Projective pseudodifferential analysis and harmonic analysis

Abstract

We consider pseudodifferential operators on functions on R n+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 G n /H n = 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 P n (R): these spaces are the representation spaces of the maximal degenerate series (π iλ,ε ) of G n . This new approach to the quantization of G n /H n , 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 L 2 (G n /H n ) under the quasiregular action of G n . We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation π iλ,ε .

Invariant trilinear forms for spherical degenerate principal series of complex symplectic groups

“…Similarly, closed formulas were found in the case of real symplectic groups in [CKØP11], where the methods used here find their origin. See also the composition formulas obtained in pseudo-differential calculus related to the quantization of certain homogeneous spaces [PU07,Unt03].…”

On the degenerate principal series of complex symplectic groups

“…The equivalences between the induced, non-compact and non-standard models of π µ,δ are summed up in the following diagram: (15) α µ,δ : V µ,δ ∼ restrict.…”

Covariant quantization: spectral analysis versus deformation theory

“…The case of SL(2, R) was studied in detail by R. Howe and E.-C. Tan [68] Ch.II §2 and by A. El Gradechi [50]. [131] Let us finish this survey by pointing out two interesting phenomena that occur in the context of covariant quantization of the rank-one para-Hermitian symmetric space G n /H n = SL(n + 1, R)/GL(n, R), which are an argument in favour of the spectral approach to quantization.…”

“…In general, in the references just given, the problem was dealt with by the use of the H n -spherical distribution method. Our approach in the paper [131] under discussion was based on the square-root method.…”