Langlands Dual - Definition For Groups Over More General Fields

Definition For Groups Over More General Fields

Now suppose that G is a reductive group over some field k with separable closure K. Over K, G has a root datum, and this comes with an action of the Galois group Gal(K/k). The identity component LGo of the L-group is the connected complex reductive group of the dual root datum; this has an induced action of the Galois group Gal(K/k). The full L-group LG is the semidirect product

LG = LGo×Gal(K/k)

of the connected component with the Galois group.

There are some variations of the definition of the L-group, as follows:

• Instead of using the full Galois group Gal(K/k) of the separable closure, one can just use the Galois group of a finite extension over which G is split. The corresponding semidirect product then has only a finite number of components and is a complex Lie group.
• Suppose that k is a local, global, or finite field. Instead of using the absolute Galois group of k, one can use the absolute Weil group, which has a natural map to the Galois group and therefore also acts on the root datum. The corresponding semidirect product is called the Weil form of the L-group.
• For algebraic groups G over finite fields, Deligne and Lusztig introduced a different dual group. As before, G gives a root datum with an action of the absolute Galois group of the finite field. The dual group G* is then the reductive algebraic group over the finite field associated to the dual root datum with the induced action of the Galois group. (This dual group is defined over a finite field, while the component of the Langlands dual group is defined over the complex numbers.)