Transfer of unitary representations (Q2484145)

In the present paper, the authors explain and give applications of the notion of transfer between real forms of a semi-simple Lie group \(G^{\mathbb C}\) over \(\mathbb C\), which first appeared in [\textit{N. R. Wallach}, Contemp. Math. 177, 181--216 (1994; Zbl 0833.22021)]. The main idea is that some representation of a real form of \(G^{\mathbb C}\) can be better understood in the context of another real form. This happens for instance for minimal representations (that are annihilated by the Joseph ideal) of a split group over \({\mathbb R}\). If the complexification admits a Hermitian symmetric real form, then minimal representations of such real form are part of the analytic continuation of the holomorphic discrete series (as in [\textit{T. Enright, R. Howe, N. R. Wallach}, Proc. Conf., Park City/Utah 1982, Prog. Math. 40, 97-143 (1983; Zbl 0535.22012)]). One can do a similar transfer also if there is no Hermitian symmetric real form, since there is always a quaternionic real form. Moreover the authors present examples which give more evidence of a possible deep connection between the notion of transfer and Howe's theory of dual pairs.