In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author’s 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if Γ \ M is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold M, then Γ \ M is homogeneous if and only if each isometry γ∈ Γ is of constant displacement. The description of bounded isometries in this paper gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford’s use of quaternions in non-Euclidean geometry, and we sketch that in a historical note.

In a recent paper we found conditions for a nilpotent Lie group N to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit together nicely to give an explicit construction of Plancherel for almost all representations of N. The prototype for this sort of group is the group of upper triangular real matrices with 1’s down the diagonal. More generally, this class of groups contains the nilradicals of minimal parabolic subgroups of all (finite-dimensional) reductive real or complex Lie groups, in other words, all groups N in Iwasawa decompositions of reductive real or complex Lie groups. The construction of stepwise square integrable representations resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. Here we extend those results to direct limits of stepwise square integrable nilpotent Lie groups. There are two keys to this extension. The first is to set up the corresponding direct system so that it respects the construction at every finite level. In the case of simple (or more generally reductive) groups this means that the restricted root Dynkin diagrams increase in a particular manner. The second is to follow Schwartz space theory through the direct limit process, develop a Schwartz space theory for certain direct limit nilpotent groups, and use it to study stepwise square integrability for coefficients of direct limits of stepwise square integrable nilpotent Lie groups. This leads to the main result, an explicit Fourier inversion formula for that class of infinite-dimensional Lie groups. One important consequence is the Fourier inversion formula for nilradicals of classical minimal parabolic subgroups of finitary real reductive Lie groups such as GL(∞; ℝ), Sp(∞; ℂ) and SO(∞, ∞).

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form (Formula presented.) where, in all but three cases, the nilpotent group (Formula presented.) has irreducible unitary representations whose coefficients are square integrable modulo the center (Formula presented.) of (Formula presented.). Here we show that, in those three “exceptional” cases, the group (Formula presented.) is a semidirect product (Formula presented.) or (Formula presented.) where the normal subgroup (Formula presented.) contains the center (Formula presented.) of (Formula presented.) and has irreducible unitary representations whose coefficients are square integrable modulo (Formula presented.). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.

There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups.Many of these consequences are still under development. In this note I’ll survey a fewof these newaspects of representation theory for nilpotent Lie groups and parabolic subgroups.

Let G be a complex simple direct limit group, specifically SL(∞; C) , SO(∞; C) or Sp(∞; C). Let F be a (generalized) flag in C∞. If G is SO(∞; C) or Sp(∞; C) we suppose further that F is isotropic. Let Z denote the corresponding flag manifold; thus Z= G/ Q where Q is a parabolic subgroup of G. In a recent paper (Penkov and Wolf in Real group orbits on flag ind-varieties of SL∞(C) , to appear in Proceedings in Mathematics and Statistics) we studied real forms G0 of G and properties of their orbits on Z. Here we concentrate on open G0-orbits D⊂ Z. When G0 is of hermitian type we work out the complete G0-orbit structure of flag manifolds dual to the bounded symmetric domain for G0. Then we develop the structure of the corresponding cycle spaces MD. Finally we study the real and quaternionic analogs of these theories. All this extends results from the finite-dimensional cases on the structure of hermitian symmetric spaces and cycle spaces (in chronological order: Wolf in Bull Am Math Soc 75:1121–1237, 1969; Wolf et al. in Ann Math 105:397–448, 1977; Wolf in Ann Math 136:541–555, 1992; Wolf in Compact subvarieties in flag domains, 1994; Wolf and Zierau in Math Ann 316:529–545, 2000; Huckleberry et al. in Journal für die reine und angewandte Mathematik 2001:171–208, 2001; Huckleberry and Wolf in Cycle spaces of real forms of SLn(C) , Springer, New York, pp 111–133, 2002; Wolf and Zierau in J Lie Theory 13:189–191, 2003; Huckleberry and Wolf in Ann Scuola Norm Sup Pisa Cl Sci (5) 9:573-580, 2010).

In a series of recent papers ([19], [20], [22], [23]) 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. 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 L2 space.

In earlier papers we studied direct limits $${(G,\,K) = \varinjlim\, (G_n,K_n)}$$ of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which $${G_n = N_n\rtimes K_n}$$ with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections $${\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}$$ for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit $${L^2(G/K) := \varinjlim L^2(G_n/K_n)}$$ is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces $${\mathcal{A}(G_n/K_n)}$$ of regular functions on G n /K n and injections $${\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}$$ for m ≧ n related to restriction by $${\nu_{m,n}(f)|_{G_n/K_n} = f}$$ . Thus the direct limit $${\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}$$ sits as a particular G-submodule of the much larger inverse limit $${\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}$$ . Further, we define a pre Hilbert space structure on $${\mathcal{A}(G/K)}$$ derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space $${\mathcal{A}(G/K)}$$ , and also defines a G-invariant inner product on $${\mathcal{A}(G/K)}$$ for which the left regular representation of G is multiplicity-free.

We study Riemannian coverings [Formula presented] where M~ is a normal homogeneous space G/K1 fibered over another normal homogeneous space M=G/K and K is locally isomorphic to a nontrivial product K1×K2. The most familiar such fibrations [Formula presented] are the natural fibrations of Stiefel manifolds SO(n1+n2)/SO(n1) over Grassmann manifolds SO(n1+n2)/[SO(n1)×SO(n2)] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings [Formula presented] 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 M~. In other words we show that [Formula presented] is homogeneous if and only if every γ∈Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M~ we work out the full isometry group of M~ extending Élie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.

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 p is solvable. When G is a linear group we verify that p is solvable if and only if M is commutative. In the general case M is abelian modulo the center ZG , 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.

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.