On the lattice model of the Weil representation and the Howe duality conjecture (Q2927749)

The Howe conjecture for a dual reductive pair over a non-Archimedean local field \(F\) of characteristic different from 2 has been proved when \(F\) has residual characteristic different from 2. For the particular case of an unramified dual pair a proof is given in [\textit{C. Moeglin} et al., Howe correspondences on a \(p\)-adic field. (French) Lecture Notes in Mathematics, 1291. Subseries: Mathematisches Institut der Universität und Max-Planck-Institut für Mathematik, Bonn, Vol. 11 (1987; Zbl 0642.22002)]. This proof makes use of the lattice model of the Weil representation (cf. loc. cit., p.42). The present article is an attempt to prove the Howe conjecture for an unramified dual pair when F has residual characteristic 2, following [loc. cit., Chap. 5]. The main theorem gives a partial result for the pair \((\mathrm{O}(2n)\), \(\mathrm{Sp}(2n))\) with unramified \(\mathrm{O}(2n)\). The result is partial in so far as there is a condition on the representation of \(\mathrm{O}(2n)\) (existence of vectors invariant under certain congruence subgroups).NEWLINENEWLINEThe references to Weil in section 2 are erroneous. This has no influence on the rest of the paper.