ELLIPTIC REPRESENTATIONS FOR SP(2N) AND SO(N)

Let G be a connected, reductive p-adic group and let G(e) denote the s et of regular elliptic elements of G. Let pi be an irreducible, temper ed representation of G with character THETA(pi), and write THETA(pi)e for the restriction of THETA(pi) to G(e) . We say pi is elliptic if TH ETA(pi)e is non-zero. In this paper we will characterize the elliptic representations for the p-adic groups Sp(2n) and SO(n). We will show f or Sp(2n) and SO(2n + 1) that every irreducible, tempered representati on is either elliptic or can be irreducibly induced from an elliptic r epresentation. We will then show that this fails for the groups SO(2n) . In this case there are irreducible tempered representations which c annot be irreducibly induced and are not elliptic.