# How does the Bernstein-Zelevinsky construction of irreducibles from supercuspidals parallel the representations of the Weil-Deligne group?

In the Corvallis article Number Theoretic Background, here is what John Tate has to say on the local Langlands correspondence for a $p$-adic field $F$:

So, granting a correspondence between irreducible supercuspidal representations of $\operatorname{GL}_n(F)$ and continuous irreducible $n$-dimensional complex representations of the local Weil group $W_F$, the general correspondence between irreducible representations and $\Phi$-semisimple representations of the Weil-Deligne group $W_F'$ should follow.

I understand the (statements of the) Bernstein-Zelevinsky classification of smooth irreducible representations of $\operatorname{GL}_n(F)$. But how are $\Phi$-semisimple representations of $W_F'$ built out of continuous irreducible representations of $W_F$?

Let $$\rho$$ be a representation of the Weil-Deligne group which is semi-simple on $$W_F$$ and algebraic on $$\mathbb C$$. Then $$\rho$$ is a direct sum of indecomposable such representations (uniquely up to isomorphism and ordering of the factors). If $$\rho$$ is indecomposable, then it is (isomorphic to) the tensor product of a unique irreducible representation of $$W_F$$ with the representation $$(\mathbb C^{t+1},\rho_{q^{t/2},t},N_t)$$ for some $$t$$ (where as usual $$\rho_{q^{t/2},t}$$ sends the Frobenius to the diagonal matrix with shifted eigenvalue by increasing powers of $$q$$ and $$N_t$$ is the maximal monodromy matrix of the correct size). Hence, taking the representations $$(\mathbb C^{t+1},\rho_{q^{t/2},t},N_t)$$ and the irreducible representations of $$W_F$$ as building blocks, you build everything.
Such representations are sums of tensor products of irreducible representations of $W_F$ with the $n$-dimensional representation in which the monodromy operator acts by the nilpotent matrix with a single Jordan block (in Jordan normal form) and the Frobenius acts diagonally by elements $q^{n/2},q^{n/2-1},\dots, q^{-n/2}$. The tensor product corresponds to the construction of discrete series representations, and the sum to principal series.
The Weil-Deligne group is sometimes defined as just $W_F \times SL_2$, in which case this is an irreducible representation of $W_F$ tensor the $n$-dimensional irreducible representation of $SL_2$. In this case, the classification follows immediately from the classification of irreducible representations of $SL_2$.