Weil Representation, Howe Duality, and the Theta correspondence

Conjecture 2 (Howe): The representation σ 0 (π) of ˜G′ has a unique irreduciblequotient denoted by σ(π), and the mapping π → σ(π) is a bijection betweenR ψ ( ˜G) and R ψ ( ˜G ′ ).Remark 4.3: The Howe conjecture is now proved for non-archimedean localfields of residue characteristic not 2. Howe himself proved it in many cases and itwas recently completed in residue characteristic not 2 by Waldspurger [Wa2].The conjecture also makes sense in the archimedean case and was proved byHowe (we take up the archimedean case in the next section).Though the conjecture is technically speaking false for finite fields (as wasobserved by Howe), it should essentially be true there too.It is a theorem of Kudla [Ku1] that σ 0 (π) has finite length and that if π issupercuspidal then σ o (π) is irreducible.The following lemma is very useful in the study of Howe correspondence, i.e.the correspondence given by conjecture 2.Lemma 4.4: For a dual reductive pair (G, G ′ ) in Sp(W ), the metaplecticcovering ˜ Sp ψ (W ) splits on G unless (G, G ′ ) is the pair (Sp(U), O(V )) in Sp(U ⊗V ),with dim V odd.Remark 4.5: The lemma 4.4 is not true for the two sheeted cover of thesymplectic group in place of Spψ ˜ (W ).4.6 Examples of the Howe Correspondence : We will be looking at certainexamples of the Howe correspondence for the dual pair (Sp(W ), O(V )) ⊂ Sp(W ⊗V ), mostly for dim W = 2 in which case Sp(W ) ∼ = SL(2, k).4.6.1 : dim V = 1, q(x) = ax 2 . Therefore O(V ) = {±1} and has two representations.Both of these appear in the Weil representation and the correspondingrepresentation of SL(2, ˆ k) are on the even and odd functions on k. The representationof SL(2, ˆ k) on odd functions is supercuspidal, cf. [Ge1], thm.5.19(c).4.6.2 : dim V = 2, q(x) = a · (norm form of a quadratic field extension K of k).In this case SO(V ) ∼ = K 1 = norm one elements of K, and O(V ) is the semi-directproduct of K 1 with a group of order 2 acting on K 1 by x → ¯x = x −1 . Therefore therepresentations of O(V ) are constructed from the characters of K 1 in the followingway:12