How does the conjectural Langlands group fit into the Tannakian point of view?

2011-09-07T00:48:26Z

I've read that one way to formulate the Langlands program is the following:

Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) representations of $\mathcal{L}_{\mathbb{Q}}$ that are algebraic(!) is equivalent to the category of motives over $\mathbb{Q}$ with coefficients in $\overline{\mathbb{Q}}$.

This is peculiar to me, since the category of motives is Tannakian, and so is equivalent to the category of (all!) representations of some affine group scheme.

How does one think of the condition that the representations must be algebraic? Does this mean that the Langlands group is not meant to be the motivic Galois group (the group guaranteed by Tannakian formalism applied to the category of motives, using some fiber functor)? Is there a way to reconcile the two approached in an insightful way?

Answer by Emerton for How does the conjectural Langlands group fit into the Tannakian point of view?

2011-09-07T02:38:32Z

The Langlands group is not meant to be the motivic Galois group; rather, it is larger (in Langlands's original formulation), or alternatively not an algebraic group, but a locally compact group which has some kind of underlying algebraic avatar (this is the more recent, indeed current, formulation, due to Kottwitz), so that one can speak of both continuous and algebraic representations.

A toy model to think about is the group $\mathbb C^{\times}$, and the difference between representations of $\mathbb C^{\times}$ just as a topological group, as opposed to $\mathbb C^{\times}$ thought of as a real algebraic group (i.e. thought of as the restriction of scalars of $\mathbb G_m$ from $\mathbb C$ to $\mathbb R$).

To see this example arising in real life, one can think about the difference between arbitrary and algebraic (type $A_0$ in Weil's terminology) Hecke characters for some number field $F$. (This is the theory for $1$-dimensional reps. of the Langlands group $\mathcal L_F$.) The relevant topological group is the idele class group of $L$, while the corresponding algebraic group is what Langlands calls the Serre group in his Ein Maerchen paper (and which is studied, but with different notation and terminology, in Serre's Abelian $\ell$-adic reps. book).

A harder example can be obtained by comparing the global Weil group over a number field to the Taniyama group over this field. This is discussed in Ein Maerchen, and in the book of Deligne, Milne, Ogus, and Shih, Hodge cycles, motives, and Shimura varieties. (This is the theory obtained by combining $1$-dimensional reps. with finite image reps. of arbitrary dimension.) (See this answer for more on Weil groups.)

How does the conjectural Langlands group fit into the Tannakian point of view? - MathOverflow

How does the conjectural Langlands group fit into the Tannakian point of view?

Answer by Emerton for How does the conjectural Langlands group fit into the Tannakian point of view?