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Joseph A. Wolf

Publications and source records attributed to Joseph A. Wolf.

At least 19 recordsLinked to original sources
arXiv2023

Pseudo-Riemannian geodesic orbit nilmanifolds of signature $\boldsymbol{(n-2,2)}$

The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds $M = G/H$ of signature $(n-2,2)$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. For that we suppose that there is a reductive decomposition $\g = \h \oplus \n \text{ (vector space direct sum) with } [\h,\n] \subset \n$ and $\n$ nilpotent. When the metric is nondegenerate on $[\n,\n]$ we show that $\n$ is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on $[\n,\n]$ we show that $\n$ is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature.

math.DG↗
arXiv2023

On the Homogeneity Conjecture

Consider a connected homogeneous Riemannian manifold $(M,ds^2)$ and a Riemannian covering $(M,ds^2) \to Γ\backslash (M,ds^2)$. If $Γ\backslash (M,ds^2)$ is homogeneous then every $γ\in Γ$ is an isometry of constant displacement. The Homogeneity Conjecture suggests the converse: if every $γ\in Γ$ is an isometry of constant displacement on $(M,ds^2)$ then $Γ\backslash (M,ds^2)$ is homogeneous. We survey the cases in which the Homogeneity Conjecture has been verified, including some new results, and suggest some related open problems.

math.DG↗
arXiv2022

Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

Two homogeneous pseudo-riemannian manifolds $(G/H, ds^2)$ and $(G'/H', ds'^2)$ belong to the same {\it real form family} if their complexifications $(G_{\mathbb C}/H_{\mathbb C}, ds_{\mathbb C}^2)$ and $(G'_{\mathbb C}/H'_{\mathbb C}, ds'^2_{\mathbb C})$ are isometric. The point is that in many cases a particular space $(G/H, ds^2)$ has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if $(G/H, ds^2)$ is a geodesic orbit space with a reductive decomposition $\mathfrak{g} = \mathfrak{h} + \mathfrak{m}$, then the same holds all the members of its real form family. In particular our understanding of compact geodesic orbit riemannian manifolds gives information on geodesic orbit pseudo-riemannian manifolds. We also prove similar results for naturally reductive spaces, for commutative spaces, and in most cases for weakly symmetric spaces. We end with a discussion of inclusions of these real form families, a discussion of D'Atri spaces, and a number of open problems.

math.DG↗
arXiv2022

The Structure of Geodesic Orbit Lorentz Nilmanifolds

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $M = G/H$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. Suppose that there is a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{n}$ (vector space direct sum) with $\mathfrak{n}$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$ we show that $\mathfrak{n}$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case we construct examples to show that the number of nilpotency steps is unbounded.

math.DG↗
arXiv2022

Partial Dirac Cohomology and Tempered Representations

The tempered representations of a real reductive Lie group $G$ are naturally partitioned into series associated with conjugacy classes of Cartan subgroups $H$ of $G$. We define partial Dirac cohomology, apply it for geometric construction of various models of these $H$--series representations, and show how this construction fits into the framework of geometric quantization and symplectic reduction.

math.RT↗
arXiv2021

On the Geometric Orbit Property for Lorentz Manifolds

The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric Riemannian manifolds are geodesic orbit spaces. Here we define "naturally reductive" for pseudo-Riemannian manifolds and note that they are geodesic orbit spaces. A few years ago two of the authors proved that weakly symmetric pseudo-Riemannian manifolds are geodesic orbit spaces. In particular these results apply to pseudo-Riemannian Lorentz manifolds. There our main results are Theorems 4.2 and 5.1. In the Riemannian case the nilpotent isometry group for a geodesic orbit nilmanifold is abelian or $2$-step nilpotent. Examples show that this fails dramatically in the pseudo-Riemannian case. Here we concentrate on the geodesic orbit property for Lorentz nilmanifolds $G/H$ with $G = N \rtimes H$ and $N$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$, Theorem 4.2 shows that $N$ either is at most $2$-step nilpotent as in the Riemannian situation, or is $4$-step nilpotent, but cannot be $3$-step nilpotent. Examples show that these bounds are the best possible. Surprisingly, Theorem 5.1 shows that $N$ is at most $2$-step nilpotent when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$. Both theorems give additional structural information and specialize to naturally reductive and to weakly symmetric Lorentz nilmanifolds. Key Words: Geodesic Orbit Space; Lorentz nilmanifold; Weakly Symmetric Space; Naturally Reductive Space; Pseudo-Riemannian Manifold.

math.DG↗
arXiv2020

Local and Global Homogeneity for Three Obstinate Spheres

In this note we complete a study of globally homogeneous Riemannian quotients $Γ\backslash (M,ds^2)$ in positive curvature. Specifically, $M$ is a homogeneous space $G/H$ that admits a $G$-invariant Riemannian metric of strictly positive sectional curvature, and $ds^2$ is a $G$--invariant Riemannian metric on $M$, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that $Γ\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $γ\in Γ$ is of constant displacement on $(M,ds^2)$. In an earlier paper we verified that conjecture for all homogeneous spaces that admit an invariant Riemannian metric of positive curvature -- with three exceptions, all odd dimensional spheres, which surprisingly did not yield to the earlier approaches. Here we develop some methods that let us verify the Homogeneity Conjecture for those three obstinate spheres. That completes verification of the Homogeneity Conjecture in positive curvature.

math.DG↗
arXiv2020

Weakly Symmetric Pseudo-Riemannian Nilmanifolds

In an earlier paper we developed the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans--lorentzian signature $(n-2,2)$. Here we work out the classification of weakly symmetric pseudo--riemannian nilmanifolds $G/H$ from the classification for the case $G = N\rtimes H$ with $H$ compact and $N$ nilpotent. It turns out that there is a plethora of new examples that merit further study. Starting with that riemannian case, we see just when a given involutive automorphism of $H$ extends to an involutive automorphism of $G$, and we show that any two such extensions result in isometric pseudo--riemannian nilmanifolds. The results are tabulated in the last two sections of the paper.

math.DG↗
arXiv2019

Unitary Representations, $L^2$ Dolbeault Cohomology, and Weakly Symmetric Pseudo--Riemannian Nilmanifolds

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction of discrete series representations on spaces of square integrable harmonic forms with values in holomorphic vector bundles over flag domains. Some special cases had been described by Satake in 1971 and the author in 1975. Here we develop a theory of pseudo--riemannian nilmanifolds of complex type and the nilmanifold versions of flag domains. We construct the associated square integrable (modulo the center) representations on holomorphic cohomology spaces over those domains and note that there are enough such representations for the Plancherel and Fourier Inversion Formulae there. Finally, we note that the most interesting such spaces are weakly symmetric pseudo-riemannian nilmanifolds, so we discuss that theory and give classifications for three basic families of weakly symmetric pseudo--riemannian nilmanifolds of complex type.

math.RT↗
arXiv2019

Local and Global Homogeneity for Manifolds that admit a Positive Curvature Metric

In this note we study globally homogeneous Riemannian quotients $Γ\backslash (M,ds^2)$ of homogeneous Riemannian manifolds $(M,ds^2)$. The Homogeneity Conjecture is that $Γ\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $γ\in Γ$ is of constant displacement on $(M,ds^2)$. We provide further evidence for that conjecture by (i) verifying it for normal homogeneous Riemannian manifolds that also admit an invariant Riemannian metric of strictly positive sectional curvature and (ii) showing that in most (three or less) cases the normality condition can be dropped.

math.DG↗
arXiv2018

Semisimple Weakly Symmetric Pseudo--Riemannian Manifolds

We develop the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derive the classification from the cases where $G$ is compact, and then we discuss the (isotropy) representation of $H$ on the tangent space of $G/H$ and the signature of the invariant pseudo--riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans--Lorentz (conformal Lorentz) signature $(n-2,2)$.

math.DG↗
arXiv2017

Representations on Partially Holomorphic Cohomology Spaces, Revisited

This is a semi--expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so many years, much of that Memoir is up to date, but of course there have been a number of refinements, advances and new developments, most of which have applied to smaller classes of real reductive Lie groups. Here we rewrite that AMS Memoir in in view of these advances and indicate the ties with some of the more recent (or at least less classical) approaches to geometric realization of unitary representations.

math.RT↗
arXiv2017

Real group orbits on flag ind-varieties of $\mathrm{SL}(\infty,\mathbb{C})$

We consider the complex ind-group $G=\mathrm{SL}(\infty,\mathbb{C})$ and its real forms $G^0=\mathrm{SU}(\infty,\infty)$, $\mathrm{SU}(p,\infty)$, $\mathrm{SL}(\infty,\mathbb{R})$, $\mathrm{SL}(\infty,\mathbb{H})$. Our main objects of study are the $G^0$-orbits on an ind-variety $G/P$ for an arbitrary splitting parabolic ind-subgroup $P\subset G$. We prove that the intersection of any $G^0$-orbit on $G/P$ with a finite-dimensional flag variety $G_n/P_n$ from a given exhaustion of $G/P$ via $G_n/P_n$ for $n\to\infty$, is a single $(G^0\cap G_n)$-orbit. We also characterize all ind-varieties $G/P$ on which there are finitely many $G^0$-orbits, and provide criteria for the existence of open and closed $G^0$-orbits on $G/P$ in the case of infinitely many $G^0$-orbits.

math.AG↗
arXiv2017

Stepwise Square Integrability for Nilradicals of Parabolic Subgroups and Maximal Amenable Subgroups

In a series of recent papers we extended the notion of square integrability, for representations of nilpotent Lie groups, to that of stepwise square integrability. There we discussed a number of applications based on the fact that nilradicals of minimal parabolic subgroups of real reductive Lie groups are stepwise square integrable. Here, in Part I, we prove stepwise square integrability for nilradicals of arbitrary parabolic subgroups of real reductive Lie groups. This is technically more delicate than the case of minimal parabolics. We further discuss applications to Plancherel formulae and Fourier inversion formulae for maximal exponential solvable subgroups of parabolics and maximal amenable subgroups of real reductive Lie groups. Finally, in Part II, we extend a number of those results to (infinite dimensional) direct limit parabolics. These extensions involve an infinite dimensional version of the Peter-Weyl Theorem, construction of a direct limit Schwartz space, and realization of that Schwartz space as a dense subspace of the corresponding $L^2$ space.

math.RT↗
arXiv2017

Solvability, Structure and Analysis for Minimal Parabolic Subgroups

We examine the structure of the Levi component $MA$ in a minimal parabolic subgroup $P = MAN$ of a real reductive Lie group $G$ and work out the cases where $M$ is metabelian, equivalently where $\mathfrak{p}$ is solvable. When $G$ is a linear group we verify that $\mathfrak{p}$ is solvable if and only if $M$ is commutative. In the general case $M$ is abelian modulo the center $Z_G$, we indicate the exact structure of $M$ and $P$, and we work out the precise Plancherel Theorem and Fourier Inversion Formulae. This lays the groundwork for comparing tempered representations of $G$ with those induced from generic representations of $P$.

math.RT↗
arXiv2016

Homogeneity for a Class of Riemannian Quotient Manifolds

We study riemannian coverings $φ: \widetilde{M} \to Γ\backslash \widetilde{M}$ where $\widetilde{M}$ is a normal homogeneous space $G/K_1$ fibered over another normal homogeneous space $M = G/K$ and $K$ is locally isomorphic to a nontrivial product $K_1\times K_2$. The most familiar such fibrations $π: \widetilde{M} \to M$ are the natural fibrations of Stieffel manifolds $SO(n_1 + n_2)/SO(n_1)$ over Grassmann manifolds $SO(n_1 + n_2)/[SO(n_1)\times SO(n_2)]$ and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings $φ: \widetilde{M} \to Γ\backslash \widetilde{M}$ are the universal riemannian coverings of spherical space forms. When $M = G/K$ is reasonably well understood, in particular when $G/K$ is a riemannian symmetric space or when $K$ is a connected subgroup of maximal rank in $G$, we show that the Homogeneity Conjecture holds for $\widetilde{M}$. In other words we show that $Γ\backslash \widetilde{M}$ is homogeneous if and only if every $γ\in Γ$ is an isometry of constant displacement. In order to find all the isometries of constant displacement on $\widetilde{M}$ we work out the full isometry group of $\widetilde{M}$, extending Elie Cartan's determination of the full group of isometries of a riemannian symmetric space. We also discuss some pseudo-riemannian extensions of our results.

math.DG↗