# Which Weil group over a $p$-adic field?

For simplicity, call the Weil group of a local nonarchimedean field $F_v$ to be the following extension: $$1\longrightarrow F^\times_v\longrightarrow W_{F_v}\longrightarrow\text{Gal}(F_v/\mathbb Q_v)\longrightarrow 1.$$

1. Finding this definition to be lacking in terms of describing $\ell$-adic representations, Deligne modified it to what is known as the Weil-Deligne group, $WD_{F_v}=W_{F_v}\ltimes\mathbb G_a.$

2. Langlands in the Corvallis proceedings suggests that one should take instead $W'_{F_v}=W_{F_v}\times SL(2,\mathbb C)$.

3. Kottwitz (and Arthur, citing him) later writes that it is known that (the Langlands group) should be $L_{F_v}=W_{F_v}\times SU(2,\mathbb R)$, to give a natural bijection between the irreducible n-dimensional complex representations of $L_{F_v}$ irreducible admissible representations of $GL_n(F_v)$.

It seems that (3) is the accepted form of the Langlands group, is (2) then no longer relevant? Is (1) now subsumed under (3), so that one should really be thinking of the latter?

As a follow up question, why is it that $L_F$ should be an extension of $W_F$ by a compact group (Kottwitz)? Or even a reductive, proalgebraic group (Langlands)?

• I just noticed that your definition of the Weil group is very weird. This is not the right definition. – Joël May 8 '14 at 23:42
• I realize that it's a nonstandard definition; but I simply lifted it from langlands' paper sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/autoreps-ps.pdf p.20. – TA Wong May 9 '14 at 19:12
• It isn't that this definition is non-standard I think. It's that this is the definition of the (relative) "finite level" Weil group of the class formation of local class field theory for the "layer" $F_v/\mathbf{Q}_p$. If one takes the inverse limit over all finite extensions of $\mathbf{Q}_p$, then one gets the usual Weil group $W_{\mathbf{Q}_p}$ of $\mathbf{Q}_p$. – Keenan Kidwell May 10 '14 at 14:47
• Ah, I see what you mean. Your point is well taken. – TA Wong May 11 '14 at 3:26