Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case)

TADICG, and the second one is to identify all unitarizable classes in G.The first step is called the problem of non-unitary dual, and the second one is called the unitarizability problem.The first problem has been studied much more then the second one.It has been completely solved for the case of archimedean F, by Langlands classification.Langlands classification is done also for the nonarchimedean case, but here remains to classify the Langlands parameters.The second problem is carried out in the case of non-archimedean F only for the groups SL (2) and closely related group GL (2) (possible reference is [10]) ( 1 ).In the case of archimedean F, despite the complete knowledge of the non-unitary dual Q the second problem was carried out completely only for a few groups of lower ranks.For a survey of this case one may consult the paper [17] of A. W. Knapp and B. Speh and therefore we are not going into further details (see also [31]).We shall give one more remark about the problem of unitary dual for groups SL (n, C), which are closely related to the groups GL (n, C).In 1950, I. M. Gelfand and M. A. Neumark constructed a family of irreducible unitary representations of SL (n, C) for which they presumed they exhausted the unitary dual [13].In 1967 E. M. Stein showed that the constructed family of representations was not complete, by constructing a new complementary series [25].G. Olshanskii generalized in [20] this result.He constructed some complementary series for GL (n) over division algebras (archimedean and non-archimedean).In [2] J. N. Bernstein constructed a much wider family of complementary series for GL (n) over a non-archimedean field.This complementary series will be discussed later.J. N. Bernstein paper contains some very important general results about unitarizability in the case of GL (n) over non-archimedean fields.In this paper we give a solution of the unitarizability problem for the groups GL (n) over a local non-archimedean field F.More precisely, Zeievinsky parameters and Langlands parameters of all unitarizable classes in GL (n, F) ^ are determined.Moreover, an explicit formula connecting Zeievinsky and Langlands parameters of GL(n, F) is proved.We prove also the Bernstein conjecture on complementary series from [2].The results and techniques we need in this paper on non-unitary dual of GL (n) over non-archimedean field, which are characteristic for this case, belong mainly to I. M. Gelfand, D. A. Kazhdan, J. N. Bernstein and A. V. Zeievinsky ([4], [5], [12], [33]).Concerning the facts about the unitary representations required for this paper, in the first stage of development of ideas of this paper, the results of D. Milicic from [18] and also results of [27] (obtained using [18]) had an important role.In the second stage, the results of J. N. Bernstein in [2] played an important role.F. Rodier pointed out to me possibility of studying the unitary duals of GL (n) over non-archimedean fields using these two groups of ideas.Now we shall describe the main results of this paper.( 1 ) Added in proof: the authors