… It was clear from [12] that a direct argument could be given once a combinatorial identity
was proved. We remarked that Tadic’s formula for the characters of Speh representations is
given by the determinant of a matrix, and that the combinatorial identity needed can be
reinterpreted as a curious identity of determinants. We proved this identity, learning later that it
was already known under the name of Lewis Caroll identity [1], [6]. We finish this introduction
by remarking that in the case of GL(m, C), the argument can be reversed, and that it gives a …

Let F be a non archimedian local field of characteristic zero, let A be a central division
algebra over F and let Gm= GL (m, A). The unitary dual of G was described in [14]([13] in the
case A= F), and the missing part of the proof (conjectures U0 and U1 in [14]) were obtained
respectively in [11] and [2]. Speh representations are building blocks for the unitary dual of
Gm, in the sense that an irreducible unitary representation of Gm is always fully parabolically
induced from a tensor product of Speh representations or complementary series starting …