Systems of Differential Operators and Generalized Verma Modules

In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems.Comment: Part of Section 2 of this article overlaps with that of Section 2 of previous paper arXiv:1209.551

Classification of differential symmetry breaking operators for differential forms

We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of principal series representations of the Lie group $O(n+1,1)$. 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

Conformal symmetry breaking operators for differential forms on spheres

We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. 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]

Conformally Invariant Systems of Differential Operators Associated to Two-step Nilpotent Maximal Parabolics of Non-heisenberg Type

The main work of this thesis concerns systems of differential operators that are equivariant under an action of a Lie algebra. We call such systems conformally invariant. The main goal of this thesis is to construct such systems of operators for a homogeneous manifold G_0/Q_0 with G_0 a Lie group and Q_0 a maximal two-step nilpotent parabolic subgroup. We use the invariant theory of a prehomogeneous vector space to build such systems. We determined the complex parameters for the line bundles L_{-s} on which our systems of differential operators are conformally invariant. The systems that we construct yield explicit homomorphisms between appropriate generalized Verma modules. We also determine whether or not these homomorphisms are standard.Department of Mathematic