A simple construction of Stein’s complementary series representations

We given an elementary construction of Stein's complementary series for GL(2n) over an arbitrary local field F , and determine their restrictions to the "mirabolic" subgroup P2n _ GL(2n 1, F) K F2nI . Taken together with the results in [S], this allows one to calculate the adduced representation Az for an arbitrary irreducible, unitary representation 7r of GL(n, C) . The purpose of this paper is (1) to give an elementary construction of Stein's complementary series representations [St] for GL(2n , F), and (2) to explicitly identify their restriction to the "mirabolic" subgroup P2n GL(2n 1, F) x 2n-1 F. The main point is that the Bernstein-Zelevinsky theory of the (highest) derivative [B] gives an answer to (2) for nonarchimedean F (where analogous representations were constructed by Godement); namely that the restriction corresponds to the Stein (Godement) representation for GL(2n 2, F) with the same parameter. For archimedean fields it is not known (at least to the author) how to extend the above arguments to answer (2). However, it is reasonable to expect that the answer should be the same for all fields. Having "guessed" the answer to (2), it seems clear that one should try an inductive argument (on n ) to prove it. Such an argument is given in ?2 and leads to the construction referred to in (1) above. The interesting feature of this argument is that it uses almost nothing from [B] or [St], and reduces the problem to the representation theory of GL(2, F) where the corresponding facts are well known (and easy to prove). 1. STATEMENT OF THE MAIN THEOREM 1.1. Let F be a local field. We begin by describing the degenerate series studied by Stein and Godement. Received by the editors October 17, 1988 and, in revised form, February 20, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 22E46, 22E45, 22E50.