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Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups
In this work we investigate solvable and nilpotent Lie groups with special
metrics. The metrics of interest are left-invariant Einstein and algebraic
Ricci soliton metrics. Our main result shows that the existence of a such a
metric is intrinsic to the underlying Lie algebra. More precisely, we show how
one may determine the existence of such a metric by analyzing algebraic
properties of the Lie algebra in question and infinitesimal deformations of any
initial metric.
Our second main result concerns the isometry groups of such distinguished
metrics. Among the completely solvable unimodular Lie groups (this includes
nilpotent groups), if the Lie group admits such a metric, we show that the
isometry group of this special metric is maximal among all isometry groups of
left-invariant metrics. We finish with a similar result for locally
left-invariant metrics on compact nilmanifolds.Comment: 28 page
Shintani functions, real spherical manifolds, and symmetry breaking operators
For a pair of reductive groups , we prove a geometric criterion
for the space of Shintani functions to be finite-dimensional
in the Archimedean case.
This criterion leads us to a complete classification of the symmetric pairs
having finite-dimensional Shintani spaces.
A geometric criterion for uniform boundedness of is
also obtained.
Furthermore, we prove that symmetry breaking operators of the restriction of
smooth admissible representations yield Shintani functions of moderate growth,
of which the dimension is determined for .Comment: to appear in Progress in Mathematics, Birkhause
Invariant Yang-Mills connections over Non-Reductive Pseudo-Riemannian Homogeneous Spaces
We study invariant gauge fields over the 4-dimensional non-reductive
pseudo-Riemannian homogeneous spaces G/K recently classified by Fels & Renner
(2006). Given H compact semi-simple, classification results are obtained for
principal H-bundles over G/K admitting: (1) a G-action (by bundle
automorphisms) projecting to left multiplication on the base, and (2) at least
one G-invariant connection. There are two cases which admit nontrivial examples
of such bundles and all G-invariant connections on these bundles are
Yang-Mills. The validity of the principle of symmetric criticality (PSC) is
investigated in the context of the bundle of connections and is shown to fail
for all but one of the Fels-Renner cases. This failure arises from degeneracy
of the scalar product on pseudo-tensorial forms restricted to the space of
symmetric variations of an invariant connection. In the exceptional case where
PSC is valid, there is a unique G-invariant connection which is moreover
universal, i.e. it is the solution of the Euler-Lagrange equations associated
to any G-invariant Lagrangian on the bundle of connections. This solution is a
canonical connection associated with a weaker notion of reductivity which we
introduce.Comment: 34 pages; minor typos corrected; to appear in Transactions of the AM
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