Spherical generalized functions induced on \(G_{\mathbb{C}}/G_{\mathbb{R}}\) and applications
Let \(G\) be a complex semi-simple Lie group, and \(H\) a connected subgroup which is a real form of \(G\). By induction one constructs a spherical generalized function \(p-\text{ind}^ G_ L\theta\) on the symmetric space \(G/H\) from a spherical generalized function \(h\) on the smaller symmetric space \(L/L \cap H\), where \(L\) is the centralizer of the split part of a Cartan subalgebra of the Lie algebra \(\mathfrak h\) of \(H\). The spherical generalized function \(p -\text{ind}^ G_ L\theta\) is an analytic function on the set of regular elements, and one gives an explicit formula for it. When \(\theta\) is the coefficient of a representation of \(L\), then \(p - \text{ind}^ G_ L\theta\) is a coefficient of a representation induced from a parabolic subgroup. The example of the symmetric space \(GL(2,\mathbb{C})/GL(2,\mathbb{R})\) is studied in detail. As an application one proves that the inversion formula for \(GL(n,\mathbb{C})/GL(n,\mathbb{R})\) due to Sano is actually a Plancherel formula, by proving that the spherical generalized functions involved in Sano's inversion formula are positive definite and extremal.
scientific article; zbMATH DE number 35783
scientific article