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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)?

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    $\begingroup$ I just noticed that your definition of the Weil group is very weird. This is not the right definition. $\endgroup$ – Joël May 8 '14 at 23:42
  • $\begingroup$ 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. $\endgroup$ – TA Wong May 9 '14 at 19:12
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    $\begingroup$ 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 $\mathbf{Q}_p$. $\endgroup$ – Keenan Kidwell May 10 '14 at 14:47
  • $\begingroup$ Ah, I see what you mean. Your point is well taken. $\endgroup$ – TA Wong May 11 '14 at 3:26
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First, (2) and (3) are functionally equivalent. What matters are not the group themselves, but their categories of finite-dimensional complex representations (continuous in case (3), algebraic in the case (2)). But the two are the same: that's the simplest case of the famous Weyl's unitary trick. So working with (2) or (3) is just a question of taste, or of context. (2) is more natural in the context of Tannakian formalism, where we work with pro-algebraic group, while (3) is simpler.

Second, yes (1) is subsumed in (3). To be precise the category of Weil-Deligne representations of (1) is equivalent in a simple way to the category of representations of (3), so we can forget about (1). Most people work with (3) nowadays.

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  • $\begingroup$ Thanks, I see the unitarian trick now. Is there a reference or quick argument to the second, how to pass from (1) to (3)? $\endgroup$ – TA Wong May 9 '14 at 19:19
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The equivalence between (1) and (2) is obtenaid by a theorem of Jacobson-Morozov-Kostant. You can find a proof in Gross, Benedict; Reeder, Mark Arithmetic invariants of discrete Langlands parameters. Duke Math. J. 154 (2010), no. 3, 431–508, section 2.1

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