In Reductive Groups Part I: Over Algebraically Closed Fields we learned about how reductive groups over algebraically closed fields are classified by their root datum, and how the based root datum helps us understand their automorphisms. In this post, we consider reductive groups over more general, not necessarily algebraically closed fields. We will discuss how they can be classified, and also define the Langlands dual of a reductive group, which will allow us to state the local Langlands correspondence (see The Local Langlands Correspondence for General Linear Groups) for certain groups other than just the general linear group.
Let be a field. We will say an algebraic group over is a reductive group if , the base change of to the algebraic closure , is a reductive group. Similarly, we say that is a torus if the base change is a torus. This means that after base change to the algebraic closure it becomes isomorphic to the product of copies of the multiplicative group . However, if over it is already isomorphic to the product of copies of , without the need for a base change, then we say that it is a split torus. If a reductive group contains a maximal split torus, we say that is split. We note that a “maximal split torus” is different from a “split maximal torus”!
The classification of split reductive groups is the same as that of reductive groups over algebraically closed fields – they are classified by their root datum. As such they will provide us with the first step towards classifying reductive groups over more general fields.
If is a reductive group over , a form of is some other reductive group over such that after base change to the algebraic closure , and are isomorphic. It happens that any reductive group is a form of a split reductive group. This follows from the fact that any abstract root datum is the root datum associated to some reductive group and some split maximal torus contained in it.
The forms of a split group are classified using Galois cohomology. Suppose we have an isomorphism . The Galois group (henceforth shortened to just acts on the isomorphism by conjugation, giving rise to another isomorphism . Composing this with the inverse of we get an automorphism of . This automorphism is an example of a –cocycle in Galois cohomology.
More generally, in Galois cohomology, for some group with a Galois action (for instance in our case ), a -cocycle is a homomorphism such that . Two -cocycles are cohomologous if there is an element such that . The set of -cocycles, modulo those which are cohomologous, is denoted .
By the above construction there is a map between the set of isomorphism classes of forms of and the Galois cohomology group . This map actually happens to be a bijection!
Let be a based root datum corresponding to together with a pinning. We have mentioned in Reductive Groups Part I: Over Algebraically Closed Fields the group of automorphisms of , the pinned automorphisms of , and the outer automorphisms of are all isomorphic to each other. We have the following exact sequence
and when we are provided the additional data of a pinning this gives us a splitting of the exact sequence (i.e. a way to decompose the middle term into a semidirect product of the other two terms).
When the pinning is defined over , we obtain a homomorphism
where is in bijection with the set of conjugacy classes of group homomorphisms . But we have said earlier that is in bijection with the set of isomorphism classes of forms of . Therefore, any form of gives us such a homomorphism .
We say that a reductive group is quasi-split if it contains a Borel subgroup. Split reductive groups are automatically quasi-split.
An inner form of a reductive group is another reductive group related by an isomorphism such that the composition is an inner automorphism of .
Once we have a split group , and given the data of a pinning, we can now use any morphism together with the given pinning to obtain an element of , which in turn will give us a quasi-split form of . Now it happens that any reductive group has a unique quasi-split inner form!
Therefore, in summary, the classification of reductive groups over general fields proceeds in the following three steps:
- Classify the split reductive groups using the root datum.
- Classify the quasi-split forms using the homomorphisms .
- Classify the inner forms of the quasi-split forms.
Let us now discuss the Langlands dual (also known as the L-group) of a reductive group. Since every abstract root datum corresponds to some reductive group (say, over a field ), we can interchange the roots and coroots and get another reductive group , which we refer to as the dual group of .
The Langlands dual of is the group (an honest to goodness group, not an algebraic group) given by the semidirect product . In order to construct this semidirect product we need an action of on , and in this case this action is via its action on the based root datum of together with a Borel subgroup , which is the same as a pinned automorphism of . We denote the Langlands dual of by .
Let us recall that in The Local Langlands Correspondence for General Linear Groups we stated the local Langlands correspondence, in the case of where is a local field, as a correspondence between the irreducible admissible representations of (over ) and the F-semisimple Weil-Deligne representations of the Weil group of .
With the definition of the Langlands dual in hand, we can now state the local Langlands correspondence more generally, not just for , and in this case it will not even be a one-to-one correspondence between irreducible admissible representations and Weil-Deligne representations anymore!
First, we will need the notion of a Langlands parameter, also called an L-parameter, which takes the place of the F-semisimple Weil-Deligne representation. It is defined to be a continuous homomorphism such that, as a homomorphism from to , it is semisimple, the composition is just the usual inclusion of into , and as a function of to it comes from a morphism of algebraic groups from to .
And now for the statement: The local Langlands correspondence states that, for a reductive group over a local field , the irreducible admissible representations of are partitioned into a finite disjoint union of sets, called L-packets, labeled by (equivalence classes of, where the equivalence is given by conjugation by elements of ) L-parameters. In other words, letting be the set of isomorphism classes of irreducible admissible representations of , and letting be the set of equivalence classes of L-parameters, we have
where is the L-packet, a set of irreducible admissible representations of . In the case that is p-adic and , each of these L-packets have only one element and this reduces to the one-to-one correspondence which we saw in The Local Langlands Correspondence for General Linear Groups.
The L-group and L-parameters are also expected to play a part in the global Langlands correspondence (in the case of function fields over a finite field, the construction of L-parameters was developed by Vincent Lafforgue using excursion operators). There is also much fascinating theory connecting the representations of the L-group to the geometry of a certain geometric object constructed from the original reductive group called the affine Grassmannian. We will discuss more of these topics in the future.
Reductive group on Wikipedia
Root datum on Wikipedia
Inner form on Wikipedia
Langlands dual group on Wikipedia
Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)
Representation theory and number theory (notes by Chao Li from a course by Benedict Gross)
Algebraic groups by J. S. Milne