Supercuspidal representations and the theta correspondence (Q1208093)

Howe's local theta correspondence is carefully re-examined for the dual pairs \(((SL_ 2(F), O(F))\), where \(F\) is a nonarchimedean local field of odd residual characteristic, and \(O(F)\) is the orthogonal group of a quadratic form in one or two variables; in particular, the representations of \(O(F)\) which occur in this correspondence are described, and the corresponding representations of \(SL_ 2(F)\) (if super-cuspidal) are determined. As the author himself emphasizes, the correspondences studied in this paper have been studied already elsewhere [cf. the reviewer's survey paper in Proc. Symp. Pure Math. 33, 1, 287-296 (1979; Zbl 0425.22024)]; therefore, the thrust of this paper is not the results obtained, but rather the techniques used. Indeed, the two main methods used are the lattice model for the oscillator (or ``Weil'') representation, and the parametrization of supercuspidal representations via induction from compact open subgroups.