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

http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf

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 an archimedean 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.]

2 added 142 characters in body

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

http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf

discusses the mysterious automorphic Langlands group'. This is the mysterious group whose complex representations should correspond to automorphic forms, generalizing '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 an 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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# Why do we see SU(2,R) in the Local automorphic Langlands group?

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

http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf

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 an 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)?