Indag. Mathem., N.S., 1 (4), 417-434 December 17, 1990 Explicit realization of a higher metaplectic representation by Yuval Z. Flicker’ Jepartment of Mathematics, Ohio State University, Columbus OH 43210, USA Communicated by Prof. T.A. Springer at the meeting of May 28, 1990 0. Let F# C be a local field of characteristic # 2, and n an integer 2 1. Denote by p : S, + 1+SL(n + 1, F) the unique non-trivial topological double covering group of SL(n + 1, F). Choose a section 5 : SL(n + 1, F)-S,,, , corresponding to a choice of a two-cocycle p’: S, + I x S,,, 1--t ker p which defines the group law on Sn+,. Put G; =p ‘(t(G,?)), where z is the embedding G,, = GL(n, F)-+ SL(n + 1, F), by gc (“0 deti-I)’ Let(.,.):FXxFX-t{t-I} betheHilbertsymbol.Identifykerpwith{+l}. Put /3(g,g’) =p’(g, g’)(det g, det g’) (g, g’E G,,). Denote by G, the group which is equal to G; as a set, whose product rule is given by s(g)@(g’)[‘= s(gg’)[cp(g, g’). Let A and B be the groups of diagonal and upper-triangular matrices in G’,, and A and B their preimages in G,. The section 2 : C’,+G, is a homomorphism on the group A of upper-triangular unipotent matrices. Put N=?(R). Let 2 be the center of G’,, and 2 the center of G,. Put A2=pP1(ki2), where A2 is the group of squares in A. Then ZA2 is the center of A. Put z =2(z) for z in Z=FX, and a=$(~) for a=diag(ai,...,a,) in A. Note that ’ Partially supported by Seed, Nato and NSF-grants. 417