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Pierre Julg

Publications and source records attributed to Pierre Julg.

7 recordsLinked to original sources
arXiv2024

Poisson transforms, the BGG complex, and discrete series representations of SU(n+1,1)

The aim of this article is to construct a specific Poisson transform mapping differential forms on the sphere $S^{2n+1}$ endowed with its natural CR structure to forms on complex hyperbolic space. The transforms we construct have values that are harmonic and co-closed and they descend to the BGG (Rumin) complex and intertwine the differential operators in that complex with the exterior derivative. Passing to the Poincar\'e ball model, we analyze the boundary asymptotics of the values of our transforms proving that they admit a continuous extension to the boundary in degrees $\leq n$. Finally, we show that composing the exterior derivative with the transform in degree $n$, one obtains an isomorphism between the kernel of the Rumin operator in degree $n$ and a dense subspace of the $L^2$-harmonic forms on complex hyperbolic space. These are well known to realize the direct sum of all discrete series representations of $SU(n+1,1)$, which we therefore realize on spaces of differential forms on the compact manifold $S^{2n+1}$. The developments in this article are motivated by a program of the third author to prove some instances of the Baum-Connes conjecture. The first part of the article is valid in a much more general setting, and is also relevant for cases in which the conjecture is still open.

math.DG↗
arXiv2022

Slow exponential growth representations of Sp(n, 1) at the edge of Cowling's strip

We obtain a slow exponential growth estimate for the spherical principal series representation rho_s of Lie group Sp(n, 1) at the edge (Re(s)=1) of Cowling's strip (|Re(s)|<1) on the Sobolev space H^alpha(G/P) when alpha is the critical value Q/2=2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy rho_s (s in [0, 1]) of the spherical principal series which is required for the first author's program for proving the Baum--Connes conjecture with coefficients for Sp(n,1).

math.RT↗
arXiv2020

A Poisson transform adapted to the Rumin complex

Let $G$ be a semisimple Lie group with finite center, $K\subset G$ a maximal compact subgroup, and $P\subset G$ a parabolic subgroup. Following ideas of P.Y.\ Gaillard, one may use $G$-invariant differential forms on $G/K\times G/P$ to construct $G$-equivariant Poisson transforms mapping differential forms on $G/P$ to differential forms on $G/K$. Such invariant forms can be constructed using finite dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on $G/P$ to the associated Bernstein-Gelfand-Gelfand (or BGG) complex in a well defined sense. The main part of the article is devoted to an explicit construction of such transforms with additional favorable properties in the case that $G=SU(n+1,1)$. Thus $G/P$ is $S^{2n+1}$ with its natural CR structure and the relevant BGG complex is the Rumin complex, while $G/K$ is complex hyperbolic space of complex dimension $n+1$. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

math.DG↗
arXiv2014

The Geometry of the Osculating Nilpotent Group Structures of the Heisenberg Calculus

We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this paper is to provide insight in the intrinsic geometry that underlies these coordinate formulas. First, we introduce `parabolic arrows' as a generalization of tangent vectors. The definition of parabolic arrows involves a mix of first and second order derivatives. Parabolic arrows can be composed, and the group of parabolic arrows can be identified with the nilpotent groups of the (generalized) Heisenberg calculus. Secondly, we formulate a notion of exponential map for the fiber bundle of parabolic arrows, and show how it explains the coordinate formulas of osculating structures found in the literature on the Heisenberg calculus. The result is a conceptual simplification and unification of the treatment of the Heisenberg calculus.

math.AP↗