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Langlands’ functoriality and the Weil representation. (English) Zbl 0532.22016

If \((G_ 1,G_ 2)\) is a reductive dual pair in some symplectic group over a local field, the Weil representation defines a corresondence between representations of \(\tilde G_ 1\) and representations of \(\tilde G_ 2\) (\(\tilde G_ i\) being the inverse image of \(G_ i\) in the two- fold cover of the symplectic group). Take \(G_ 1\) to be the orthogonal group of a quadratic form Q and \(G_ 2=Sp_ n\). In the present paper a surjective homomorphism from the Hecke algebra of \(G_ 1\) to that of \(G_ 2\) which is compatible with the local Weil representation is explicitly constructed (if index \(Q\geq n)\). In the case when Q is split it is shown that this homomorphism comes from an L-homomorphism between the L-groups of \(G_ 1\) and \(G_ 2\), so that we my speak of a ”lifting”, in the sense of Langlands’ functoriality, of local spherical representations. As for the global case, an example is given where the corresponence is not functorial. In the author’s paper [”On the Howe duality conjecture” (to appear)] a correction to the proof of Proposition 2.2 of the present paper is given.
Reviewer: J.G.M.Mars

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
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