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On the local theta-correspondence. (English) Zbl 0583.22010
Let $$^{\sim}({\mathbb{W}})$$ be the non-trivial 2-fold central extension of the symplectic group Sp($${\mathbb{W}})$$ for a non-degenerate symplectic vector space $${\mathbb{W}}$$ over a non-Archimedean local field k of characteristic 0 and ($$\omega$$,S) a smooth oscillator representation of $$^{\sim}({\mathbb{W}})$$ corresponding to a fixed non-trivial character of k. Then, for any reductive dual pair (G,G’) in Sp($${\mathbb{W}})$$, ($$\omega$$,S) can be considered as a representation of $$\tilde G\times \tilde G'$$ where $$\tilde G,\tilde G'$$ are the inverse images of G,G’ in $$^{\sim}({\mathbb{W}})$$. Let Irr(.) denote the set of equivalence classes of irreducible smooth representations. For $$\pi\in Irr(\tilde G)$$, let $$\theta(\pi;\tilde G)=\{\pi'\in Irr(\tilde G')|$$ $$Hom_{\tilde G\times \tilde G'}(\omega,\pi \otimes \pi')\neq 0\}$$ and $$\theta(\pi';\tilde G),$$ likewise for $$\pi'\in Irr(\tilde G')$$. Under this local theta correspondence, $$\pi$$ and $$\pi'$$ are said to ’correspond’, if $$\pi'\in \theta (\pi;\tilde G')$$ or equivalently, if $$\pi\in \theta(\pi';\tilde G)$$; according to the ”local Howe duality conjecture”: $$| \theta (\pi;\tilde G')| \leq 1$$, $$| \theta (\pi';\tilde G)| \leq 1$$, the local theta correspondence is a bijection for all $$\pi\in Irr(\tilde G)$$ and $$\pi'\in Irr(\tilde G').$$
For a reductive dual pair of the type $$(G=O(V_ m)$$, $$G'=Sp(W_ n))$$ in Sp($${\mathbb{W}})$$ with $${\mathbb{W}}=V_ m\otimes W_ n$$, the local theta correspondence is shown here to be compatible with induction (via irreducible representations of Levi subgroups); a unique cuspidal $$\theta(\pi)$$ in $$Irr(\tilde G')$$ turns out to correspond to a cuspidal $$\pi$$ in $$Irr(\tilde G)$$ and vice versa. The arguments used (are intricate and) involve restrictions of induced representations to subgroups H of the form $$Sp(W')\times Sp(W'')$$ with $$W_ n=W'+W''$$ and a study of the orbit structure of such H on a flag manifold; the latter reminds one of similar methods (but in a global situation) in some significant recent work of Garrett and of Böcherer on Eisenstein series.
Reviewer: S.Raghavan

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 11S99 Algebraic number theory: local fields 11F85 $$p$$-adic theory, local fields
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