On Newforms for Split Special Odd Orthogonal Groups

Abstract

The theory of local newforms has been studied for the group of (PGL_n) and recently (PGSp_4) and some other groups of small ranks. In this dissertation, we develop a newform theory for generic supercuspidal representations of (SO_{2n+1}) over non-Archimedean local fields with odd characteristic by defining a family of open compact subgroup (K(p^m)), (m \geq 0) (up to conjugacy) which are analogous to the groups (\Gamma(p^m)) in the classical theory of modular forms. We give lower bounds on the dimension of the fixed subspaces of (K(p^m)) in terms of the conductor of the generic representation, and give a conjectural description of the space of old forms. These results generalize the known cases for n = 1,2 by Casselman [4] and Roberts and Schmidt [23].