut}(\Gamma[\underline{m}])}\big{(}\operatorname{sgn}\otimes\widetilde{H}_{\ell(\underline{S})-3}(\Delta(\overline{\Pi}_{\omega_{\underline{S}}}))\otimes\bigg{(}\bigotimes^{\ell(\underline{S})}_{i=1}\mathcal{H}_{a}^{2c(\sigma_{i},g)}(\mathscr{S}_{\\{\sigma_{i}\\}}|_{N_{S_{i}}})\bigg{)}\big{)},$

Comparing with (29), we conclude that the
$\operatorname{Aut}(\Gamma[\underline{n}])$-representations

$\mathscr{L}_{\underline{n}}(d)_{a}\simeq\operatorname{sgn}\otimes\widetilde{H}_{\ell(\underline{n})-3}(\Delta(\overline{\Pi}_{\omega_{\underline{n}}(d)}))$

are isomorphic by induction on the length of $\underline{m}$. ∎

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