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!