Analysis in matrix space and Speh's representations

The irreducible unitary representations of GL(n, ~) have been classified by D. Vogan in [V2]. In his work, four families of representations play a special role. They are (i) the one dimensional unitary representations and (ii) their complementary series; (iii) Speh's representations and (iv) their complementary series. These representations serve as "building blocks" in the sense that every irreducible unitary representation is obtained, via unitary parabolic induction, from a tensor product of these representations. Vogan's techniques and results are completely algebraic in nature; indeed they represent a considerable triumph for the algebraic method. What Vogan actually classifies are the irreducible, admissible (g, K)-modules which possess an invariant, positive-definite Hermitian form. However, by a well known theorem of HarishChandra [HI , every such module is the space of K-finite vectors of a unique irreducible unitary representation. For many purposes, the (g, K)-module is an adequate substitute for the unitary representation itself. However, for certain other problems, especially those of an analytic nature, it is important to have an explicit construction of the unitary representation together with its Hilbert space. For representations of type (i), this is a trivial problem. For those of type (ii), there is an analytic construction of the type carried out in [St] for the case of GL(n, C); similar results are true for any local field. An alternative, uniform approach to this problem is described in [$2]. This brings us to Speh's representations which are the subject of this paper. They occur for each GL(n, R) with n even; and for each such group there is, up to tensoring by a one-dimensional unitary representation, exactly one Speh representation for each positive integer m. The existence of these representations was conjectured as early as 1956 by Gelfand and Graev in [GG]; and they are mentioned again in [B]. However, all