# The operator \boxtimes and \boxplus in automorphic representations

Given two automorphic representations $\pi_1, \pi_2$ of $GL_2(\mathbb A_Q)$ and $GL_3(\mathbb A_Q)$ respectively. Let $\pi_i =\otimes_v \pi_{i, v}$.

Now, for each $v$, let $\pi_{1, v}\boxtimes \pi_{2, v}$ and $\pi_{1, v}\boxplus \pi_{2, v}$ be the admissible representation of $GL_6(Q_v)$, which, via local Langlands, corresponds tensor product and direct sum of the Weil-Deligne representations.

My question is, are there explicit construction of the $\boxtimes$ and $\boxplus$ operators?

In fact, is there any basic introduction to these operators? I see them quite a lot, but have not found any explanatory definitions...

Thanks!

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What do you mean here? The exterior tensor product $\boxtimes$ is a priori a rep of $GL(2) \times GL(3)$. It is rather trivial, that it will remain automorphic. What is $\boxplus$? – Marc Palm Feb 11 '13 at 13:45
boxtimes. You may like to see the paper by Langlands "einmaerchen" paper in Corvallis where these operations were discussed for the first time. – Venkataramana Feb 11 '13 at 15:33
Sorry; the sentence got mangled. I meant the paper titled "...., Einmarchen" by Langlands, in the Corvallis volume on automorphic forms, Shimura Varieties, L functions". This introduces the operations which you have mentioned. – Venkataramana Feb 11 '13 at 15:40
Here is the link to the "Ein Maerchen" paper: sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/autoreps-ps.pdf – Venkataramana Feb 11 '13 at 15:52
@Aakumadula: okay, thx for the explanation. $\boxtimes$ is not the exterior tensor. – Marc Palm Feb 14 '13 at 8:57