The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if somebody could give me a reference.

Let $\Gamma_n$ be the kernel of the natural morphism $\mathrm{SL}_2(\mathbf Z)\to \mathrm{SL}_2(\mathbf Z/n)$.

Let $X_n$ be the quotient of the Poincaré half-plane $\mathcal H$ by the usual action of $\Gamma_n$.

Then for $n$ greater than $3$, $\mathrm H_1(X_n,\mathbf Z)$ is a free module of rank $$\frac{\sharp \mathrm{SL}_2(\mathbf Z/n)}{12}+1\ \ \ .$$

Thus we have a sequence of representations $\rho_n$ (of $\mathrm{SL}_2(\mathbf Z/n)$ on $\mathrm H_1(X_n,\mathbf Z)$) together with morphisms $\rho_n\to\rho_m$ for $m\vert n$, and at the limit a representation $\hat{\rho}$ of $\mathrm{SL}_2(\hat{\mathbf Z})$ on $\lim_n\mathrm{H}_1(X_n,\mathbf Z)$.

The question is the following : is there an algebraic construction of this representation $\hat \rho$ ?

(Something along the lines of "the second exterior power of the adjoint representation" would qualify (but it's obviously false ...)).

(Of course the question could be declined in many ways : replace $\mathrm{SL}_2$ by another group, $\mathcal H$ by the associated symmetric space, $\mathrm{H}_1()$ by another homology theory and so on ...)


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