The horospherical duality
The author discusses the horospherical duality as a geometrical background of harmonic analysis on semisimple symmetric spaces. \textit{A. Martineau} gave a remarkable construction of an analytic duality on the base of complex projective duality [C. R. Acad. Sci., Paris 255, 2888--2890 (1962; Zbl 0195.36603); Math. Ann. 163, 62--88 (1966; Zbl 0138.38101)]. He defined (complex) linear concave domains (compacts) as unions of hyperplanes in the complex projective space \(\mathbb CP^n\) and their complimentaries as linear convex compacts (domains). The constructions considered in the paper, are in many points similar to these constructions of Martineau. The objects of classical harmonic analysis are complex semisimple Lie groups, but it is natural to work with more general objects -- symmetric Stein manifolds with complex groups of automorphisms. They have the form \(Z = G/H\) where \(G\) is a simply connected complex semisimple Lie group and \(H\) is its involutive subgroup corresponding to a holomorphic involution. Let \(A\) be a maximal Abelian subgroup transversal to \(H\) and \(N\) be a corresponding unipotent subgroup such that \(G^0 = HAN\) is a Zariski open set in \(G\) (complex Iwasawa decomposition). Let \(M\) be the centralizer of \(A\) in \(H\). The homogeneous manifold \(\xi=G/MN\) is called the horospherical manifold. The geometrical duality between horospheres on \(Z\) and pseudospheres on \(\xi\) is called complex horospherical duality. If we are to compare this situation with the Martineau's construction, let us note that in the case we have duality between nonisomorphic manifolds \(Z\), \(\xi\) and the dual families are different. The manifold \(Z\) is a Stein manifold; \(\xi\) is not Stein but it admits an extension up to a Stein space with singularities. In the example of the quadratic cone, we need to add the vertex. It delivers a dramatic difference with respect to duality of compact manifolds (projective spaces). The biggest surprise is that in this situation the geometric duality already delivers an analytic duality for holomorphic functions. The spaces of holomorphic functions \(\mathcal O(Z)\), \(\mathcal O(\xi)\) are isomorphic as \(G\)-modules. Let us emphasize that the manifolds \(Z\), \(\xi\) are not isomorphic neither as \(G\)-homogeneous manifolds nor as complex manifolds.
scientific article; zbMATH DE number 5323340
scientific article