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Weil group, Weil-Delinge Weil-Deligne group scheme and conjectural Langlands group

I was reading a series of article from the Corvallis volume. There are couple of questions which came to my mind:

  1. Why do we need to consider representation of Weil-Delinge Weil-Deligne group? That is what is an example of irreducible admissible representation of $ Gl(n,F)$ which does not correspond to a representation of $W_F$ of dimension $n$ ? An example for $ n=2 $ will be of great help.

  2. In the setting of global Langlands conjecture, why extension of $W_F$ by $G_a$ or products of $W'_{F_v}$ does not work?

Thank you.
show/hide this revision's text 1

Weil group, Weil-Delinge group scheme and conjectural Langlands group

I was reading a series of article from the Corvallis volume. There are couple of questions which came to my mind:

  1. Why do we need to consider representation of Weil-Delinge group? That is what is an example of irreducible admissible representation of $ Gl(n,F)$ which does not correspond to a representation of $W_F$ of dimension $n$ ? An example for $ n=2 $ will be of great help.

  2. In the setting of global Langlands conjecture, why extension of $W_F$ by $G_a$ or products of $W'_{F_v}$ does not work?

Thank you.