A comparison of the automorphic representations of GL(3) and its twisted forms

Abstract

0* Introduction* The theorems are deduced as consequences of the Arthur-Selberg trace formula. The proofs have been patterned after those used in [17] in comparing the representation theory of the groups GL (2) over two distinct fields. The two main theorems of this thesis are as follows. Let F be a nonarchimedean local field of characteristic zero, let G — GL (3, F), and let G be the group of invertible elements in a central division algebra of rank 3 over F. Define admissible irreducible representations π of G and π' of G to be related, and write π ~ π', if Θπ(g) = Θπ,(g') for all pairs of elements g e G and g' e G' which have the same irreducible characteristic polynomial, where Θπ (resp., Θπ,) is the character of π (resp., π ).