Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $F$ be a local non-Archimedean field. Let $\rho$ be an irreducible smooth representation of $G(F)$. How does one define the conductor of $\rho$? 


The conductor should be an integer canonically associated with the representation $\rho$ so that it matches up with the conductor of the corresponding Galois representation on the Langlands' dual side. 


An example of an answer that I'm looking for is given in <http://www2.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf>
Here, Kevin Buzzard gives a nice simple algebraic definition of the conductor for representations $\mathrm{GL}\_2$ using the mirabolic subgroup (see middle of page six of his notes). Does his definition generalize to $\mathrm{GL}\_n$? Is there a simple algebraic definition for the conductor of representations of general $G$? What is a good reference for this subject? 



Conductor of a representation of a $p$-adic group
for GL\_n you could look at:


H.Jacquet,I.I.Piatetski-Shapiro,J.Shalika:Conducteur des representations du groupe lineaire, Math. Annalen 256, 199–214 (1981).


