Michael Pevzner

Vector-Valued Covariant Differential Operators for the Möbius Transformation

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups \((SL(2, \mathbb{C}),SL(2, \mathbb{R}))\) that arise from conformal geometry.

Berezin kernels and analysis on Makarevich spaces

Abstract Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels. We consider the conformal group Conf(V) of a simple real Jordan algebra V. The maximal degenerate representations πs (s e ℂ) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf(V). These representations πs can be realized on a space Is of smooth functions on V. There is an invariant bilinear form ℬs on the space Is. The problem we consider is to diagonalize this bilinear form ℬs, with respect to the action of a symmetric subgroup G of the conformal group Conf(V). This bilinear form can be written as an integral involving the Berezin kernel Bv an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of Bv. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: D(ν)Bν=b(ν)Bν+1, where D(ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I−s to Is. Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V).

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.

H*-ALGEBRAS AND QUANTIZATION OF PARA-HERMITIAN SPACES

In the present note we describe a family of H � -algebra structures on the set L 2 (X) of square integrable functions on a rank-one para-Hermitian symmetric space X.

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.