TITLE: What is a tamely-ramified Weil-Deligne representation?
QUESTION [28 upvotes]: Let $W_F$ denote the Weil group of a finite extension of $\mathbb{Q}_p$. Let $I$ denote the inertia subgroup and $I^{>0}$ the (pro-$p$) subgroup of wild inertia. (I hope I've got my notation right...apologies if I haven't.)
We have $W_F/ I = \mathbb{Z}$ (canonically), and $I/I^{>0} \cong \prod_{\ell \ne p} \mathbb{Z}_\ell$ (non-canonically).
Consider a Weil-Deligne representation $(V, \rho, N)$, where $V$ is a finite-dimensional complex vector space, $\rho : W_K \to GL(V)$ is a rep of $W_K$, and $N$ is a nilpotent operator satisfying a well-known condition.
Naively, I would have thought that a Weil-Deligne representation $(V,\rho,N)$ is tamely ramified if

$\rho$ vanishes on $I^{>0}$, no conditions on $N$

However, when I read literature on the Deligne-Langlands conjecture (e.g. Kazhdan-Lusztig, Chriss-Ginzburg) it seems that tamely ramified actually means

$\rho$ vanishes on $I$, no conditions on $N$

[Under this assumption an $F$-semi-simple, tamely ramified, Weil-Deligne representation becomes a (conjugacy class of a) pair $(s,N)$ consisting of a semi-simple element $s$, a nilpotent element $N$, such that $sNs^{-1} = qN$, which is what one expects from affine Hecke algebras, and what we are told the Deligne-Langlands conjecture predicts that the answer should be. [EDIT (following Ben Zvi's comments below): Actually we also need an irreducible representation of the component group of the centraliser of this pair. This extra piece of data is absent for $GL_n$ as this group is always connected.]
Passing back through Grothedieck's equivalence, the above seems to give that a tamely-ramified and continuous representation of $W_K$ in a $\mathbb{Q_\ell}$-vector space, should have each $\mathbb{Z}_{\ell'}$ factor (for $\ell' \ne \ell$) acting trivially, which seems a little strange. (I.e. the finite monodromy which could come from the other factors should magically be trivial.)
Finally, an admittedly lazy question. I can read in several places that, under the local Langlands correspondence, tamely ramified should correspond to representations admitting an Iwahori fixed vector.

Is this explained somewhere? Why should tamely ramified (whatever that actually means, c.f. the above question) translate into having an Iwahori fixed vector?

I'm sure this is easy for the pros! Thanks in advance.

REPLY [4 votes]: This is more of an extended comment. If I'm understanding the question correctly, I don't believe it's correct to say that tamely ramified Galois representations with unipotent monodromy correspond precisely to representations with Iwahori fixed vector, outside of say $GL_n$ -- it seems to me one can't characterize reps with an Iwahori-fixed vector in terms of only their Langlands parameter. 
[Edit: More precisely, there are strictly more unipotent tamely ramified representations than reps with an Iwahori-fixed vector. This is precisely analogous to the situation with Chevalley groups: there are more representations which are unipotent in the Deligne-Lusztig sense, i.e. are attached the trivial semisimple conjugacy class in the dual group, than there are unipotent principal series, i.e., reps with a Borel-fixed vector. This is accounted for by Lusztig's generalized Springer correspondence -- geometrically, it's the statement that not all equivariant perverse sheaves on the nilpotent cone are generated by the Springer sheaf, outside of $GL_n$.]
Kazhdan-Lusztig's result identifies representations with Iwahori-fixed vector as having unipotent Langlands parameter - i.e. q-commuting pairs of a semisimple and nilpotent in the dual group. However representations aren't classified by their Langlands parameter - rather one should think of them as suitable sheaves on a stack of Langlands parameters, and one has to identify the action of the stabilizer (i.e. the L-packet). (This I think is a modern form of local Langlands correspondence, one seeks to identify categories of reps of groups over local fields as or inside suitable categories of sheaves on stacks of Langlands parameters). Kazhdan-Lusztig say we only get SOME of the objects in the corresponding L-packets: the representations of stabilizers that appear are only those that take part in a form of the Springer correspondence. This has to do with the existence of cuspidal local systems on Levis (as in the generalized Springer correspondence) and hence is invisible for $GL_n$.
Lusztig then proved [Classification of unipotent representations of simple p-adic groups. Internat. Math. Res. Notices 1995, no. 11, 517–589] an "affine generalized Springer correspondence". To paraphrase: unipotent Langlands parameters exactly correspond to unipotent representations -- a generalization of reps with an Iwahori fixed vector, in which the notion of Iwahori fixed vector (given by the trivial representation of the finite torus, hence of the finite Borel, hence of Iwahori) is replaced by a unipotent cuspidal representation of a Levi of the corresponding Chevalley group.  
This distinction (unipotent vs Iwahori-fixed) does not arise in geometric Langlands -- it appears that the categorical analogs of unipotent representations have Iwahori-fixed vectors. This is e.g. well understood for reductive groups (rather than loop groups), where all unipotent representations come from decategorifying the finite Hecke category. Likewise, if you formally take the categorical trace of Frobenius on a (suitably mixed!!) version of Bezrukavnikov's Langlands duality for the Iwahori-Hecke category (cited by Dr. Evil), you get (up to singular support issues) the full category of coherent sheaves on the stack of unipotent Langlands parameters -- i.e. the home (thanks to Lusztig's theorem) of all unipotent reps of our original group.