Publications (5)0 Total impact
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Article: Rankin-Cohen Operators for Symmetric Pairs
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ABSTRACT: Rankin-Cohen bidifferential operators are the projectors onto irreducible summands in the decomposition of the tensor product of two particular representations of SL(2,R). We consider the general problem to find explicit formulas for such projectors in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose we develop a new method (F-method) based on an algebraic Fourier transform for generalized Verma modules, which enables us to characterize those projectors by means of certain systems of partial differential equations of second order. We discover explicit formulas for new equivariant holomorphic differential operators in the six different complex geometries arising from real symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of Jacobi polynomials appear in these operators including the classical Rankin-Cohen brackets as a special case.01/2013; -
Article: Generalized Bernstein--Reznikov integrals
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ABSTRACT: We find a closed formula for the triple integral on spheres in $\mathbb{R}^{2n}\times\mathbb{R}^{2n}\times\mathbb{R}^{2n}$ whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein--Reznikov integral formula in the $n=1$ case. Our method also applies for linear and conformal structures.06/2009; -
Article: Geometric analysis on small unitary representations of GL(N,R).
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ABSTRACT: The most degenerate unitary principal series representations $\pi_{i\lambda,\delta}$ ($\lambda\in \R,\,\delta\in\mathbb Z/2\mathbb Z$) of $G = GL(N,\R)$ attain the minimum of the Gelfand--Kirillov dimension among all irreducible unitary representations of $G$. This article gives an explicit formula of the irreducible decomposition of the restriction $\pi_{i\lambda,\delta}|_H$ (\textit{branching law}) with respect to all symmetric pairs $(G,H)$. For $N=2n$ with $n \ge 2$, the restriction $\pi_{i\lambda,\delta}|_H$ remains irreducible for $H=Sp(n,\R)$ if $\lambda\ne0$ and splits into two irreducible representations if $\lambda=0$. The branching law of the restriction $\pi_{i\lambda,\delta}|_H$ is purely discrete for $H = GL(n,\C)$, consists only of continuous spectrum for $H = GL(p,\R) \times GL(q,\R)$ $(p+q=N)$, and contains both discrete and continuous spectra for $H=O(p,q)$ $(p>q\ge1)$. Our emphasis is laid on geometric analysis, which arises from the restriction of `small representations' to various subgroups. -
Article: Projective pseudodifferential analysis and harmonic analysis
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ABSTRACT: We consider pseudodifferential operators on functions on Rn+1 which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. The symbols of such restrictions can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space Gn/Hn=SL(n+1,R)/GL(n,R), and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space Pn(R): these spaces are the representation spaces of the maximal degenerate series (πiλ,ε) of Gn. This new approach to the quantization of Gn/Hn, 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 L2(Gn/Hn) under the quasiregular action of Gn. We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation πiλ,ε.Journal of Functional Analysis. -
Article: Composition formulas in the Weyl calculus
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ABSTRACT: In pseudodifferential analysis, the usual composition formula, which has asymptotic value, extends that valid for differential operators. The one developed here is based instead on the decomposition of symbols (functions in Rn × Rn) as integral superpositions of homogeneous ones, of degrees lying on the complex line with real part −n. It extends the one known in the one-dimensional case in connection with automorphic pseudodifferential analysis.
