Every unitary $\infty$-dim'l irreducible representation can be writen as inducing a square-integrable representation from a parabolic subgroup with Levi subgroup $G' =G_{r_1} \times \dots G_{r_2}$. Googling for Bernstein center might help.
I guess $[\pi_0]$ is the family, where you tensor by unramified one-dimensional representations of $G'$, so that is not a fancy action but does not preserve unitarity.
The character formula for a prabolic induced rep looks for parabolic $P_r = G_r N_r$ and Iwasawa decomposition $G = P_r K_r$ like $$ tr\; \pi(\phi) = tr\; \pi(\phi^{N,K}) , \qquad \phi^{N,K}(g) := \int\limits_{N}\int\limits_{K} \pi(k^{-1}gnk) d k\; d n\;(g \in G_r).$$ This is an exercise in functional analysis.
I can not address the algebraic geometry stuff, but it is probably not that difficult. I would need a definition of $Im[\pi_0]$ and the finite group though, but I have no idea what that should be. May be the finite group means permuting $G_k$'s of equal dimension, that is, the relative Weyl group?
All what I am mentioning holds for arbitrary local fields, but the polynomial statement will only be true for non-archimedean fields.
I would check Laumon - Cohomology of Drinfeld modules I and probably II. Most likely, he does the local things without specifying the characteristic of the local field, but of course does not address the archimedean situation at all.