On Newforms for Split Special Odd Orthogonal Groups

The theory of local newforms has been studied for the group of PGLn and recently PGSp4 and some other groups of small ranks. In this dissertation, we develop a newform theory for generic supercuspidal representations of SO2n+1 over non-Archimedean local fields with odd characteristic by defining a family of open compact subgroup K(p), m ≥ 0 (up to conjugacy) which are analogous to the groups Γ0(p ) in the classical theory of modular forms. We give lower bounds on the dimension of the fixed subspaces of K(p) 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].