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].