Characters of Nonlinear Groups

Two of the primary methods of constructing automorphic forms are the Langlands program and Howe's theory of dual pairs. The Langlands program concerns a reductive linear group G deened over a number eld. Associated to G is its dual group L G. The con-jectural principle of functoriality says that a homomorphism L H ! L G should provide a \transfer" of automorphic representations from H to those of G. On the other hand Howe's theory of dual pairs, the theta correspondence , starts with the oscillator representation of the non-linear meta-plectic group Mp(2n), the twofold cover of Sp(2n). Restricting this automorphic representation to a commuting pair of subgroups (G; G 0) of Mp(2n) gives a relationship between the automorphic representations of G and G 0. This suggests a natural question: is the theta-correspondence in some sense \functorial". As Langlands points out in 16]: \the connection between theta series and functoriality is quite delicate, and therefore quite fascinating : : : ". Now G and G 0 may be non-linear groups, and so even to deene the notion of functoriality requires some work. In particular the L-groups of G and G 0 are not deened. Nevertheless it is reasonable to ask that theta-lifting be given by some sort of data on the \dual" side. This can be done in some cases in which the non-linearity of G and G 0 do not play an essential role. Nevertheless a proper understanding of the relationship between theta-lifting (and its generalizations) and functoriality requires bringing the representation theory of non-linear groups into the Langlands program. Some discussion of the relation of the theta-correspondence to func-toriality may be found in 15], 21], and 2]. The case of U(3) has been discussed in great detail in 8].