Characters of Non-Linear Groups

Abstract

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 e defined over a number field.Associated to e is its dual group Le.The conjectural principle of functoriality says that a homomorphism L H-----+ Le should provide a "transfer" of automorphic representations from H to those of e.On the other hand Howe's theory of dual pairs, the theta correspondence, starts with the oscillator representation of the non-linear metaplectic group Mp(2n), the two-fold cover of Sp(2n).Restricting this automorphic representation to a commuting pair of subgroups (e, e 1 ) of Mp(2n) gives a relationship between the automorphic representations of e and e'.This suggests a natural question: is the theta-correspondence in some sense "functorial".As Langlands points out [16]: "the connection between theta series and functoriality is quite delicate, and therefore quite fascinating ... ".Now e and e' may be non-linear groups, and so even to define the notion of functoriality requires some work. .In particular the L-groups of e and e' are not defined.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 nonlinearity of e and e' 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 functoriality may be found in [15], [21], and [2].The case of U(3) has been

Locations

• Advanced studies in pure mathematics - View
• Project Euclid (Cornell University) - View - PDF