A question on the proof of Gelfand-Kazhdan theorem.

Question

Consider $F=\Bbb Q_p$ and $G=GL(n, F)$ and $\theta \in \operatorname{Aut(G)}$ s.t $\theta(g)=(g^t)^{-1}$

Now Gelfand-Kazhdan theorem says that:

For any representation $(\pi, V)$ of $G$, if we construct another representation $\pi^{\theta}(g)=\pi(\theta(g))$ then that will be equivalent to $\pi^{\vee}$, where $(\pi^{\vee}, V^{\vee})$ is the representation of $V^{\vee})$ i.e $\pi^{\vee}:G \to \operatorname{GL}(V^{\vee})$ such that $\pi^{\vee}(g)v^{\vee})(v)=v^{\vee}(\pi(g^{-1}v)$.

1. Now if I have any finite field $\Bbb F_p$ will it be possible to give me a particular intertwining operator on that because in general the proof goes using character theory in finite dimensional case.
2. Next I was wandering that if there would be any simple proof of Gelfand-Kazhdan theorem using basic linear algebra.

My question might seem to be odd, but I am in general trying to give generalisation of the theorem in the covering space of a group. So if I someone have some simple idea to prove it, please give that link or proof.

Answer 1

As with the earlier link you can also have a look

1) A work of Alan Roche and Steven Spallone 2) Work of K Balasubramanian 3) Work of K Balasubramanian 4) A work of Alan Roche and Steven Spallone

Answer 2

I think you can have a look at this. You will get a partial answer to your question.