The product formula for the spherical functions on symmetric spaces in the complex case. (Q701264)

A product formula for spherical functions on symmetric spaces in the complex case is proved, and the properties of the corresponding integral kernels of the formula are studied in this note. Let \(G\) be a semisimple noncompact Lie group with finite center, \(K\) a maximal compact subgroup of \(G\), and \(X=G/K\) the corresponding Riemannian symmetric space of noncompact type. Suppose that \(G\) has the Cartan decomposition \(G=KAK\) and Iwasawa decomposition \(G=KAN\). The main results are: 1. The product formula for complex Lie groups. Theorem. Suppose that \(G\) is a complex Lie group, then the following product formula holds \[ \varphi_\lambda(e^X) \varphi_\lambda(e^Y)= \int_\alpha \varphi_\lambda (e^H)k (H,X,Y) \delta(H)\,dH \] with kernel \[ k(H,X,Y)=\frac{1} {\delta^{1/2}(H) \delta^{ 1/2} (Y)}\frac{1}{| W |}\sum_{w\in W}\varepsilon(w)K(X,w\cdot H-Y), \] where the expressions of \(\delta^{1/2}\), \(\varepsilon,K,\dots\) are as usual; the reader can refer to the paper. Moreover, four corollaries are obtained. 2. The explicit product formula for complex groups. Theorem. Suppose that \(G\) is a complex Lie group, then the following product formula holds \[ \varphi_\lambda (e^X) \varphi_\lambda (e^Y)=\int_\alpha \varphi_\lambda (e^H)k(H,X,Y)\delta(H)\,dH \] with kernel \[ k(H,X,Y)= \frac{\pi(\rho)}{| W|}\frac{1} {\delta^{1/2} (H) \delta^{1/2} (X)\delta^{1/2}(Y)} \sum_{v,w\in W}\varepsilon(v) \varepsilon(w) T(vX + wY-H). \] Also two corollaries are obtained. 3. The support in the case of \(G=SL(3,C)\). In this part, necessary and sufficient conditions for \(k(H, X,Y)\neq 0\) are proved. The proofs of three lemmas and one proposition are technical. 4. The function \(T\) in the case of \(SL(n,C)\). Theorem. The function \(T\) for \(SL(n,C)\) is given by \[ T(y_1 \alpha_1+\cdots +y_{n-1} \alpha_{n-1})=1_{\{0\leq y_1\}} \delta_0(dy_2,\dots, dy_{n-1})* 1_{\{y_1\leq y_2\}}\delta_0 (dy_3,\dots,dy_{n-1})* \] \[ *\cdots*1_{\{y_1\leq \cdots\leq y_{n-2}\}} \delta_0(dy_{n-1})* 1_{\{y_1 \leq \cdots \leq y_{n-1}\}}. \]