We study representations of the classical infinite dimensional real simple Lie groups G induced f... more We study representations of the classical infinite dimensional real simple Lie groups G induced from factor representations of minimal parabolic subgroups P. This makes strong use of the recently developed structure theory for those parabolic subgroups and subalgebras. In general parabolics in the infinite dimensional classical Lie groups are are somewhat more complicated than in the finite dimensional case, and are not direct limits of finite dimensional parabolics. We extend their structure theory and use it for the infinite dimensional analog of the classical principal series representations. In order to do this we examine two types of conditions on P: the flag-closed condition and minimality. We use some riemannian symmetric space theory to prove that if P is flag-closed then any maximal lim-compact subgroup K of G is transitive on G /P . When P is minimal we prove that it is amenable, and we use properties of amenable groups to induce unitary representations τ of P up to contin...
We study the structure of minimal parabolic subgroups of the classical infinite dimensional real ... more We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite dimensional parabolics. We then discuss the use of that structure theory for the infinite dimensional analog of the classical principal series representations. We look at the unitary representation theory of the classical lim--compact groups U(∞), SO(∞) and Sp(∞) in order to construct the inducing representations, and we indicate some of the analytic considerations in the actual construction of the induced representations.
Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry... more Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry. Here I'll describe how this goes for nilpotent Lie groups and for a class of Riemannian manifolds closely related to a nilpotent Lie group structure. There are also some infinite dimensional analogs but I won't go into that here. The analytic ideas are not so different from those of the classical Fourier transform and Fourier inversion theories in one real variable.
Let G be a complex simple direct limit group, specifically SL(∞;C), SO(∞;C) or Sp(∞;C). Let F be ... more 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 with Ignatyev and Penkov, we studied real forms G_0 of G and properties of their orbits on Z. Here we concentrate on open G_0--orbits D ⊂Z. When G_0 is of hermitian type we work out the complete G_0--orbit structure of flag manifolds dual to the bounded symmetric domain for G_0. Then we develop the structure of the corresponding cycle spaces M_D. Finally we study the real and quaternionic analogs of these theories. All this extends an large body of results from the finite dimensional cases on the structure of hermitian symmetric spaces and related cycle spaces.
We develop the classification of weakly symmetric pseudo--riemannian manifolds G/H where G is a s... more 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).
Killing vector fields of constant length correspond to isometries of constant displacement. Those... more Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of research on constant length Killing vector fields to a class of Riemannian normal homogeneous spaces.
In this paper we develop the basic tools for a classification of Killing vector fields of constan... more In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo--riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs (M,ξ) where M = G/H is a Riemannian normal homogeneous space, G is a compact simple Lie group, and ξ∈g defines a nonzero Killing vector field of constant length on M. The method there was direct computation. Here we make use of the moment map M →g^* and the flag manifold structure of Ad(G)ξ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo--riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where ξ is elliptic and G is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.
We study direct limits (G,K) = (G_n,K_n) of compact Gelfand pairs. First, we develop a criterion ... more We study direct limits (G,K) = (G_n,K_n) of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits G/K = G_n/K_n of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of G = G_n on a certain function space L^2(G_n/K_n) is multiplicity free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The first, based on "parabolic direct limits" and "defining representations", extends the method used in the symmetric space case. The second uses some (new) branching rules from finite dimensional representation theory. In both cases we define function spaces (G/K), (G/K) and L^2(G/K) to which our multiplicity--free criterion applies.
A compact Riemannian homogeneous space G/H, with a bi--invariant orthogonal decomposition g=h+m i... more A compact Riemannian homogeneous space G/H, with a bi--invariant orthogonal decomposition g=h+m is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in T_eH(G/H) spanned by a linearly independent commuting pair in m. In this paper,we will prove that on the coset space Sp(2)/U(1), in which U(1) corresponds to a short root, admits positively curved metrics for commuting pairs. B. Wilking recently proved that this Sp(2)/U(1) can not be positively curved in the general sense. This is the first example to distinguish the set of compact coset spaces admitting positively curved metrics, and that for metrics positively curved only for commuting pairs.
We study representations of the classical infinite dimensional real simple Lie groups G induced f... more We study representations of the classical infinite dimensional real simple Lie groups G induced from factor representations of minimal parabolic subgroups P. This makes strong use of the recently developed structure theory for those parabolic subgroups and subalgebras. In general parabolics in the infinite dimensional classical Lie groups are are somewhat more complicated than in the finite dimensional case, and are not direct limits of finite dimensional parabolics. We extend their structure theory and use it for the infinite dimensional analog of the classical principal series representations. In order to do this we examine two types of conditions on P: the flag-closed condition and minimality. We use some riemannian symmetric space theory to prove that if P is flag-closed then any maximal lim-compact subgroup K of G is transitive on G /P . When P is minimal we prove that it is amenable, and we use properties of amenable groups to induce unitary representations τ of P up to contin...
We study the structure of minimal parabolic subgroups of the classical infinite dimensional real ... more We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite dimensional parabolics. We then discuss the use of that structure theory for the infinite dimensional analog of the classical principal series representations. We look at the unitary representation theory of the classical lim--compact groups U(∞), SO(∞) and Sp(∞) in order to construct the inducing representations, and we indicate some of the analytic considerations in the actual construction of the induced representations.
Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry... more Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry. Here I'll describe how this goes for nilpotent Lie groups and for a class of Riemannian manifolds closely related to a nilpotent Lie group structure. There are also some infinite dimensional analogs but I won't go into that here. The analytic ideas are not so different from those of the classical Fourier transform and Fourier inversion theories in one real variable.
Let G be a complex simple direct limit group, specifically SL(∞;C), SO(∞;C) or Sp(∞;C). Let F be ... more 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 with Ignatyev and Penkov, we studied real forms G_0 of G and properties of their orbits on Z. Here we concentrate on open G_0--orbits D ⊂Z. When G_0 is of hermitian type we work out the complete G_0--orbit structure of flag manifolds dual to the bounded symmetric domain for G_0. Then we develop the structure of the corresponding cycle spaces M_D. Finally we study the real and quaternionic analogs of these theories. All this extends an large body of results from the finite dimensional cases on the structure of hermitian symmetric spaces and related cycle spaces.
We develop the classification of weakly symmetric pseudo--riemannian manifolds G/H where G is a s... more 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).
Killing vector fields of constant length correspond to isometries of constant displacement. Those... more Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of research on constant length Killing vector fields to a class of Riemannian normal homogeneous spaces.
In this paper we develop the basic tools for a classification of Killing vector fields of constan... more In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo--riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs (M,ξ) where M = G/H is a Riemannian normal homogeneous space, G is a compact simple Lie group, and ξ∈g defines a nonzero Killing vector field of constant length on M. The method there was direct computation. Here we make use of the moment map M →g^* and the flag manifold structure of Ad(G)ξ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo--riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where ξ is elliptic and G is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.
We study direct limits (G,K) = (G_n,K_n) of compact Gelfand pairs. First, we develop a criterion ... more We study direct limits (G,K) = (G_n,K_n) of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits G/K = G_n/K_n of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of G = G_n on a certain function space L^2(G_n/K_n) is multiplicity free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The first, based on "parabolic direct limits" and "defining representations", extends the method used in the symmetric space case. The second uses some (new) branching rules from finite dimensional representation theory. In both cases we define function spaces (G/K), (G/K) and L^2(G/K) to which our multiplicity--free criterion applies.
A compact Riemannian homogeneous space G/H, with a bi--invariant orthogonal decomposition g=h+m i... more A compact Riemannian homogeneous space G/H, with a bi--invariant orthogonal decomposition g=h+m is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in T_eH(G/H) spanned by a linearly independent commuting pair in m. In this paper,we will prove that on the coset space Sp(2)/U(1), in which U(1) corresponds to a short root, admits positively curved metrics for commuting pairs. B. Wilking recently proved that this Sp(2)/U(1) can not be positively curved in the general sense. This is the first example to distinguish the set of compact coset spaces admitting positively curved metrics, and that for metrics positively curved only for commuting pairs.
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Papers by Joseph Wolf