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 $n,n' > 1$ ?



Langlands product
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". 


