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


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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