Spherical Functions on Euclidean Space

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space $E^n = G/K$ where $G$ is the semidirect product $R^n \cdot K$ of the translation group with a closed subgroup $K$ of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K)$--spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where $K$ is transitive on the unit sphere in $E^n$.Comment: 10 page

Stepwise Square Integrability for Nilradicals of Parabolic Subgroups and Maximal Amenable Subgroups

In a series of recent papers we extended the notion of square integrability, for representations of nilpotent Lie groups, to that of stepwise square integrability. There we discussed a number of applications based on the fact that nilradicals of minimal parabolic subgroups of real reductive Lie groups are stepwise square integrable. Here, in Part I, we prove stepwise square integrability for nilradicals of arbitrary parabolic subgroups of real reductive Lie groups. This is technically more delicate than the case of minimal parabolics. We further discuss applications to Plancherel formulae and Fourier inversion formulae for maximal exponential solvable subgroups of parabolics and maximal amenable subgroups of real reductive Lie groups. Finally, in Part II, we extend a number of those results to (infinite dimensional) direct limit parabolics. These extensions involve an infinite dimensional version of the Peter-Weyl Theorem, construction of a direct limit Schwartz space, and realization of that Schwartz space as a dense subspace of the corresponding $L^2$ space.Comment: The proof of Theorem 5.9 is improved, several statements are clarified, and a certain number of typographical errors are correcte

Cycle Spaces of Infinite Dimensional Flag Domains

Let $G$ be a complex simple direct limit group, specifically $SL(\infty;\mathbb{C})$, $SO(\infty;\mathbb{C})$ or $Sp(\infty;\mathbb{C})$. Let $\mathcal{F}$ be a (generalized) flag in $\mathbb{C}^\infty$. If $G$ is $SO(\infty;\mathbb{C})$ or $Sp(\infty;\mathbb{C})$ we suppose further that $\mathcal{F}$ is isotropic. Let $\mathcal{Z}$ denote the corresponding flag manifold; thus $\mathcal{Z} = G/Q$ where $Q$ is a parabolic subgroup of $G$. In a recent paper with Ignatyev and Penkov, we studied real forms $G_0$ of $G$ and properties of their orbits on $\mathcal{Z}$. Here we concentrate on open $G_0$--orbits $D \subset \mathcal{Z}$. When $G_0$ is of hermitian type we work out the complete $G_0$--orbit structure of flag manifolds dual to the bounded symmetric domain for $G_0$. Then we develop the structure of the corresponding cycle spaces $\mathcal{M}_D$. Finally we study the real and quaternionic analogs of these theories. All this extends an large body of results from the finite dimensional cases on the structure of hermitian symmetric spaces and related cycle spaces.Comment: This revision improves the exposition and corrects a number of typos. Earlier revisions had clarified the ordering of subspaces in a flag relative to a given ordered basis of the ambient $\mathbb{C}^\infty$ as well as the product structure of the base cycles for flag domains of $Sp(\infty;R)$ and $SO^*(\infty)$. These revisions had no effect on the results for the structure of the cycle space