On the representation of the determinant of Harish-Chandra's \(C\)-function of \(\text{SL}(n,\mathbb{R})\) (Q1207875)

Let \(G\) be a connected semisimple Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). Let \(P\) be a minimal parabolic subgroup of \(G\) and \(\overline{P}\) its opposite. In his book [``Analytic Theory of the Harish-Chandra \(C\)-function'' (Lect. Notes Math. 429, 1974; Zbl 0342.33026)] \textit{L. Cohn} has proved that the determinant of the Harish- Chandra \(C\)-function \(C_{\overline{P}\mid P}(1:\nu)\) can be expressed as a quotient of products of \(\Gamma\)-functions involving some undetermined rational numbers. He also formulated a conjecture related to these numbers. The author establishes an explicit formula for the \(C\)-function in the case \(G = \text{SL}(n,\mathbb{R})\) and establishes Cohn's conjecture in this case. Also, as a corollary of this result he reproves a necessary and sufficient condition for the irreducibility of the corresponding principal series representations.