Toshihisa Kubo

Conformal Symmetry Breaking Operators for Anti-de Sitter Spaces

For a pseudo-Riemannian manifold X and a totally geodesic hypersurface Y, we consider the problem of constructing and classifying all linear differential operators ℰi (X) → ℰj (Y ) between the spaces of differential forms that intertwine multiplier representations of the Lie algebra of conformal vector fields. Extending the recent results in the Riemannian setting by Kobayashi–Kubo–Pevzner [Lecture Notes in Math. 2170, (2016)], we construct such differential operators and give a classification of them in the pseudo-Riemannian setting where both X and X are of constant sectional curvature, illustrated by the examples of anti-de Sitter spaces and hyperbolic spaces.

Construction of Differential Symmetry Breaking Operators

We proved in Proposition 5.19 that there exist nonzero differential symmetry breaking operators from the G-representation I(i;l)a to the G’-representation J(j, n)b only if

Identities of Scalar-Valued Differential Operators \mathfrak{D}_\mathrm{l}^\mu

Uncorrected Proof In this chapter, we derive identities for the (scalar-valued) differential operators ( mathfrak{D}_mathrm{l}^mu )(see (1.2) for the definition) systematically from those for the Gegenbauer polynomials given in Appendix. We note that some of the formulae here were previously known up to the restriction map Restxn=0, see [11, 16, 21, 24].

Matrix-Valued F-method for O(n+1, 1)

This chapter summarizes a strategy and technical details in applying the F-mehod to find matrix-valued symmetry breaking operators in the setting where (G, G’) = (O(n+1, 1), O(n, 1)).

Appendix: Gegenbauer Polynomials

This chapter collects some properties of the Gegenbauer polynomials that we use throughout this work, in particular, in the proof of the explicit formulae for differential symmetry breaking operators (Theorems 1.5, 1.6, 1.7, and 1.8) and the factorization identities for special parameters (Theorems 13.1, 13.2, and 13.3).

Symmetry Breaking Operators and Principal Series Representations of G = O(n+1, 1)

The conformal compactification Sn of Rn may be thought of as the real flag variety of the indefinite orthogonal group G = O(n+1, 1), and the twisted action v(i) u, d of G on E i(Sn) is a special case of the principal series representations of G.

Solutions to Problems A and B for (Sn, Sn–1)

In this chapter, we complete the proof of Theorem 1.1 and Theorems 1.5–1.8, which solve Problems A and B of conformal geometry for the model space (X, Y) = (Sn, Sn–1), respectively.

F-method for Matrix-Valued Differential Operators

In this chapter we recall from [14, 15, 21, 22] a method based on the Fourier transform (F-method) to find explicit formulae of differential symmetry breaking operators. For our purpose we need to develop the F-method for matrix-valued operators. A new ingredient is a canonical decomposition of the algebraic Fourier transform of the vector-valued principal series representations into the “scalar part” involving differential operators of higher order and into the “vector part” of first order. This is formulated and proved in Sect. 3.4.

Application of Finite-Dimensional Representation Theory

In this chapter we prepare some results on finite-dimensional representations that will be used in applying the general theory developed in Chaps. 3 and 4 to symmetry breaking operators for differential forms.

Basic Operators in Differential Geometry and Conformal Covariance

In this chapter we collect some elementary properties of basic operators such as the Hodge star operator, the codifferential d*, and the interior multiplication iNY (X) by the normal vector field for hypersurfaces Y in pseudo-Riemannian manifolds X.