The diagonal distribution for the invariant measure of a unitary type symmetric space (Q882632)

From the text: Suppose that \(K\) is a simply connected compact Lie group, and let \(G\) denote the complexification. Given a triangular decomposition \(\mathfrak g = \mathfrak n^- \oplus \mathfrak h \oplus n^+\), a generic \(g \in K\) has a unique `LDU decomposition', \(g=lmau\), where \(l \in N^-\) (lower triangular), \(u \in N^+\) (upper triangular), and \(ma \in H\), with \(m \in H \cap K\) (unitary) and \(a \in \exp (\mathfrak h_{\mathbb R} = \mathfrak h \cap i\mathfrak k)\) (positive). A formula of Harish-Chandra (closely related to the Weyl dimension formula) asserts that, for \(\lambda \in \mathfrak h_{\mathbb R}^*\), \[ \int_K a^{-i\lambda} = c(2\delta - i\lambda) = \prod_{\alpha > 0} \frac{\langle 2\delta, \alpha \rangle}{\langle 2\delta - i\lambda, \alpha \rangle}, \] where the integral is with respect to normalized Haar measure, the product is over positive complex roots, \(\langle \cdot, \cdot \rangle\) denotes the Killing form (on the dual), and \(2\delta\) is the sum of the positive complex roots. The purpose of this paper is to present a generalization of this formula, and some of the related geometry, in which \(K\) is replaced by a compact symmetric space. Let \(\Theta\) denote an involution for a simply connected compact Lie group \(U\), let \(K\) denote the fixed point set, and let \(\mu\) denote the \(U\)-invariant probability measure on \(U/K\). Consider the geodesic embedding \(\varphi :U/K\to U:u\mapsto uu^{ -\Theta}\) of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for \(\varphi_{*}\mu,\) relative to a compatible triangular decomposition of \(G\), the complexification of \(U\). This boils down to a Duistermaat-Heckman exact stationary phase calculation, involving a Poisson structure on the dual symmetric space \(G_{0}/K\) discovered by Evens and Lu.