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In his 'Märchen' Langlands considers for a local field $F$ a certain abelian category $\Pi(F)$ whose objects are given by isomorphisms classes of irreducible admissible representations of $GL_n(F)$, where $n \in \mathbf{N}$ runs over all natural numbers. For $[\pi],[\pi']$ represented by cuspidal reps $\pi,\pi'$ of $GL_n(F)$ and $GL_{n'}(F)$ he introduces the sum $[\pi] \boxplus [\pi]$ as the class of the unique irreducible quotient of the parabolic induction of $\pi\otimes \pi'$ where $GL_n(F)\times GL_{n'}(F)$ is viewed as a Levi component of the obvious standard parabolic of $GL_{n+n'}(F)$. Since he wants to recognize $\Pi(F)$ as the category of representations of some proalgebraic group via the Tannaka formalism he poses the problem of defining a tensor product $[\pi]\boxtimes [\pi']$ as a class of a representation of $GL_{n\cdot n'}(F)$. Moreover $\boxplus$, $\boxtimes$ and $\oplus$, $\otimes$ should correspond to each other under the Langlands correspondence. Of course since it has now been proven one could define $\boxtimes$ via the Local Langlands correspondence.

My question is: Is there any known elementary construction of $[\pi]\boxtimes [\pi']$ for any example with $n,n' > 1$ ?

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up vote 6 down vote accepted

Basically, no. Marc Palm's answer addresses L-functions, but that is a long long way from determining the irrep -- you'd need L-functions and epsilon-factors of twists, plus an impressive ability to translate such information into a construction of a smooth irrep (if you want more than existence).

To me, your question is the most important outstanding problem in the local Langlands program. I anticipate that a deeper understanding of types (redoing Bushnell-Kutzko in such a way that the type of a putative Langlands tensor product is clear) will be needed. Whatever it is, it won't be "elementary".

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That confirms my suspicion. I had hoped however that an explicit construction of $[\pi]\boxtimes [\pi']$ was known at least in some special (nontrivial) cases. – Nicolas Schmidt May 23 '13 at 13:05

I am not an expert on that matter and I don't know what qualifies as elementary here. I think that the Rankin Selberg method is an instance for the symmetric tensor of what you want to look at. It is one of the crucial method for proving nontrivial bounds towards the Ramanujan Petersson conjecture. I believe that locally it is pretty well understood, so you can write down the L function, since you know all local L factors. So yes, you know what you get explicitly, and the only issue is whether it is automorphic. Cogdell has some pretty good lecture notes on his homepage. The global function field and the local case is probably implied by the known correspondences with Galois representations.

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