On level-raising congruences

In this paper we rewrite a work of Sorensen to include nontrivial types at the infinite places. This work extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^x, where D is a definite quaternion algebra over a totally real field F. We do this for any automorphic representations π of an arbitrary reductive group G over F which is compact at infinity. We do not assume π∞ is trivial. If λ is a finite place of Q, and ω is a place where π_ω is unramified and π _ω (mod λ), then under some mild additional assumptions (we relax requirements on the relation between ω and l which appear in previous works) we prove the existence of a π ≡ π (mod λ) such that π_ω has more parahoric fixed vectors than π_ω. In the case where G_ω has semisimple rank one, we sharpen results of Clozel, Bellaiche and Graftieaux according to which π _ω is Steinberg. To provide applications of the main theorem we consider two examples over F of rank greater than one. In the first example we take G to be a unitary group in three variables and a split place ω. In the second we take G to be an inner form of GSp(2). In both cases, we obtain precise satisfiable conditions on a split prime ω guaranteeing the existence of a π ≡ π (mod λ) such that the component π_ω is generic and Iwahori spherical. For symplectic G, to conclude that π _ω is generic, we use computations of R. Schmidt. In particular, if π is of Saito-Kurokawa type, it is congruent to a π which is not of Saito-Kurokawa type.