By Cassen B.
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We assume, as is often the 26 Jeffrey Adams case, that (M, M ) is itself a dual pair in some Sp(2m, R). Let ω be an oscillator representation for Sp(2n, R) restricted to G × G , and let ωM be an oscillator representation for M × M . The Jacquet module ωN ×N of ω is a representation of M × M , and a calculation shows it has ωM ζ as a quotient for some character ζ of M × M . Suppose there is a non–zero M × M map ˇ φ : ωM → σ ⊗σ for some representation σ ⊗ σ of M × M . That is θ(ψ, M, M )(σ) = σ .
2q + τ ) + 2). Example 8. (O(p, q), Sp(2n, R)) with p + q = 2n, 2n + 1, 2n + 2. In these examples the groups are the same “size”, and are of particular interest from the point of view of L–functions (cf. the lectures of Steve Kudla). This is the opposite extreme of the stable range . ). The cases (O(p, q), Sp(2n, R)) with p + q = 2n, 2n + 2 and p, q even are in , p, q odd are only missing because of covering group technicalities. Finally (O(p, q), Sp(2nR)) witih p + q = 2n + 1 is in , this is similar to  except that the covering groups are unavoidable.
Pages 179– 207. Amer. Math. , Providence, RI, 1985.  Roger Howe. The oscillator semigroup. In The mathematical heritage of Hermann Weyl (Durham, NC, 1987), volume 48 of Proc. Sympos. , pages 61–132. Amer. Math. , Providence, RI, 1988. The Theta Correspondence over R 37  Roger Howe. Remarks on classical invariant theory. Trans. Amer. Math. , 313(2):539–570, 1989.  Roger Howe. Transcending classical invariant theory. J. Amer. Math. , 2(3):535–552, 1989.  Roger Howe. Perspectives on invariant theory: Schur duality, multiplicityfree actions and beyond.