The theta correspondence for similitudes
Let \(X\) be a nondegenerate finite dimensional quadratic space and \(Y\) a nondegenerate finite dimensional symplectic space over a nonarchimedean local field. The restriction of the oscillator (or Weil-) representation of the metaplectic cover of \(Sp(X\otimes Y)\) to the inverse image of \(O(X)Sp(Y)\) under the covering map gives rise to the local theta correspondence of admissible representations of \(H=O(X)\) and of \(G=Sp(Y)\). This correspondence has been extensively studied because of its importance in the (global) theory of automorphic forms as well as in the local representation theory. In particular, the fundamental conjectures of Roger Howe about (strong) duality and multiplicity preservation have been proven by Howe and Waldspurger with the exception of the case of residual characteristic \(2\). In applications to the theory of automorphic forms and to Shimura varieties, however, one often needs corresponding statements for groups of orthogonal and symplectic similitudes instead of isometries. The attempt to generalize the results to this situation faces the difficulty that the inverse images of the groups of similitudes of \(X\) and \(Y\) in the metaplectic cover of the group of similitudes of \(X\otimes Y\) do not commute. Barthel studied this problem and found a correspondence between certain sets of representations of either group. The present article reconsiders the problem with a different method, studying representations \(\Omega\) of \(G'\times H\) and \(\omega\) of the subgroup of pairs of equal similitude norm, where \(G'\) is either the symplectic similitude group of \(Y\) or its metaplectic cover, depending on the parity of the dimension of \(X\). Both representations have been studied previously. The detailed investigation of the resulting duality properties that is undertaken here has been missing so far and is an extremely useful tool for further investigations in this area. The author first gives a proof of the fact that Howe duality and multiplicity preservation together imply strong Howe duality (which was generally accepted but lacked a rigorous proof so far) and then describes the precise extent to which the Howe conjectures generalize to both situations considered, assuming their validity for the isometry situation. In particular, concerning \(\omega\) it is shown that the analogues of Howe duality and multiplicity preservation hold provided one restricts to representations with multiplicity free restriction to the respective groups of isometries. The results for \(\Omega\) are more complicated.
scientific article; zbMATH DE number 902626
scientific article