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Let $F$ be a p-adic field, with p a prime denoting the residue field characteristic. Let $\mathcal{W}_F$ be the Weil group. In the local Langlands correspondence for $GL(n,F)$, it is important to know parametrizations of irreducible finite dimensional continuous representations of $\mathcal{W}_F$.

When $(n,p)=1$,that is the socalled tame case, the description of irreducible n-dimensional continuous representations of $\mathcal{W}_F$ is relatively easier to describe. For example, in section 2 of Moy's paper

'local constants and the tame Langlands correspondence', American Journal of Math. vol 108, No.4, 1986, 863-929

the 'admissible characters of degree n extensions' parametrize the irreducible n-dimensional continuous representations of $\mathcal{W}_F$.

Now I'm wondering what's happening in the general case? Do we still have some 'nice' description of irreducible n-dimensional continuous representations of $\mathcal{W}_F$ without assuming $(n,p)=1$?

To my knowledge, in the book 'the local Langlands conjecture for GL(2)' by Bushnell and Henniart, such a description is given when $n=2$ and $F$ a finite extension of $\mathbb{Q}_2$. But that's much complicated than tame case.

Also, I wanna know if such a parametrization exists, does it play a role in the proof of local Langlands correspondence by Harris and Taylor?

Much appreciated for any answer and reference.

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