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The Langlands correspondence over a function field matches irreducible $n$-dimensional Galois representations with cuspidal irreducible automorphic representations.

My question is: Is there any idea or conjecture around, how this could extend to reducible representations?

As automorphic representations are semisimple, they don't allow for non-trivial extension, which however on the Galois-side can happen. Maybe there are ideas from geometric Langlands (of which I don't know much) that can help?

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    $\begingroup$ Over number fields, one can construct non-semisimple Galois representations by putting a "reducible" automorphic form (e.g. Eisenstein series) into a family of "irreducible" forms (cusp forms). The original idea goes back to Ribet. However, I don't know a criterion for which extensions can be constructed in this way, or even what is expected. $\endgroup$
    – Kevin Ventullo
    Commented Jun 7, 2013 at 23:15

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