Weight distribution of Iwasawa projection (Q2399774)

Let \(G\) be a connected semisimple real Lie group with finite center. Let \(G=KAN\) be the Iwasawa decomposition. Let \(\mathfrak{a}\) be the Lie algebra of \(A\), and let \(H: G\to \mathfrak{a}\) be the map defined by \(g=k \exp[H(g)] n\). Fix a positive open Weyl chamber \(\mathfrak{a}^+\) of \(\mathfrak{a}\) and let \(\overline{\mathfrak{a}^+}\) be its closure. A result of \textit{B. Kostant} [Ann. Sci. Éc. Norm. Supér. (4) 6, 413--455 (1973; Zbl 0293.22019)] says that the map \[ F_2^X: K\to \mathfrak{a} \] defined by \(k\mapsto H(\exp(X)k)\) has image \(\mathrm{CH}(WX)\) -- the convex hull of the Weyl group orbit of \(X\). The main result of the paper under review is the interesting observation that almost all the weight of \(F_2^X \rho\) is concentrated near the point \(X\). Here \(\rho\) is the unique Haar measure on \(K\) such that \(\rho(K)=1\).