Action of intertwining operators on pseudospherical \(K\)-types (Q2520789)

The paper under review is devoted to the study of intertwining operators between certain principal series representations of double covers of real reductive groups. The first half of the article provides a detailed description of the structure of such groups (maximal compact and Borel subgroups) as Chevalley groups, following [\textit{R. Steinberg}, Lectures on Chevalley groups. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1361.20003)] and definitions of pseudospherical representations, following [\textit{J. Adams} et al., J. Am. Math. Soc. 20, No. 3, 701--751 (2007; Zbl 1114.22009)]. In that context, standard intertwining integrals for pseudospherical principal series can be defined in a canonical way and the main result of the paper is an explicit calculation of the scalar by which these intertwiners act on pseudospherical \(K\)-types, appearing with multiplicity one. The result is formulated in terms of certain \(c\)-functions and proved by reduction to the \(\mathrm{SL}_2\) case. It mirrors similar results obtained for \(p\)-adic groups in [\textit{H.-Y. Loke} and \textit{G. Savin}, Trans. Am. Math. Soc. 362, No. 9, 4901--4920 (2010; Zbl 1214.22003)].