149 research outputs found
Projective Pseudodifferential Analysis and Harmonic Analysis
We consider pseudodifferential operators on functions on which
commute with the Euler operator, and can thus be restricted to spaces of
functions homogeneous of some given degree. Their symbols can be regarded as
functions on a reduced phase space, isomorphic to the homogeneous space
, and the resulting calculus is a
pseudodifferential analysis of operators acting on spaces of appropriate
sections of line bundles over the projective space : these spaces are
the representation spaces of the maximal degenerate series
of . This new approach to the quantization of
, already considered by other authors, has several advantages: as an
example, it makes it possible to give a very explicit version of the continuous
part from the decomposition of under the quasiregular action of
. We also consider interesting special symbols, which arise from the
consideration of the resolvents of certain infinitesimal operators of the
representation
Conformal symmetry breaking operators for differential forms on spheres
We give a complete classification of conformally covariant differential
operators between the spaces of -forms on the sphere and -forms on
the totally geodesic hypersphere . Moreover, we find explicit
formul{\ae} for these new matrix-valued operators in the flat coordinates in
terms of basic operators in differential geometry and classical orthogonal
polynomials. We also establish matrix-valued factorization identities among all
possible combinations of conformally covariant differential operators. The main
machinery of the proof is the "F-method" based on the "algebraic Fourier
transform of Verma modules" (Kobayashi-Pevzner [Selecta Math. 2016]) and its
extension to matrix-valued case developed here. A short summary of the main
results was announced in [C. R. Acad. Sci. Paris, 2016]
Classification of differential symmetry breaking operators for differential forms
We give a complete classification of conformally covariant differential
operators between the spaces of differential -forms on the sphere and
-forms on the totally geodesic hypersphere by analyzing the
restriction of principal series representations of the Lie group .
Further, we provide explicit formul\ae{} for these matrix-valued operators in
the flat coordinates and find factorization identities for them.Comment: This note was published in C. R. Acad. Sci. Paris, Ser. I, (2016),
http://dx.doi.org/10.1016/j.crma.2016.04.01
A generating operator for Rankin-Cohen brackets
Motivated by the classical ideas of generating functions for orthogonal
polynomials, we initiate a new line of investigation on "generating operators"
for a family of differential operators between two manifolds. We prove a novel
formula of the generating operators for the Rankin-Cohen brackets by using
higher-dimensional residue calculus. A validity of the formula is established
by a different method based on infinite-dimensional representation theory as
well
Inversion of Rankin-Cohen operators via Holographic Transform
The analysis of branching problems for restriction of representations brings
the concept of symmetry breaking transform and holographic transform. Symmetry
breaking operators decrease the number of variables in geometric models,
whereas holographic operators increase it. Various expansions in classical
analysis can be interpreted as particular occurrences of these transforms. From
this perspective, we investigate two remarkable families of differential
operators: the Rankin-Cohen operators and the holomorphic Juhl conformally
covariant operators. Then we establish for the corresponding symmetry breaking
transforms the Parseval-Plancherel type theorems and find explicit inversion
formul{\ae} with integral expression of holographic operators.
The proof uses the F-method which provides a duality between symmetry
breaking operators in the holomorphic model and holographic operators in the
-model, leading us to deep links between special orthogonal polynomials
and branching laws for infinite-dimensional representations of real reductive
Lie groups.Comment: To appear in Annales de l'Institut Fourie
Strategies of Loop Recombination in Ciliates
Gene assembly in ciliates is an extremely involved DNA transformation
process, which transforms a nucleus, the micronucleus, to another functionally
different nucleus, the macronucleus. In this paper we characterize which loop
recombination operations (one of the three types of molecular operations that
accomplish gene assembly) can possibly be applied in the transformation of a
given gene from its micronuclear form to its macronuclear form. We also
characterize in which order these loop recombination operations are applicable.
This is done in the abstract and more general setting of so-called legal
strings.Comment: 22 pages, 14 figure
Equivariant resolutions over Veronese rings
Working in a polynomial ring where
is an arbitrary commutative ring with , we consider the
Veronese subalgebras , as well as natural -submodules
inside . We develop and use characteristic-free theory
of Schur functors associated to ribbon skew diagrams as a tool to construct
simple -equivariant minimal free -resolutions for the
quotient ring and for these modules . These also lead to
elegant descriptions of for all and
for any pair of these modules .Comment: 37 pages. Further minor edits. Version to appear in J. London Math.
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