5,208 research outputs found
Spherical Functions on Euclidean Space
We study special functions on euclidean spaces from the viewpoint of
riemannian symmetric spaces. Here the euclidean space where is
the semidirect product of the translation group with a closed
subgroup of the orthogonal group O(n). We give exact parameterizations of
the space of --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 is
transitive on the unit sphere in .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 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 be a complex simple direct limit group, specifically
, or .
Let be a (generalized) flag in . If is
or we suppose further that
is isotropic. Let denote the corresponding flag
manifold; thus where is a parabolic subgroup of . In
a recent paper with Ignatyev and Penkov, we studied real forms of and
properties of their orbits on . Here we concentrate on open
--orbits . When is of hermitian type we work
out the complete --orbit structure of flag manifolds dual to the bounded
symmetric domain for . Then we develop the structure of the corresponding
cycle spaces . 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 as well as the
product structure of the base cycles for flag domains of and
. These revisions had no effect on the results for the
structure of the cycle space
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