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On the Howe duality conjecture. (English) Zbl 0624.22011
The author shows that Howe’s global duality conjecture would follow from the local duality conjecture at all places, and he proves the latter in a number of new cases. Moreover, he conjectures a refinement in the global conjecture, concerning multiplicities.
Let K be a number field with adèle ring $${\mathbb{A}}$$, Q a nondegenerate quadratic form in m variables defined over K. Let $$G=O(Q)$$, and let $$(\pi)\subset L^ 2_{cusp}(G(K)\setminus G({\mathbb{A}}))$$ be the isotypic subspace for an irreducible cuspidal representation $$\pi$$ of G($${\mathbb{A}})$$. Now $$G\times Sp_ r\subset Sp_{mr}$$ for any positive integer r. By integrating (over G(K)$$\setminus G({\mathbb{A}}))$$ every $$f\in (\pi)$$ against every theta-kernel for the oscillator representation of $$Sp_{mr}({\mathbb{A}})$$ one generates a complex subspace $${\mathcal A}_ r(\pi)$$ of functions on $$Sp_ r(K)\setminus Sp_ r({\mathbb{A}}).$$
Let $$L_ r=L_ r(Q)$$ be the subspace of $$L^ 2_{cusp}(G(K)\setminus G({\mathbb{A}}))$$ orthogonal to all theta-kernels for $$Sp_{mr}$$, and let $$R_ i=R_ i(Q)$$ be the orthogonal complement of $$L_ 1\cap \cdot \cdot \cdot \cap L_ i$$ in $$L_ 1\cap \cdot \cdot \cdot \cap L_{i- 1}.$$
Theorem: (1) $$L^ 2_{cusp}(G(K)\setminus G({\mathbb{A}}))=R_ 1\oplus \cdot \cdot \cdot \oplus R_ m$$, orthogonal direct sum.
(2) If $$\pi \in R_ i$$, then $${\mathcal A}_ i(\pi)\cap L^ 2_{cusp}(Sp_ i(K)\setminus Sp_ i({\mathbb{A}}))\neq (0).$$
Let $${\mathcal A}^ 0(\pi)$$ denote the intersection in (2). Then the author states Howe’s global duality conjecture in the following form: Conjecture: Given $$\pi$$ occuring in $$R_ i$$, there is an irreducible representation $${\mathcal B}(\pi)$$ of $$Sp_ i({\mathbb{A}})$$ occuring in $$L^ 2_{cusp}(Sp_ i(K)\setminus Sp_ i({\mathbb{A}}))$$ such that $${\mathcal A}^ 0(\pi)$$ is a multiple of $${\mathcal B}(\pi)$$. Moreover, the map $$\pi\mapsto {\mathcal B}(\pi)$$ is injective. To this he adds a refinement: Conjecture: The multiplicity of $$\pi$$ in $$R_ i$$ is equal to that of $${\mathcal B}(\pi)$$ in $${\mathcal A}^ 0(\pi).$$
The refinement would follow from a local multiplicity one supplement to Howe’s local duality conjecture at all places. The author verifies the local multiplicity one conjecture in certain cases. Besides the orthogonal decomposition of $$L^ 2_{cusp}$$ of O(Q) there is an analogous one for $$L^ 2_{cusp}$$ of $$Sp_ n$$. For $$n=2$$, the Saito- Kurakawa subspace defined by Piatetski-Shapiro is properly contained in one piece of the orthogonal decomposition.

##### MSC:
 2.2e+56 Representations of Lie and linear algebraic groups over global fields and adèle rings 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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