# Langlands product

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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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