Lifting of characters on orthogonal and metaplectic groups (original) (raw)

1998, Duke Mathematical Journal

An important principle in representation theory and automorphic forms is that of lifting or transfer of representations between reductive algebraic groups. Endoscopic transfer and base change are primary examples. Another type of example is provided by theta-lifting between members of a reductive dual pair. In this paper we study lifting, defined directly on characters, between special orthogonal groups SO(2n + 1) over R and the non-linear metaplectic group Sp(2n, R). This is closely related both to endoscopy and theta-lifting, and is an aspect of the duality between root systems of type B n and C n. Let π be an irreducible representation of SO(p, q), the special orthogonal group of a symmetric bilinear form in p + q = 2n + 1 real variables; π has a non-zero theta-lift to a representation π ′ of Sp(2n, R). A natural question is: what is the relationship, if any, between the global characters of π and π ′ ? When n = 1 this is closely related to the Shimura correspondence, which has been the subject of extensive study. Evidence for such a relation is provided by the orbit correspondence, which provides a matching of semisimple conjugacy classes of SO(p, q) and Sp(2n, R). This is analogous to the matching of stable conjugacy classes in the theory of endoscopy. In fact there is a natural bijection between (strongly) regular, semisimple, stable conjugacy classes in the split groups SO(n + 1, n) and Sp(2n, R). In elementary terms two such conjugacy classes correspond if they have the same non-trivial eigenvalues. Alternatively there is a bijection between conjugacy classes of Cartan subgroups in these two groups. The main ideas are best illustrated by the example of the discrete series. Let π SO (λ) be a discrete series representation of SO(n+1, n). We have fixed a compact Cartan subgroup T , and λ ∈ t * is a Harish-Chandra parameter. In the usual coordinates λ = (a 1 ,. .. , a k ; b 1 ,. .. , b ℓ) (1.1)(a) with a i , b j ∈ Z + 1 2 , a 1 > • • • > a k > 0, b 1 > • • • > b ℓ > 0. Fix a compact Cartan subgroup T ′ of Sp(2n, R), with inverse imageT ′ in Sp(2n, R). The theta-lift of π SO (λ) to Sp(2n, R) is the discrete series representation π Sp (λ ′) with Harish-Chandra parameter