Representations of abelian algebraic groups

Abstract

There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension