Let $p$ be a prime number, and $F$ be a finite extension of $Q_p$. To any smooth irreducible representation $\pi$ of $G = Gl_n(F)$ we may associate a sort of ``dual´' representation, called the Zelevinsky dual or Aubert dual, constructed as follows. Let $R$ be the Grothendieck group of smooth $G$-representations of finite length. For any standard parabolic subgroup $P$ with Levi-decomposition $P = MN$ we have the functor of induction $Ind_P^G$ and restriction $Res^G_P$ (both normalized so that they send unitary reps to unitary reps, as in the notes of Casselman on p-adic reps). These functors are both exact, and yield morphisms between the Grothendieck group of $G$ and the Grothendieck group of $M$. Thus, one can define for any $X \in R$ the following object $$ i(X) = \sum_P \varepsilon_P Ind_P^G(Res^G_P(X)) \in R, $$ where the sum ranges over the standard parabolic subgroups of $G$. Then $i(X)$ is the Aubert dual of $X$. The $\eps_P$ is a sign: $(-1)$ to the rank of $P$. The Zelevinsky dual is almost the same thing as the Aubert dual. Zelevinsky's dual is only defined on the isom classes of smooth irreducible reps. If one applies $i$ to an irreducible representation $\pi$ viewed as an element of the Grothendieck group $R$ of $G$ then there is a sign $\varepsilon$ such that $\varepsilon \cdot i(\pi) \in R$ comes from an irreducible $G$-representation $\iota(\pi)$. This representation $\iota(\pi)$ is the Zelevinsky dual of $\pi$.
The Zelevinsky dual has an explicit description in terms of the Zelevinsky segment classification for smooth irreducible $G$-representations. Historically it came before the Aubert dual.
References: Aubert's paper duality dans le group de Grothendieck de la categorie des representations lisses de longueur finie dún groupe reductif p-adique´'
and Zevinsky's paper:Induced representations of reductive p-adic groups II´'.
See also the IHES paper of Schneider and Stuhler.
The operator $i$ commutes with parabolic induction, and leaves cuspidal representations invariant (look at the formula above). So it is ``interesting´' for $p$-adic representations of $G$ that occur as a proper irreducible subquotient of some parabolic induction. For example, it interchanges the Steinberg representation with the trivial representation. More generally, if $\pi$ is a Speh representation associated to the numbers $(a, b)$, then $\iota(\pi)$ is equal to the Speh representation associated to $ (b, a)$, $n = ab$.
Observe also that, because it interchanges Steinberg with the trivial, the involution $i$ may change the ramification of a representation (in my example, ramified became unramified), but there is a bound to it: for example, semi-stable representations will stay semi-stable.
By the Local langlands correspondence, the Zelevinsky dual corresponds to an involution on the set of Weil-Deligne representations. Let us call this involution $i$. I would like to understand this involution in as many different ways as possible.
To fix normalizations, for me the trivial representation of $G$ corresponds to the Weil Deligne representation with trivial $N$, unramified $\rho$ for which geometric Frobenius acts with eigenvalues $q^{(n-1)/2}, q^{(n-3)/2}, \ldots, q^{(1-n)/2}$ ($q$ is the cardinality of the residue field of $F$).
The first question is also the one of the title. It suffices to think only of semi-stable representations. Thus, let $\pi$ be a semi-stable $G$-representation which occurs in some $Ind_B^G(\chi)$, $B$ being the standard borel of $G$. Because $Ind$ and $Res$ are functors we see that $\varepsilon \cdot i(X)$ also occurs in $Ind_B^G(\chi)$. On the Galois side this means that, if $(\rho, N)$ is a Weil-Deligne representation, then $\iota(\rho, N) =: (\rho', N')$ has $\rho = \rho'$. Thus, in some sense, we have made second new monodromy operator $N'$ on the space $(\rho, N)$. So my question: What is $N'$? Is there a direct construction of $N'$ without using Local langlands and Aubert/Zel duality? How do $N$ and $N'$ relate (I only know this for examples...)?
The second question: Is there a geometric construction for $N'$ for representations that occur in the cohomology of varieties?
