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For a reductive group $G$ over a number field $K$, an automorphic representation for $(G, K)$ is an irreducible admissible constituent of the right regular representation of $G(\mathbb{A}_K)$ in the function space $L^2(G(K)\backslash G(\mathbb{A}_K))$ satisfying certain conditions, $\mathbb{A}_K$ being the ring of adeles of $K$.

It's tempting to see how this construction would develop if $G$ were replaced by "Gal", i.e., if we were to perversely take $G(R)$ to mean the absolute Galois group of commutative ring R, in the sense of $\pi_1^{et}(Spec \text{ } R)$.

Of course, one runs straight into a basic problem: inclusion $K \hookrightarrow \mathbb{A}_K$ leads to the reverse inclusion $Gal(\mathbb{A}_K) \hookrightarrow Gal(K)$ here. (I am assuming this, but is this true? In fact, is there a place where one can find studied $Gal(\mathbb{A}_K)$ and its relation to absolute Galois groups of localizations of $K$?)

If, even in face of this reversal, we insist on continuing with the construction in a way that is still sensible and look at the right regular action of $Gal(K)$ on $L^2(Gal(\mathbb{A}_K) \backslash Gal(K))$ instead (or on some other appropriate space of functions on the coset space, $\bar{\mathbb{Q}}_l$-valued rather than $\mathbb{C}$-valued, or valued in vector spaces over these, or sheafified perhaps) and see how it decomposes, which irreducible Galois representations do we get?

All this has to be non-sensical, trivial or useless since it doesn't seem to be considered in the literature as far as I know unless it is in some other guise. The question is: why?

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  • $\begingroup$ Nice question, although I am not even sure that the definitions make sense ($L^{2}$ in which sense? what is this space?). Out of curiosity, is there any mathematical reason to be interested in this space? If we consider the classical analogue $G(\mathbb{R})/G(\mathbb{Z})$, I don't see any significance. Moreover, those groups $G$ didn't simply appear out of nowhere, for $G_{a}$, it is simply Fourier analysis, for $G=PSL_{2}$, this (classical) space is the space of unimodular lattices, what that $Gal$ space would be? $\endgroup$
    – Asaf
    Apr 9, 2018 at 17:52
  • $\begingroup$ @Asaf, I have left the choice of measure on the coset space vague to allow for flexibility of interpretation. I don’t know much about Haar measures on $Gal(\mathbb{Q})$, nor about the coset space in question. $L^2$ would have the usual sense for complex-valued functions once a measure is chosen, though there may not be enough functions in this case I’m afraid. $\endgroup$
    – user122285
    Apr 10, 2018 at 21:06
  • $\begingroup$ @Asaf, the significance or utility of considering the scenario would lie in having a repository of certain Galois representations to which tools of harmonic analysis may apply. Which if any interesting ones is, of course, the original question. The question is purely speculative, and may well turn out to be meaningless. Motivation for asking comes from parallels with automorphic reps. which are related, often conjecturally, to Galois reps. via Langlands program. $\endgroup$
    – user122285
    Apr 10, 2018 at 21:11
  • $\begingroup$ You need to choose a basepoint to define a fundamental group (or Galois group). Otherwise, you just have a groupoid. On the other hand, the finite étale property is preserved under base change, so maps of fundamental groupoids are covariant with respect to maps of schemes. See stacks.math.columbia.edu/tag/0BL6 $\endgroup$
    – S. Carnahan ♦
    Apr 11, 2018 at 6:58
  • $\begingroup$ @ShD , saying Haar measure/whatever is not an easy way out, as first - you need to consider what kind of top. grp is $Gal(A_{K})$, and even then, check whether $Gal(K)$ is a lattice (or at-least discrete) subgroup to make sense out of measure on the quotient. The significance of $L^{2}(G/\Gamma)$ arises as $G$ acts properly-discontinuously on $G/\Gamma$ and then you may induce a unitary representation by that and study that using tools of harmonic analysis. Just bluntly changing $G$ does not mean that the construction still makes sense. So I ask again, what is $Gal(A_{K})/Gal(K)$? $\endgroup$
    – Asaf
    Apr 11, 2018 at 18:05

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