Conductor of a representation of a $p$-adic group

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?

• The case of $\mathrm{GL}_2$ was first dealt with Casselman in his paper "On some results of Atkin and Lehner". However, it seems that he uses a different subgroup from Jacquet, Piatetski-Shapiro, Shalika [JP-SS]. Namely, Casselman uses the subgroup of $\mathrm{GL_2}(\mathcal{O})$ whose reduction modulo $t^n$ is the Borel subgroup, where as, [JP-SS] use the subgroup whose reduction modulo $t^n$ is the mirabolic subgroup. Can someone explain the discrepancy? Dec 5, 2013 at 3:59