A convexity property for the \(\text{SO}(2,\mathbb{C})\)-double coset decomposition of \(\text{SL}(2,\mathbb{C})\) and applications to spherical functions (Q1762696)

Let \(X= G/K\) be a Riemannian symmetric space of the non-compact type. Inside its natural complexification \(X_{\mathbb{C}}= G_{\mathbb{C}}/K_{\mathbb{C}}\) there is a certain \(G\)-stable domain \(\Xi\supset X\), called the complex crown of \(X\), which plays an important role in the harmonic analysis of \(G\) as it is a natural domain for solutions of invariant differential equations. It is known that \(\Xi\) coincides with the natural domain of the Iwasawa decomposition but the same assertion fails for the polar decomposition. In this paper the authors propose a convexity assertion which describes the polar projection of a given element of \(\Xi\). They prove this assertion in the simplest case \(G= \text{SL}^2(\mathbb{R})\) by explicit calculations.