Invariant vectors and level raising operators in representations of the p-adic group GL(3)

Abstract

Each irreducibel, admissible representation (&pi,V) of GL(n) over a non-archimedean local field F has associated fixed vector spaces VK(m), for K(m) a compact-open subgroup of GL(n). It is known that there exists some non-negative integer m such that dim(VK(m))=1 and, if m'<m, then dim(VK(m'))=0. Such an m is called the conductor of &pi which is denoted by c(&pi). If the represetation is also generic, an equation is given by Reeder for calculating the dimension of VK(m). In this paper, the dimension of VK(m) is determined when (&pi,V) is a non-generic representation of GL(3).

For m&gec(&pi), an element of VK(m) is consideered to have level m. A non-zero element in VK(c(&pi)) is a local newform, elements of higher level are known as oldforms. Level raising operators are maps from VK(m) to VK(m+1) that lift an element from one level to the next. In this paper, level raising operators are presented for VK(m) associated to representations of GL(3) and the main theorem proves that, when applied to a local newform, these level raising opertors cna be used to obtain a set of basis elements for each level.

In the generic case, the proof uses Whittaker functions, zeta integrals, Hecke operators and Satake parameters. For the non-generic case, it is shown that unramified characters of F play a role and the matrix of each level raising operator is used.