The paper "A note on the automorphic Langlands group" by J. Arthur,


discusses the mysterious `automorphic Langlands group'. This is the mysterious group whose complex representations should correspond to automorphic forms, 'generalizing' the way in which l-adic representations of the Galois group (de Rham at all places dividing l, and so on) correspond to automorphic forms with algebraic character at infinity.

Although the global automorphic Langlands group is mysterious, it is stated on p2 of the paper cited above that the local versions of it are understood. In particular, in the second displayed equation on p2 we are told that at a non-archimedean place, the Local version of the automorphic langlands group the product of the local Weil group and SU(2,R), but the author does not explain why, apart from a reference to a paper of Kottwitz which also does not (seem to) give an explanation. Indeed, I'm not even sure what is meant by SU(2,R) since to take SU I think I need a field with an automorphism of degree 2, and R doesn't have one.

Does anyone know of a reference which explains why SU(2,R) appears (or even what it is)?

[Edit - added scare quotes around the word 'generalizing', since as KB points out in a comment below, it's not really a generalization.]

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    $\begingroup$ $SU(2,\mathbf{R})$ is the $\mathbf{R}$-points of the special unitary group associated to $\mathbf{C}/\mathbf{R}$ (since the latter is an algebraic group over $\mathbf{R}$). This group is the standard group of two-by-two unitary matrices with complex entries and determinant 1 that you might see in a first course on linear algebra. A paper you can try looking at would be Arthur's 2002 paper "A Note on the Automorphic Langlands Group" where he gives a conjectural construction of a global Langlands group which could help you understand why the local group is what it is. $\endgroup$ – Rob Harron Apr 4 '11 at 16:35
  • $\begingroup$ @Rob - thanks! I thought $SU(2,\mathbf{R})$ was probably what I would call $SU(2,\mathbf{C})$, but I wasn't sure. Thanks for the Arthur paper reference---that's actually the paper I was looking at that prompted the question. If mathoverflow fails, I will indeed try to see if the rest of the paper sheds any light on where the SU(2,C) comes from. $\endgroup$ – blt Apr 4 '11 at 17:17
  • $\begingroup$ Picky point: I am not sure about your use of the word "generalizing" in the second sentence above, because one traditionally uses complex reps of the Langlands group but $\ell$-adic representations of Galois groups and it's not at all clear (to me) how to pass from one story to the other (in either direction). $\endgroup$ – Kevin Buzzard Apr 4 '11 at 18:18
  • $\begingroup$ @Kevin - You're absolutely right; I had the same thought as I was writing the question, but decided that brevity was the soul of wit and that I couldn't be more accurate without being a lot more verbose. But perhaps I will add scare quotes around 'generalizing'. (Indeed I've never really known what the conjectural picture for what an $l$-adic repn's of the automorphic Langlands group is meant to relate to, or more generally how the automorphic Langlands group fits into the $p$-adic Langlands programme.) $\endgroup$ – blt Apr 4 '11 at 18:37
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    $\begingroup$ "...we are told that at an archimedean place..." you mean a non-archimedean place. $\endgroup$ – TSG Apr 6 '11 at 11:30

Dear blt,

Just to rephrase Rob H.'s answer in a slightly different way: adding the $SU(2)$ is just another way of adding the $N$ operator, the explanation for which I think you're familiar with. In other words, the local Langlands group does the same job as the Weil--Deligne group, but just uses $SU(2)$ rather than $N$. (In fact, I presume that some people just refer to $W_F \times SU(2)$ as the Weil--Deligne group.) It it to some extent a question of taste whether you prefer to add the $N$ operator or the $SU(2)$, but the latter has some advantages: e.g. one can just talk directly about the locally compact group $W_F \times SU(2)$, rather than have to talk about $W_F$-representations with an extra $N$ operator satisfying some axioms.



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Given local Langlands, you're asking why the complex finite-dimensional semi-simple representations of $W_F\times SU(2,\mathbf{R})$ are in correspondence with Frobenius semi-simple representations of the Weil–Deligne group, $W_F^\prime$, of a non-archimedean field. In fact, you can replace the $SU(2,\mathbf{R})$ with $SL(2,\mathbf{C})$. In this latter case, the correspondence is described in section 6 of Rohrlich's very nice paper Elliptic curves and the Weil–Deligne group. Basically, the indecomposable Frobenius semi-simple Weil–Deligne representations of $W_F$ are all of the form $\rho\otimes sp(n)$ where $\rho$ is an irreducible representation of $W_F$ and $sp(n)$ is the $n$-dimensional "special" representation. Then $\rho\otimes sp(n)$ corresponds to $\rho\otimes \text{Sym}^n\mathbf{C}^2$.

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There is a very nice explanation of the back-and-forth between Weil-Deligne representations and representations of $W_F \times SL_2$ in section 2.1 of the paper "Arithmetic invariants of discrete Langlands parameters" by Gross and Reeder.

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