The product of exponentials in the definition of canonical kernel function. (Q1858318)

In this paper the authors obtain an explicit Campbell-Baker-Hausdorff series for the product of exponentials in the definition of Satake's canonical kernel function of a bounded symmetric domain \(\mathcal{D}\). This product of exponentials is of the form \(\exp -\bar w\exp z\), with \(w\) and \(z\) in \(\mathfrak{p}^+\), where \(\mathfrak{g}_c =\mathfrak{k}_c\oplus\mathfrak{p}_c =\mathfrak{k}_c\oplus\mathfrak{p}^+ \oplus\mathfrak{p}^-\) is the Cartan decomposition of the complexification of the (Hermitian) Lie algebra corresponding to \(\mathcal{D}\), and \(\mathcal{D}\subset\mathfrak{p}^+\) is in its Harish-Chandra realization. To perform the calculation, the authors first note that \(w\) and \(z\) can be assumed to lie either in the same root space with respect to a compact Cartan subalgebra of \(\mathfrak{g}_c\), or to lie in root spaces corresponding to strongly orthogonal roots. Thus the calculation reduces to the \(\text{SL}(2)\)-case, which is handled directly.