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

Question

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 $W_F$?

Answer 1

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.

Compared to the Bernstein-Zelevinsky classification, building Weil-Deligne representations out of irreducible representations is the easy part.

Answer 2

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 $SL_2$.

Otherwise, the Weil-Deligne group is defined using a monodromy operator or a one-dimensional unipotent group. In this case, it is not hard to check that the same classification holds by decomposing into irreducible representations of the Weil group and examining how the monodromy operator moves between them (and this is why the two definitions can be used interchangeably)