When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. More precisely, let $F$ be a nonarchimedean local field, let $G$ be a reductive group over $F$, let $\mu$ be a Haar measure on $G(F)$, and let $f:G(F)\rightarrow\mathbb{C}$ be a locally constant function with compact support. For any admissible representation $\pi$ of $G(F)$, we have the *trace* of $f$:

$$ \operatorname{tr}(f|\pi) = \operatorname{tr}\left(v\mapsto\int_{G(F)}f(g)\pi(g)v\cdot d\mu(g)\right). $$

(One aspect of) Kazhdan's density theorem says that if $\operatorname{tr}(f|\pi)=0$ for all admissible tempered representations $\pi$ of $G(F)$, then $\operatorname{tr}(f|\pi)=0$ for *all* admissible representations $\pi$ of $G(F)$.

This is precisely direction (d)$\implies$(c) in Theorem 0 of Kazhdan's "Cuspidal Geometry of $p$-adic Groups." Kazhdan's paper simply says that it follows from Theorem XI.2.11 in Borel and Wallach's *Continuous cohomology, discrete subgroups, and representations of reductive groups*, which is some seemingly impenetrable statement about the Langlands classification for nonarchimedean local fields.

At this point, I have a few questions:

How does one prove (this part of) Kazhdan's density theorem? What is this part of Borel and Wallach's book really saying, and how does their result imply Kazhdan's?

This part of Borel and Wallach's book considers any nonarchimedean local field $F$, whereas Kazhdan's paper only considers $F$ with characteristic zero. Does (this part of) Kazhdan's density theorem hold over $F$ with positive characteristic too? Of course, this might just be an immediate generalization of the answer to 1.

Thank you in advance!