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?

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?