Infinite rank mellin transform of a global theta section

Throughout, $\mathfrak S_n$ is assumed to be associated with a rank-$(n-2)$ zeta function.

Consider $\mathfrak{S}_3 = (\mathcal{I}_3, \Gamma_3)$, composed of four orbifold sheets glued along a seam graph

$$ \mathcal{I}_3 = \left(\bigsqcup_{i=1}^4 X_i\right)\Big/\!\sim_{\Gamma_3}, \qquad X_i = S^2(\alpha, \alpha) $$

assumed to be $\Pi_3$-equivariant, where $\Pi_3\cong O_h$. Write $\mathfrak S_3:=(\mathcal I_3,\Gamma_3,\Pi_3)$.

Then

$$ \mathfrak S_n = (\mathcal I_n,\Gamma_n, \Pi_n) $$

accepts $B_n\cong\Pi_n$-equivariant objects subject to

$$ \mathcal I_n = \bigg( \bigsqcup_{i=1}^{2^{n-1}} \Sigma^{n-3}X_i \bigg)\bigg/\sim_{\Gamma_n}. $$

Define

$$ \widehat{\mathfrak S}_+ := \varprojlim_{n\ge 3} \mathfrak S_n =\varprojlim_{n\ge3} (\mathcal I_n,\Gamma_n,\Pi_n) = \bigg(\varprojlim_{n\ge 3}\mathcal I_n,\varprojlim_{n\ge 3}\Gamma_n, \varprojlim_{n\ge3}\Pi_n \bigg) $$

with bonding maps as desuspensions/projections in the stable category.

Define

$$ \Bbb L:= \widehat{\mathfrak S}_- \sqcup_{\mathfrak S_3} \widehat{\mathfrak S}_+ $$

to be the pushout over $\mathfrak S_3$, with

$$ \widehat{\mathfrak S}_- := \varprojlim_{n\ge 3} \mathfrak S_{-n} =\varprojlim_{n\ge3} (\mathcal I_{-n},\Gamma_{-n},\Pi_{-n}) = \bigg(\varprojlim_{n\ge 3}\mathcal I_{-n},\varprojlim_{n\ge 3}\Gamma_{-n}, \varprojlim_{n\ge3}\Pi_{-n} \bigg). $$

Here

$$ \mathfrak S_{-n} := \mathfrak S_{n}^\lor. $$

I'm interested in a Grothendieck fibration

$$ \zeta \longrightarrow \Theta \longrightarrow\Bbb L. $$

$\Bbb L$ encodes the geometry, $\Theta$ encodes the admissible spectra over that geometry and $\zeta$ packages the total space.

I want to understand how to Mellin transform a universal $\Theta$-section: a function that is smooth along the sheets, modular and discrete along $\Gamma$ and invariant under the $\Pi_3$-action at the pushout core. What kind of infinite rank Mellin transform, adelic spectral integral may send the universal $\Theta$-section to a section of $\zeta$ as a universal $L$-function? I understand this very well for the rank-1 case. And one can induct over this case for rank-(n-2), to stitch together the global theta section.

I'll take references in this direction because I really don't know how to approach the infinite rank Mellin transform here.