Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\pi_1$ and $\pi_2$:
$\pi_1 \times \pi_2$ (Rankin–Selberg product?)
$\pi_1 \boxplus \pi_2$ (isobaric sum)
$\pi_1 \boxtimes \pi_2$ (isobaric product)
How are these representations defined exactly? Can they be defined easily in terms of the Local Langlands correspondence?
The right person to answer this question is probably Dinakar Ramakrishnan (a good working reference is his paper "Modularity of the Rankin–Selberg $L$-series, and multiplicity one for $\mathrm{SL}(2)$"), but since he doesn't seem to use MathOverflow, here is my understanding of this. In each case, I will first discuss the local theory, then the global theory.
The isobaric sum $\pi_1 \boxplus \pi_2$ (which Langlands called the "sum operation", and by some is called the Langlands sum) is perhaps the easiest to explain. The key property is that the $L$-function of the isobaric sum is the product of the two $L$-functions: \[L(s,\pi_1 \boxplus \pi_2) = L(s,\pi_1) L(s,\pi_2).\] In terms of representations of $\mathrm{GL}_{n_1}(F)$ and $\mathrm{GL}_{n_2}(F)$, where $F$ is a local field, $\pi_1 \boxplus \pi_2$ is simply the normalised parabolic induction from the Levi subgroup $\mathrm{M} \cong \mathrm{GL}_{n_1}(F) \times \mathrm{GL}_{n_2}(F)$ to $\mathrm{GL}_{n_1 + n_2}(F)$. Via the local Langlands correspondence, in terms of the corresponding $n_1$- and $n_2$-dimensional Weil–Deligne representations $\rho_1$ and $\rho_2$, the isobaric sum simply corresponds to the direct sum $\rho_1 \oplus \rho_2$; note that this is a $n_1 + n_2$-dimensional (reducible) representation of the same group.
More precisely, by $\pi_1 \boxplus \cdots \boxplus \pi_r$, I mean the induced representation $\mathrm{Ind}_{\mathrm{P}(F)}^{\mathrm{GL}_n(F)} \bigotimes_{j = 1}^{r} \pi_j$ of $\mathrm{GL}_n(F)$, where $n = n_1 + \cdots + n_r$ and $\bigotimes_{j = 1}^{r} \pi_j$ denotes the outer tensor product of $\pi_1,\ldots,\pi_r$, which is a representation of the Levi subgroup $\mathrm{M} \cong \mathrm{GL}_{n_1}(F) \times \cdots \times \mathrm{GL}_{n_r}(F)$; this is then trivially extended to a representation of the parabolic subgroup $\mathrm{P}$ with Levi subgroup $\mathrm{M}$, then induced to a representation of $\mathrm{GL}_n(F)$. If each $\pi_j$ is essentially square-integrable, then $\pi_1 \boxplus \cdots \boxplus \pi_r$ is called an induced representation of Whittaker type (since these representations have a Whittaker model). These need not be irreducible, but the Langlands quotient theorem states that every irreducible admissible representation of $\mathrm{GL}_n(F)$ is unitarily equivalent to the quotient of an induced representation of Whittaker type.
This extends naturally to the global setting: if $\pi_1$ and $\pi_2$ are automorphic representations of $\mathrm{GL}_{n_1}(\mathbb{A}_F)$ and $\mathrm{GL}_{n_2}(\mathbb{A}_F)$, where now $F$ is a global field, then $\pi_1 \boxplus \pi_2$ is an automorphic representation of $\mathrm{GL}_{n_1 + n_2}(\mathbb{A}_F)$; the automorphic forms that for a vector space for this automorphic representation are Eisenstein series induced from the Levi subgroup $M \cong \mathrm{GL}_{n_1} \times \mathrm{GL}_{n_2}$. In particular, a classical Eisenstein series on $\mathrm{GL}_2$ has two characters associated to it (possibly the trivial characters), and this is simply induced from $\mathrm{GL}_1 \times \mathrm{GL}_1$, and the $L$-function of such an Eisenstein series is simply the product of the $L$-functions of the two characters.
(As an etymological aside, isobaric means equal pressure, which is somewhat incongruous since $\pi_1$ and $\pi_2$ can have different dimensions. Moreover, the isobaric sum is usually only used to describe the global operation, not the local one, though I personally believe it is appropriate to use it in the local setting.)
The point of the isobaric sum is that it allows one to break up (local) representations into parts that cannot be isobarically decomposed any further; these are the essentially square-integrable representations. The global analogue of these isobarically indecomposable representations are cuspidal automorphic representations.
For $\pi_1 \boxtimes \pi_2$ and $\pi_1 \times \pi_2$ (and $\pi_1 \otimes \pi_2$), the answer is a little less clear, because authors use these notations interchangeably. To me, $\boxtimes$ ought to denote the outer tensor product, so $\pi_1 \boxtimes \pi_2$ ought to denote a representation of the product of groups $\mathrm{GL}_{n_1}(F) \times \mathrm{GL}_{n_2}(F)$; more precisely, should think of $\boxtimes$ as a map from $\mathcal{R}(\mathrm{GL}_{n_1}(F)) \times \mathcal{R}(\mathrm{GL}_{n_2}(F))$ to $\mathcal{R}(\mathrm{GL}_{n_1 n_2}(F))$, where $\mathcal{R}(\mathrm{GL}_n(F))$ denotes the set of irreducible admissible representations of $\mathrm{GL}_n(F)$. In terms of $\rho_1$ and $\rho_2$, this should simply be the tensor product $\rho_1 \otimes \rho_2$, which is a $n_1 n_2$-dimensional (possibly reducible) representation of the same group. Since the local Langlands correspondence is a theorem, this means there is a representation $\pi_1 \times \pi_2$ (or $\pi_1 \otimes \pi_2$) of $\mathrm{GL}_{n_1 n_2}(F)$ corresponding to the outer tensor product $\pi_1 \boxtimes \pi_2$. Locally, this is reasonably well-understood via the Langlands correspondence: if \[\pi_1 = \boxplus_{j = 1}^{m_1} \pi_{1,j}, \qquad \pi_2 = \boxplus_{k = 1}^{m_2} \pi_{2,k},\] then \[\pi_1 \times \pi_2 = \boxplus_{j = 1}^{m_1} \boxplus_{k = 1}^{m_2} \pi_{1,j} \boxtimes \pi_{2,k}.\] Equivalently, if \[\rho_1 = \bigoplus_{j = 1}^{m_1} \rho_{1,j}, \qquad \rho_2 = \bigoplus_{k = 1}^{m_2} \rho_{2,k},\] then \[\rho_1 \otimes \rho_2 = \bigoplus_{j = 1}^{m_1} \bigoplus_{k = 1}^{m_2} \rho_{1,j} \otimes \rho_{2,k}.\] So everything is reduced to essentially square-integrable representations (in which case understanding this tensor product can be complicated).
For the global situation, things are a little murkier, since functoriality is not a theorem in general. In particular, the issue is that if $\pi_1$ and $\pi_2$ are automorphic representations of $\mathrm{GL}_{n_1}(\mathbb{A}_F)$ and $\mathrm{GL}_{n_2}(\mathbb{A}_F)$, where now $F$ is a global field, it is not known if $\pi_1 \times \pi_2$ is an automorphic representation of $\mathrm{GL}_{n_1 n_2}(\mathbb{A}_F)$. That is, we know how to define the global $L$-function $L(s,\pi_1 \times \pi_2)$ of this object (Jacquet, Piatetski-Shapiro, Shalika: "Rankin–Selberg Convolutions") as a product of local $L$-functions, as well as (more or less) how to describe the local components, but we don't know that the global object obtained by gluing together these local components is a genuine automorphic representation. This is the issue of the functorial transfer of the tensor product.
Notationally, often $\pi_1 \otimes \pi_2$ is used in place of $\pi_1 \times \pi_2$, which is, I suppose, an artefact of the fact that this corresponds to a genuine tensor product on the Weil–Deligne side. Moreover, when $n_2 = 1$, so that $\pi_2 = \chi$ is a character, then functoriality is a theorem, and we usually write $\pi_1 \otimes \chi$ (but again, $\pi_1 \times \chi$ still appears regularly in the literature).
Note that $\pi_1 \times \pi_2$ is called the Rankin–Selberg product or Rankin–Selberg convolution, but that really we only ever mean the $L$-function $L(s,\pi_1 \times \pi_2)$, since in general we don't know the hypothetical automorphic object associated to this $L$-function.
In this setting, when this global object is known to be a genuine automorphic representation of $\mathrm{GL}_{n_1 n_2}(\mathbb{A}_F)$, Ramakrishnan denotes this by $\pi_1 \boxtimes \pi_2$ . That is, $\boxtimes$ denotes a map from $\mathcal{A}(\mathrm{GL}_{n_1}(\mathbb{A}_F)) \times \mathcal{A}(\mathrm{GL}_{n_2}(\mathbb{A}_F))$ to $\mathcal{A}(\mathrm{GL}_{n_1 n_2}(\mathbb{A}_F))$, where $\mathcal{A}(\mathrm{GL}_n(\mathbb{A}_F))$ denotes the set of automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$, such that $L(s,\pi_1 \boxtimes \pi_2) = L(s,\pi_1 \times \pi_2)$. This is the only usage of $\boxtimes$ in the global setting that I have seen. (I don't believe that $\boxtimes$ is given a name, but if it is, it's definitely not the isobaric product.)
