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1. 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.

2. 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.

3. 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.

4. 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.

1. 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.

2. 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.

3. 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 $$ tr\; \pi(\phi) = tr\; \pi(\phi^{N,K}) , \qquad \phi^{N,K}(g) := \int\limits_{N}\int\limits_{K} \phi(k^{-1}gnk) d k\; d n\;(g \in G_r).$$ This is an exercise in functional analysis.

4. 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.

1. Every unitary $\infty$-dim'l irreducible representation can be writen as inducing a square-integrable representation from a parabolic subgroup with Levi subgroup $G' =G_{r_1} \times \dots G_{r_2}$  .

2. 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.

3. 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 $$ tr\; \pi(\phi) = tr\; \pi(\phi^{N,K}) , \qquad \phi^{N,K}(g) := \int\limits_{N}\int\limits_{K} \pi(k^{-1}gnk) d k\; d n\;(g \in G_r).$$

4. 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?

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.

1. 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.

2. 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.

3. 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.

4. 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.

1. Every unitary $\infty$-dim'l irreducible representation can be writen as inducing a square-integrable representation from a Levi subgroup $G' =G_{r_1} \times \dots G_{r_2}$.

2. 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.

3. 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 $$ tr\; \pi(\phi) = tr\; \pi(\phi^{N,K}) , \qquad \phi^{N,K}(g) := \int\limits_{N}\int\limits_{K} \pi(k^{-1}gnk) d k\; d n\;(g \in G_r).$$

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.

1. Every unitary $\infty$-dim'l irreducible representation can be writen as inducing a square-integrable representation from a parabolic subgroup with Levi subgroup $G' =G_{r_1} \times \dots G_{r_2}$  .

2. 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.

3. 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 $$ tr\; \pi(\phi) = tr\; \pi(\phi^{N,K}) , \qquad \phi^{N,K}(g) := \int\limits_{N}\int\limits_{K} \pi(k^{-1}gnk) d k\; d n\;(g \in G_r).$$

4. 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?

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.