19 research outputs found
Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces
Let be a direct product of two rank-one Riemannian
symmetric spaces of the noncompact type. We show that when at least one of the
two spaces is isomorphic to a real hyperbolic space of odd dimension, the
resolvent of the Laplacian of can be lifted to a holomorphic function on a
Riemann surface which is a branched covering of . In all other
cases, the resolvent of the Laplacian of admits a singular meromorphic
lift. The poles of this function are called the resonances of the Laplacian. We
determine all resonances and show that the corresponding residue operators are
given by convolution with spherical functions parameterized by the resonances.
The ranges of these operators are finite dimensional and explicitly realized as
direct sums of finite-dimensional irreducible spherical representations of the
group of the isometries of
Semisimple orbital integrals on the symplectic space for a real reductive dual pair
We prove a Weyl Harish-Chandra integration formula for the action of a
reductive dual pair on the corresponding symplectic space . As an
intermediate step, we introduce a notion of a Cartan subspace and a notion of
an almost semisimple element in the symplectic space . We prove that the
almost semisimple elements are dense in . Finally, we provide estimates for
the orbital integrals associated with the different Cartan subspaces in
Weyl calculus and dual pairs
We consider a dual pair , in the sense of Howe, with compact
acting on for an appropriate via the Weil
Representation. Let be the preimage of in the metaplectic
group. Given a genuine irreducible unitary representation of
we compute the Weyl symbol of orthogonal projection onto
, the -isotypic component. We apply the result to
obtain an explicit formula for the character of the corresponding irreducible
unitary representation of and to compute of the wave
front set of by elementary means
Symmetry breaking operators for dual pairs with one member compact
We consider a dual pair , in the sense of Howe, with G compact
acting on , for an appropriate , via the Weil
representation . Let be the preimage of G in the
metaplectic group. Given a genuine irreducible unitary representation of
, let be the corresponding irreducible unitary
representation of in the Howe duality. The orthogonal
projection onto , the -isotypic component, is the
essentially unique symmetry breaking operator in
. We study this
operator by computing its Weyl symbol. Our results allow us to recover the
known list of highest weights of irreducible representations of
occurring in Howe's correspondence when the rank of
is strictly bigger than the rank of .
They also allow us to compute the wavefront set of by elementary means.Comment: 94 pages. The present paper subsumes and extends sections 2,3,4 and 6
of the (unpublished) preprint arXiv:1405.243
The wave front set correspondence for dual pairs with one member compact
Let W be a real symplectic space and (G,G') an irreducible dual pair in
Sp(W), in the sense of Howe, with G compact. Let be
the preimage of G in the metaplectic group
. Given an irreducible unitary
representation of that occurs in the restriction
of the Weil representation to , let denote
its character. We prove that, for the embedding of
in the space of tempered distributions on
W given by the Weil representation, the distribution has
an asymptotic limit. This limit is an orbital integral over a nilpotent orbit
. The closure of the image of
in under the moment map is the wave front set of , the
representation of dual to .Comment: arXiv admin note: substantial text overlap with arXiv:1405.243
Transfer of K-types on local theta lifts of characters and unitary lowest weight modules
In this paper we study representations of the indefinite orthogonal group
O(n,m) which are local theta lifts of one dimensional characters or unitary
lowest weight modules of the double covers of the symplectic groups. We apply
the transfer of K-types on these representations of O(n,m), and we study their
effects on the dual pair correspondences. These results provide examples that
the theta lifting is compatible with the transfer of K-types. Finally we will
use these results to study subquotients of some cohomologically induced
modules
Resonances for the Laplacian on Riemannian symmetric spaces: the case of SL(3, R)/SO(3)
We show that the resolvent of the Laplacian on SL(3, R)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of C. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are infinite dimensional irreducible SL(3, R)-representations whose Langlands parameters can also be read off from the corresponding resonances. Alternatively, they are given by the differential equations which determine the image of the Poisson transform associated with the resonance
Resonances for the Laplacian: the cases and (except with odd)
Let be a Riemannian symmetric space of the noncompact type and restricted root system or (except with odd). The analysis of the meromorphic continuation of the resolvent of the Laplacian of is reduced from the analysis of the same problem for a direct product of two isomorphic rank-one Riemannian symmetric spaces of the noncompact type which are not isomorphic to real hyperbolic spaces. We prove that the resolvent of the Laplacian of can be lifted to a meromorphic function on a Riemann surface which is a branched covering of the complex plane. Its poles, that is the resonances of the Laplacian, are explicitly located on this Riemann surface. The residue operators at the resonances have finite rank. Their images are finite direct sums of finite-dimensional irreducible spherical representations of