1 Introduction
1.1 Representations of semisimple Lie groups and their lattices
Lattices (that is, discrete subgroups with finite covolume) of semisimple Lie groups may be thought of as discretizations of these Lie groups. The question of knowing how much of the ambient group is encoded in its lattices is very natural and has attracted a lot of interest in the past decades.
Among the many results, one can highlight Mostow’s strong rigidity that implies that a lattice in a higher-rank semisimple Lie group without compact factors completely determines the Lie group [Reference MostowMos73]. Later, Margulis proved his super-rigidity theorem and showed that linear representations of irreducible lattices of higher-rank semisimple algebraic groups over local fields are ruled by representations of the ambient algebraic groups [Reference MargulisMar91].
These rigidity results may be understood using a geometric object associated with the algebraic group: a Riemannian symmetric space (for a Lie group) or a Euclidean building (for an algebraic group over a non-Archimedean field).
Lattices have natural and interesting linear representations outside the finite-dimensional world, which starts with Hilbert spaces. For example, some representations may come from the principal series of the Lie group. Outside the world of unitary representations, some infinite-dimensional representations of a lattice have a very strong geometric flavor. This is the case when there is an invariant non-degenerate quadratic or Hermitian form of the finite index, that is, when the representation falls in
$\operatorname {\mathrm{PO}}_{{\mathbf K}}(p,\infty )$
where
${\textbf K}=\textbf {R}$
or
${\mathbf{C}}$
and p is finite. Then, one can consider the associated action on some infinite-dimensional Riemannian symmetric space of non-positive curvature
${\mathcal X}_{{\mathbf K}}(p,\infty )$
. For example, when
$p=1$
,
${\mathcal X}_{{\mathbf K}}(p,\infty )$
is the infinite-dimensional real or complex hyperbolic space. Gromov had the following expressive words to say about
${\mathcal X}_{\mathbf {R}}(p,\infty )$
[Reference GromovGro93, p. 121]:
These spaces look as cute and sexy to me as their finite-dimensional siblings but they have been neglected by geometers and algebraists alike.
In [Reference DuchesneDuc15b], an analog of Margulis super-rigidity has been obtained for higher-rank cocompact lattices of semisimple Lie groups using harmonic map techniques. The main result is that non-elementary representations preserve a totally geodesic copy of a finite-dimensional symmetric space of non-compact type. The finite-rank assumption, here
$p<\infty $
, may be thought of as a geometric Ersatz of local compactness.
The reader should be warned that even in the case of actions on finite-rank symmetric spaces of infinite dimension, some new baffling phenomena may appear. For example, Delzant and Py exhibited representations of
$\operatorname {\textrm {PSL}}_2(\textbf {R})$
in
$\operatorname {\textrm {O}}_{\mathbf {R}}(1,\infty )$
(and, more generally, of
$\operatorname {\mathrm{PO}}(1,n)$
in
$\operatorname {\textrm {O}}_{\mathbf {R}}(p,\infty )$
for some values of p depending on n). They found a one-parameter family of exotic deformations of
${\mathcal X}_{\mathbf {R}}(1,2)$
in
${\mathcal X}_{\mathbf {R}}(1,\infty )$
equivariant with respect to representations leaving no finite-dimensional totally geodesic subspace invariant. See [Reference Delzant and PyDP12, Reference Monod and PyMP14] for a classification. Very recently, this classification has been extended to self-representations of
$\operatorname {\textrm {O}}_{\mathbf {R}}(1,\infty )$
[Reference Monod and PyMP18]. Moreover, exotic representations of
$\operatorname {\mathrm{SU}}(1,n)$
in
$\operatorname {\textrm {O}}_{\mathbf{C}}(1,\infty )$
have also been obtained in [Reference MonodMon18].
In rank one, there is, in general, no hope for an analog of Margulis super-rigidity (even in finite dimension). For example, fundamental groups of non-compact hyperbolic surfaces of finite volume are free groups and thus not rigid. For compact hyperbolic surfaces, the lack of rigidity gives rise to the Teichmüller space and thus to a whole variety of deformations of the corresponding lattices.
For complex hyperbolic lattices, the complex structure constrains the lattices because the Kähler form implies the non-vanishing of the cohomology in degree two. Furthermore, in finite dimension, the Kähler form was successfully used to define a characteristic invariant that selects representations with surprising rigidity properties, the so-called Toledo invariant [Reference Burger, Iozzi and WienhardBIW10, Reference ToledoTol89].
The goal of this paper is to study representations of complex hyperbolic lattices in the groups
$\operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
and
$\operatorname {\mathrm{PO}}_{\mathbf {R}}(2,\infty )$
, and the associated isometric actions on the Hermitian symmetric spaces
${\mathcal X}_{\mathbf{C}}(p,\infty )$
and
${\mathcal X}_{\mathbf {R}}(2,\infty )$
. These spaces have a Kähler form
$\omega $
and this yields a class in bounded cohomology of degree two on
$G=\operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
induced by the cocycle that computes the integral of the Kähler form
$\omega $
over a straight geodesic triangle
$\Delta (g_0x,g_1x,g_2x)$
whose vertices are in the orbit of a basepoint:
$$ \begin{align*}C_{\omega}^x(g_0,g_1,g_2)=\frac{1}{\pi}\int_{\Delta(g_0x,g_1x,g_2x)}\omega.\end{align*} $$
We denote by
$\kappa ^b_G\in \text {H}_b^2(G,\textbf {R})$
the associated cohomology class where
$G=\operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
(see §5). As in finite dimension, the Gromov norm
$\|\kappa ^b_G\|_{\infty }$
is exactly the rank of
${\mathcal X}_{\mathbf{C}}(p,\infty )$
(after normalization of the metric). Let
$\rho \colon \Gamma \to \operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
be a homomorphism of a complex hyperbolic lattice. Pulling back
$\kappa ^b_G$
by
$\rho $
, one gets a bounded cohomology class for
$\Gamma $
and one can define maximal representations of
$\Gamma $
as representations maximizing a Toledo number defined as in finite dimension (see Definition 5.7).
Our main results concern maximal representations of fundamental groups of surfaces and, more generally, hyperbolic lattices. It is a continuation of previous results for finite-dimensional Hermitian targets, see [Reference Burger and IozziBI08, Reference Burger, Iozzi and WienhardBIW10, Reference Koziarz and MaubonKM17, Reference PozzettiPoz15] among other references. The meaning of Zariski density in infinite dimension is explained in the following subsection. For representations with target
$\operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
, we prove rigidity in the following way.
Theorem 1.1. Let
${\Gamma }<\operatorname {\mathrm{SU}}(1,n)$
be a complex hyperbolic lattice with n a positive integer, and let
$\rho \colon \Gamma \to \operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
be a maximal representation. If
$p\leq 2$
, then there is a finite-dimensional totally geodesic Hermitian symmetric subspace
$\mathcal Y\subset {\mathcal X}_{\mathbf{C}}(p,\infty )$
that is invariant by
$\Gamma $
. Furthermore, the representation
$\Gamma \to \operatorname {\mathrm{Isom}}(\mathcal Y)$
is maximal.
More generally, for any
$p\in \mathbf{N}$
, there is no maximal Zariski-dense representation
$\rho \colon {\Gamma }\to \operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
.
In particular, because
$\mathcal Y$
is finite dimensional, the results of Burger and Iozzi [Reference Burger and IozziBI08], the third author [Reference PozzettiPoz15] and Koziarz and Maubon [Reference Koziarz and MaubonKM17] apply.
Interestingly enough, the analogous result of Theorem 1.1 does not hold for the orthogonal group
$\operatorname {{\textrm O_{\mathbf {R}}}}(2,\infty )$
and
$n=1$
. Let
$\Sigma $
be a compact connected Riemann surface of genus one with one connected boundary component (which is a circle), that is, a one-holed torus. The fundamental group
$\Gamma _{\Sigma }$
of
$\Sigma $
is thus a free group on two generators and a lattice in
$\operatorname {\mathrm{SU}}(1,1)$
.
Theorem 1.2. There are geometrically dense maximal representations
$\rho :{\Gamma }_{\Sigma }\to \operatorname {\mathrm{PO}}_{\mathbf {R}}(2,\infty )$
.
Observe that the properties of maximal representations in
$\operatorname {\mathrm{PO}}_{\mathbf {R}}(2,\infty )$
and
$\operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
are so different because, for every p, the Hermitian Lie group
$\operatorname {{\textrm O_{\mathbf {R}}}}(2,p)$
is of tube type, while the Hermitian Lie groups
$\operatorname {\mathrm{SU}}(p,q)$
are of tube type if and only if
$p=q$
. We refer to §2.4 for more details. This allows much more flexibility, the chain geometry at infinity being trivial.
Not much is known about the representations of complex hyperbolic lattices, and even less so in infinite dimension. In the case of surface groups, instead, from the complementary series of
$\operatorname {\textrm {PSL}}_2(\textbf {R})$
, Delzant and Py exhibited one-parameter families of representations in
$\operatorname {\mathrm{PO}}_{\mathbf {R}}(p,\infty )$
for every
$p\in \mathbf{N}$
[Reference Delzant and PyDP12]. Having explicit representations in
$\operatorname {\mathrm{PO}}_{\mathbf {R}}(2,\infty )$
, it is compelling to determine if they induce maximal representations. Showing that some harmonic equivariant map is actually totally real, we conclude in Appendix A that the Toledo invariant of these representations vanishes.
Remark 1.3. The difference between
$p\leq 2$
and
$p>2$
lies in the hypotheses under which we can prove the existence of boundary maps (see §1.2). For
$p\leq 2$
, we are able to prove the existence of well-suited boundary maps under geometric density (a hypothesis to which we can easily reduce). Unfortunately, for
$p>2$
, we can only prove it under Zariski density, which is a stronger assumption.
1.2 Boundary maps and standard algebraic groups
To prove Theorem 1.1, we use, as it is now standard in bounded cohomology, boundary map techniques. Let
${\Gamma }$
be a lattice in
$\operatorname {\mathrm{SU}}(1,n)$
and P a strict parabolic subgroup of
$\operatorname {\mathrm{SU}}(1,n)$
. The space
$B=\operatorname {\mathrm{SU}}(1,n)/P$
is a measurable
${\Gamma }$
-space which is amenable and has very strong ergodic properties, and is thus a strong boundary (see Definition 4.7) in the sense of [Reference Bader and FurmanBF14]. This space can be identified with the visual boundary of the hyperbolic space
${\mathcal X}_{\mathbf{C}}(1,n)$
.
In finite dimension, for example in [Reference PozzettiPoz15], the target of the boundary map is the Shilov boundary of the symmetric space
${\mathcal X}_{\mathbf{C}}(p,q)$
. If
$p\leq q$
, this Shilov boundary can be identified with the space
$\mathcal I_p$
of isotropic linear subspaces of dimension p in
${\mathbf{C}}^{p+q}$
. In our infinite-dimensional setting, we use the same space
$\mathcal I_p$
of isotropic linear subspaces of dimension p.
A main difficulty appears in this infinite-dimensional context: this space is not compact anymore for the natural Grassmann topology. Thus the existence of boundary
$\Gamma $
-maps
$B\to \mathcal I_p$
is more involved than in finite dimension. Such boundary maps have been obtained in a non-locally compact setting when the target is the visual boundary
$\partial {\mathcal X}$
of a CAT(0) space
${\mathcal X}$
of finite telescopic dimension, on which a group
$\Gamma $
acts isometrically [Reference Bader, Duchesne and LécureuxBDL16, Reference DuchesneDuc13]. Here,
$\mathcal I_p$
is only a closed G-orbit of
$\partial {\mathcal X}_{\mathbf{C}}(p,\infty )$
. Actually,
$\mathcal I_p$
is a subset of the set of vertices in the spherical building structure on
$\partial {\mathcal X}_{\mathbf{C}}(p,\infty )$
and the previous result is not sufficient. To prove the existence of boundary maps to
$\mathcal I_p$
, we reduce to representations whose images are dense, in the sense that is explained below.
Following [Reference Caprace and MonodCM09], we say that a group
$\Gamma $
acting by isometries on a symmetric space (possibly of infinite dimension) of non-positive curvature is geometrically dense if there is no strict closed invariant totally geodesic subspace (possibly reduced to a point) or fixed point in the visual boundary. For finite-dimensional symmetric spaces, the geometric density is equivalent to Zariski density in the isometry group, which is a real algebraic group. To prove Theorem 1.7, we rely also on the theory of algebraic groups in infinite dimension introduced in [Reference Harris and KaupHK77]. Roughly speaking, a subgroup of the group of invertible elements of a Banach algebra is algebraic if it is defined by (possibly infinitely many) polynomial equations with a uniform bound on the degrees of the polynomials.
This notion of algebraic groups is too coarse for our goals and we introduce the notion of standard algebraic groups in infinite dimension. Let
${\mathcal H}$
be a Hilbert space and let
$\operatorname {\textrm {GL}}({\mathcal H})$
be the group of invertible bounded operators of
${\mathcal H}$
. An algebraic subgroup of
$\operatorname {\textrm {GL}}({\mathcal H})$
is standard if it is defined by polynomial equations in the matrix coefficients
$g\mapsto \langle ge_i,e_j\rangle $
, where
$(e_i)$
is some Hilbert base of
${\mathcal H}$
. See Definition 3.4. With this definition, we are able to show that stabilizers of points in
$\partial X_{{\mathbf K}}(p,\infty )$
are standard algebraic subgroups, and the same holds for stabilizers of proper totally geodesic subspaces.
Definition 1.4. A subgroup of
$\operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )$
is Zariski dense when it is not contained in a proper standard algebraic group. A representation
$\rho \colon {\Gamma }\to \operatorname {\mathrm{PO}}_{{\mathbf K}}(p,\infty )$
is Zariski dense if the preimage of
$\rho ({\Gamma })$
in
$\operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )$
is Zariski dense. For a short discussion about a possible Zariski topology in infinite dimension, we refer to Remark 3.2.
We show in Proposition 1.5 that Zariski density implies geometric density.
Proposition 1.5. Let
$p\in \mathbf{N}$
. Stabilizers of closed totally geodesic subspaces of
${\mathcal X}_{{\mathbf K}}(p,\infty )$
and stabilizers of points in
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
are standard algebraic subgroups of
$\operatorname {\mathrm{O}_{\mathbf {K}}}(p,\infty )$
.
In particular, a Zariski-dense subgroup of
$\operatorname {\mathrm{O}_{\mathbf {K}}}(p,\infty )$
is geometrically dense.
Question 1.6. Is it true that the converse of Proposition 1.5 holds? Namely, are geometric density and Zariski density equivalent? It is also possible that one needs to strengthen the definition of standard algebraic groups to ensure that geometric density and Zariski density are the same.
Finally, we get the existence of the desired boundary maps under geometric or Zariski density. In the following statement, two linear subspaces are said to be transverse if their intersection is trivial.
Theorem 1.7. Let
$\Gamma $
be a countable group with a strong boundary B and
$p\in \mathbf{N}$
.
If
$\Gamma $
acts geometrically densely on
${\mathcal X}_{{\mathbf K}}(p,\infty )$
with
$p\leq 2$
, then there is a measurable
$\Gamma $
-equivariant map
$\phi \colon B\to \mathcal I_p$
. Moreover, for almost all pairs
$(b,b')\in B^2$
,
$\phi (b)$
and
$\phi (b')$
are transverse.
If
$\Gamma \to \operatorname {\mathrm{PO}}_{{\mathbf K}}(p,\infty )$
is a representation with a Zariski-dense image, then there is a measurable
$\Gamma $
-equivariant map
$\phi \colon B\to \mathcal I_p$
. Moreover, for almost all pairs
$(b,b')\in B^2$
,
$\phi (b)$
and
$\phi (b')$
are transverse.
1.3 Geometry of chains
In [Reference CartanCar32], Cartan introduced a very nice geometry on the boundary
$\partial {\mathcal X}_{\mathbf{C}}(1,n)$
of the complex hyperbolic space. A chain in
$\partial {\mathcal X}_{\mathbf{C}}(1,n)$
is the boundary of a complex geodesic in
${\mathcal X}_{\mathbf{C}}(1,n)$
. It is an easy observation that any two distinct points in
$\partial {\mathcal X}_{\mathbf{C}}(1,n)$
define a unique chain; moreover, to determine if three points lie in a common chain, one can use a numerical invariant, the so-called Cartan invariant. Three points lie in a common chain if and only if they maximize the absolute value of the Cartan invariant. This invariant can be understood as an angle or the oriented area of the associated ideal triangle. See [Reference GoldmanGol99, §7.1].
As in [Reference PozzettiPoz15], we use a generalization of chains and of the Cartan invariant to prove our rigidity statements. For
$p\geq 1$
and
$q\in \mathbf{N}\cup \{\infty \}$
with
$q\geq p$
, we denote by
$\mathcal I_p(p,q)$
, or simply
$\mathcal I_p$
if the pair
$(p,q)$
is understood, the set of isotropic subspaces of dimension p in
${\mathbf{C}}^{p+q}$
. A p-chain (or simply a chain) in
$\mathcal I_p(p,q)$
is the image of a standard embedding of
$\mathcal I_p(p,p)$
in
$\mathcal I_p(p,q)$
. This corresponds to the choice of a linear subspace of
${\mathbf{C}}^{p+q}$
where the Hermitian form has signature
$(p,p)$
. A generalization of the Cartan invariant is realized by the Bergmann cocycle
$\beta \colon \mathcal I_p^3\to [-p,p]$
. Two transverse points in
$\mathcal I_p$
determine a unique chain and once again, three points in
$\mathcal I_p^3$
that maximize the absolute value of the Bergmann cocycle lie in a common chain.
The strategy of proof of Theorem 1.1 goes now as follows. We first reduce to geometrically dense representations (Proposition 5.15) if needed. Thanks to a now well-established formula in bounded cohomology (Proposition 5.10), we prove that a maximal representation of a lattice
$\Gamma \leq \operatorname {\mathrm{SU}}(1,n)$
in
$\operatorname {\textrm O}_{\mathbf{C}}(p,\infty )$
has to preserve the chain geometry and almost surely maps 1-chains to p-chains (Corollary 6.1).
1.4 Outline of the paper
Section 2 is devoted to the background on Riemannian and Hermitian symmetric spaces in infinite dimension. Section 3 focuses on algebraic and standard algebraic subgroups where Proposition 1.5 is proved. The existence of boundary maps is proved in §4. In §5, we provide a short summary of the basic definitions related to maximal representations, and adapt them in infinite dimension. Section 6 deals with representations in
$\operatorname {\mathrm{PO}}_{\mathbf{C}}(p,\infty )$
, where we prove Theorem 1.1. In §7, we study representations of fundamental groups of surfaces in
$\operatorname {\mathrm{PO}}_{\mathbf {R}}(2,\infty )$
and prove Theorem 1.2. The computation of the Toledo invariant for the variation on the complementary series is carried out in Appendix A.
2 Riemannian and Hermitian symmetric spaces of infinite dimension
2.1 Infinite-dimensional symmetric spaces
In this section, we recall definitions and facts about infinite-dimensional Riemannian symmetric spaces. By a Riemannian manifold, we mean a (possibly infinite-dimensional) smooth manifold modeled on some real Hilbert space with a smooth Riemannian metric. For a background on infinite-dimensional Riemannian manifolds, we refer to [Reference LangLan99] or [Reference PetersenPet06].
Remark 2.1. Implicitly, all Hilbert spaces considered in this paper will be separable. In particular, any two Hilbert spaces of infinite dimension over the same field will be isomorphic. The symmetric spaces studied below can be defined as well on non-separable Hilbert spaces but because we will consider representations of countable groups, we can always restrict ourselves to the separable case.
Let
$(M,g)$
be a Riemannian manifold, a symmetry at a point
$p\in M$
is an involutive isometry
$\sigma _p\colon M\to M$
such that
$\sigma _p(p) = p$
, and the differential at p is
$-\operatorname {\textrm {Id}}$
. A Riemannian symmetric space is a connected Riemannian manifold such that, at each point, there exists a symmetry. See [Reference DuchesneDuc15a, §3] for more details.
We will be interested in infinite-dimensional analogs of symmetric spaces of non-compact type. If
$(M,g)$
is a symmetric space of non-positive sectional curvature without local Euclidean factor, then for any point
$p\in M$
, the exponential
$\exp \colon T_pM\to M$
is a diffeomorphism and, if d is the distance associated to the metric g, then
$(M,d)$
is a CAT(0) space [Reference DuchesneDuc15a, Proposition 4.1]. So, such a space M has a very pleasant metric geometry and in particular, it has a visual boundary
$\partial M$
at infinity. If M is infinite-dimensional, then
$\partial M$
is not compact for the cone topology.
Let us describe the principal example of an infinite-dimensional Riemannian symmetric space of non-positive curvature.
Example 2.2. Let
${\mathcal H}$
be some real Hilbert space with orthogonal group
$\operatorname {\textrm {O}}(\mathcal {H})$
. We denote by
$\operatorname {\mathrm{L}}({\mathcal H})$
the set of bounded operators on
${\mathcal H}$
and by
$\operatorname {\textrm {GL}}({\mathcal H})$
the group of the invertible ones with continuous inverse. If
$A\in \operatorname {\mathrm{L}}({\mathcal H})$
, we denote its adjoint by
$^tA$
. An operator
$A\in \operatorname {\mathrm{L}}({\mathcal H})$
is Hilbert–Schmidt if
$\sum _{i,j}\langle Ae_i,Ae_j\rangle ^2<\infty $
, where
$(e_i)$
is some orthonormal basis of
${\mathcal H}$
. We denote by
$\operatorname {\mathrm{L}}^2({\mathcal H})$
the ideal of Hilbert–Schmidt operators and by
$\operatorname {\textrm {GL}}^2({\mathcal H})$
the elements of
$\operatorname {\textrm {GL}}({\mathcal H})$
that can be written
$\operatorname {\textrm {Id}}+A$
, where
$A\in \operatorname {\mathrm{L}}^2({\mathcal H})$
. This is a subgroup of
$\operatorname {\textrm {GL}}({\mathcal H})$
: the inverse of
$\operatorname {\textrm {Id}}+A$
is
$\operatorname {\textrm {Id}}-B$
with
$B=A(\operatorname {\textrm {Id}}+A)^{-1}=(\operatorname {\textrm {Id}}+A)^{-1}A\in \operatorname {\mathrm{L}}^2({\mathcal H})$
. We also set
$\operatorname {\textrm {O}}^2(\mathcal {H})=\operatorname {\textrm {O}}(\mathcal {H})\cap \operatorname {\textrm {GL}}^2(\mathcal {H})$
, and denote by
$\operatorname {\textrm {S}}^2(\mathcal {H})$
the closed subspace of symmetric operators in
$\operatorname {\mathrm{L}}^2({\mathcal H})$
and by
$\operatorname {\textrm {P}}^2(\mathcal {H})$
the set of symmetric positive definite operators in
$\operatorname {\textrm {GL}}^2({\mathcal H})$
.
Then
$\operatorname {\textrm {P}}^2(\mathcal {H})$
identifies with the quotient
$\operatorname {\textrm {GL}}^2({\mathcal H})/\operatorname {\textrm {O}}^2(\mathcal {H})$
under the action of
$\operatorname {\textrm {GL}}^2({\mathcal H})$
on
$\operatorname {\textrm {P}}^2(\mathcal {H})$
given by
$g\cdot x=gx^tg$
, where
$g\in \operatorname {\textrm {GL}}^2({\mathcal H})$
and
$x\in \operatorname {\textrm {P}}^2({\mathcal H})$
. The space
$\operatorname {\textrm {P}}^2(\mathcal {H})$
is actually a Riemannian manifold,
$\operatorname {\textrm {GL}}^2({\mathcal H})$
acts transitively by isometries, and the exponential map
$\exp \colon \operatorname {\textrm {S}}^2(\mathcal {H})\to \operatorname {\textrm {P}}^2(\mathcal {H})$
is a diffeomorphism. The metric at the origin
$o=\operatorname {\textrm {Id}}$
is given by
$\langle X,Y\rangle =\text {Trace}(XY)$
and it has non-positive sectional curvature. Then it is a complete simply connected Riemannian manifold of non-positive sectional curvature. This is a Riemannian symmetric space and the symmetry at the origin is given by
$x\mapsto x^{-1}$
.
A totally geodesic subspace of a Riemannian manifold
$(M,g)$
is a closed submanifold N such that for any
$x\in N$
and
$v\in T_xN\setminus \{0\}$
, the whole geodesic with direction v is contained in N. All the simply connected non-positively curved symmetric spaces that will appear in this paper are totally geodesic subspaces of the space
$\operatorname {\textrm {P}}^2(\mathcal {H})$
described in Example 2.2.
A Lie triple system of
$\operatorname {\textrm {S}}^2(\mathcal {H})$
is a closed linear subspace
$\mathfrak {p}<\operatorname {\textrm {S}}^2(\mathcal {H})$
such that for all
$X,Y,Z\in \mathfrak {p}$
,
$[X,[Y,Z]]\in \mathfrak {p}$
, where the Lie bracket
$[X,Y]$
is simply
$XY-YX$
. The totally geodesic subspaces N of
$\operatorname {\textrm {P}}^2(\mathcal {H})$
containing
$\operatorname {\textrm {Id}}$
are in bijection with the Lie triple systems
$\mathfrak {p}$
of
$\operatorname {\textrm {S}}^2(\mathcal {H})$
. This correspondence is given by
$N=\exp (\mathfrak {p})$
. See [Reference de la HarpedlH72, Proposition III.4].
All totally geodesic subspaces of
$\operatorname {\textrm {P}}^2(\mathcal {H})$
are symmetric spaces as well and satisfy a condition of non-positivity of the curvature operator. This condition of non-positivity of the curvature operator allows a classification of these symmetric spaces [Reference DuchesneDuc15a, Theorem 1.8]. In this classification, all the spaces that appear are the natural analogs of the classical finite-dimensional Riemannian symmetric spaces of non-compact type.
The isometry group of a finite-dimensional symmetric space is a real algebraic group and thus has a Zariski topology; this is no more available in infinite dimension. Let
$(M,g)$
be an irreducible symmetric space of finite dimension and non-positive sectional curvature, and let
$G\leq \operatorname {\mathrm{Isom}}(M)$
. It is well known that the group G is Zariski dense if and only if there is neither a fixed point at infinity nor an invariant totally geodesic strict subspace (possibly reduced to a point). Thus, following the ideas in [Reference Caprace and MonodCM09], we say that a group G acting by isometries on a (possibly infinite-dimensional) Riemannian symmetric space of non-positive curvature
${\mathcal X}$
is geometrically dense if G has no fixed point in
$\partial {\mathcal X}$
and no invariant closed totally geodesic strict subspace in
${\mathcal X}$
.
2.2 The Riemannian symmetric spaces
${\mathcal X}_{{\mathbf K}}(p,\infty )$
Throughout the paper,
$\textbf {H}$
denotes the division algebra of the quaternions, and
$\mathcal {H}$
is a separable Hilbert space over
${\textbf K}=\textbf {R}$
,
${\mathbf{C}}$
, or
$\textbf {H}$
of infinite dimension. In the latter case, the scalar multiplication is understood to be on the right. We denote by
$\operatorname {\mathrm{L}}(\mathcal {H})$
the algebra of all bounded
${\textbf K}$
-linear operators of
$\mathcal {H}$
, and
$\operatorname {\textrm {GL}}(\mathcal {H})$
is the group of all bounded invertible
${\textbf K}$
-linear operators with bounded inverse. Using the real Hilbert space
$\mathcal {H}_{\mathbf {R}}$
underlying
$\mathcal {H}$
, one can consider
$\operatorname {\textrm {GL}}(\mathcal {H})$
as a closed subgroup of
$\operatorname {\textrm {GL}}(\mathcal {H}_{\mathbf {R}})$
. We denote by
$A^*$
the adjoint of
$A\in \operatorname {\mathrm{L}}({\mathcal H})$
. In particular, when
${\textbf K}=\textbf {R}$
,
$A^*= ^t\!A$
.
Let
$p\in \mathbf{N}$
. We fix an orthonormal basis
$(e_i)_{i\in \mathbf{N}}$
of the separable Hilbert space
$\mathcal {H}$
, and we consider the Hermitian form
$$ \begin{align*}Q(x)=\sum_{i=1}^p\overline x_ix_i-\sum_{i>p}\overline x_ix_i,\end{align*} $$
where
$x=\sum e_ix_i$
. The isometry group of this quadratic form will be denoted
$\operatorname {{\textrm O_{\mathbf {K}}}}(Q)$
or equivalently
$\operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )$
.
The intersection of
$\operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )$
and the orthogonal group of
$\mathcal {H}$
is isomorphic to
$\operatorname {{\textrm O_{\mathbf {K}}}}(p)\times \operatorname {{\textrm O_{\mathbf {K}}}}(\infty )$
, where
$\operatorname {{\textrm O_{\mathbf {K}}}}(p)$
(respectively
$\operatorname {{\textrm O_{\mathbf {K}}}}(\infty )$
) is the orthogonal group of the separable Hilbert space of dimension p (respectively of infinite dimension). Then the quotient
$$ \begin{align*}{\mathcal X}={\mathcal X}_{{\mathbf K}}(p,\infty)= \operatorname{{\textrm O_{\mathbf{K}}}}(p,\infty)/(\operatorname{{\textrm O_{\mathbf{K}}}}(p)\times\operatorname{{\textrm O_{\mathbf{K}}}}(\infty))\end{align*} $$
has a structure of an infinite-dimensional irreducible Riemannian symmetric space of non-positive curvature. This can be seen by the identification of
${\mathcal X}_{{\mathbf K}}(p,\infty )$
with the set
$$ \begin{align*}\mathcal{V}=\{V\leq{\mathcal H},\ \dim_{{\mathbf K}}(V)=p,\ Q|_V>0\}.\end{align*} $$
The group
$\operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )$
acts transitively on
$\mathcal {V}$
(by Witt’s theorem) and the stabilizer of the span of the p first vectors is
$\operatorname {{\textrm O_{\mathbf {K}}}}(p)\times \operatorname {{\textrm O_{\mathbf {K}}}}(\infty )$
. Moreover, the subgroup
$\operatorname {\textrm {O}}_{{\mathbf K}}^2(p,\infty )=\operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )\cap \operatorname {\textrm {GL}}^2({\mathcal H})$
also acts transitively on
$\mathcal {V}$
and thus
$$ \begin{align*}{\mathcal X}_{{\mathbf K}}(p,\infty)\simeq \mathcal{V}\simeq \operatorname{\textrm{O}}_{{\mathbf K}}^2(p,\infty)/(\operatorname{{\textrm O_{\mathbf{K}}}}(p)\times\operatorname{\textrm{O}}_{{\mathbf K}}^2(\infty)).\end{align*} $$
The stabilizer of the origin in the action of
$\operatorname {\textrm {O}}_{{\mathbf K}}^2(p,\infty )$
on
$\operatorname {\textrm {P}}^2({\mathcal H}_{\mathbf {R}})$
is exactly
$\operatorname {{\textrm O_{\mathbf {K}}}}(p)\times \operatorname {\textrm {O}}_{{\mathbf K}}^2(\infty )$
and the orbit of
$\operatorname {\textrm {O}}_{{\mathbf K}}^2(p,\infty )$
in
$\operatorname {\textrm {P}}^2({\mathcal H}_{\mathbf {R}})$
is a totally geodesic subspace [Reference DuchesneDuc13, Proposition 2.3]. Thus
${\mathcal X}_{{\mathbf K}}(p,\infty )$
has a structure of a simply connected non-positively curved Riemannian symmetric space.
Observe that when
${\textbf K}=\textbf {R}$
or
${\mathbf{C}}$
, homotheties act trivially on
${\mathcal X}_{{\mathbf K}}(p,\infty )$
and thus the group
$\operatorname {\mathrm{PO}}_{{\mathbf K}}(p,\infty )$
, defined to be
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )/\{\unicode{x3bb} \operatorname {\textrm {Id}},\ |\unicode{x3bb} |=1\}$
, acts by isometries on
${\mathcal X}_{{\mathbf K}}(p,\infty )$
. Moreover, it is proved in [Reference DuchesneDuc13, Theorem 1.5] that this is exactly the isometry group of
${\mathcal X}_{{\mathbf K}}(p,\infty )$
when
${\textbf K}=\textbf {R}$
.
Definition 2.3. Let
${\mathcal X}_1,{\mathcal X}_2$
be two symmetric spaces of type
${\mathcal X}_{{\mathbf K}}(p_i,q_i)$
, where
$p_i\leq q_i\in \mathbf{N}\cup \{\infty \}$
, corresponding to Hilbert spaces
${\mathcal H}_1,{\mathcal H}_2$
and Hermitian forms
$Q_1,Q_2$
. By a standard embedding, we mean the data of a linear map
$f\colon {\mathcal H}_1\to {\mathcal H}_2$
such that
$Q_2(f(x),f(y))=Q_1(x,y)$
for all
$x,y\in {\mathcal H}_1$
. The group
$\operatorname {\textrm {O}}_{{\mathbf K}}(Q_1)$
embeds in
$\operatorname {\textrm {O}}_{{\mathbf K}}(Q_2)$
in the following way: f intertwines the actions on
${\mathcal H}_1$
and
$f({\mathcal H}_1)$
and the action is trivial on the orthogonal of
$f({\mathcal H}_1)$
, which is a supplementary of
$f({\mathcal H}_1)$
because
$Q_2$
is non-degenerate on
$f({\mathcal H}_1)$
.
Finally the totally geodesic embedding
${\mathcal X}_1\hookrightarrow {\mathcal X}_2$
is given by the orbit of the identity under the action of the orthogonal group of
$Q_1$
.
The spaces
${\mathcal X}_{{\mathbf K}}(p,\infty )$
, with p finite, are very special among infinite-dimensional Riemannian symmetric spaces: they have finite rank, which is p. This means there are totally geodesic embeddings of
$\textbf {R}^p$
in
${\mathcal X}_{{\mathbf K}}(p,\infty )$
but there are no totally geodesic embeddings of
$\textbf {R}^q$
for
$q>p$
. Furthermore, every infinite-dimensional irreducible Riemannian symmetric space of non-positive curvature operator and finite rank arises in this way [Reference DuchesneDuc15a].
This finite-rank property gives some compactness on
$\overline {{\mathcal X}}={\mathcal X}\cup \partial {\mathcal X}$
for a weaker topology [Reference Caprace and LytchakCL10, Remark 1.2]. Moreover, we have the following important property.
Proposition 2.4. [Reference DuchesneDuc13, Proposition 2.6]
Any finite configuration of points, geodesics, points at infinity, flat subspaces of
${\mathcal X}_{{\mathbf K}}(p,\infty )$
is contained in some finite-dimensional totally geodesic subspace of
${\mathcal X}_{{\mathbf K}}(p,\infty )$
which is a standard embedding of
${\mathcal X}_{{\mathbf K}}(p,q)$
with
$q\in \mathbf{N}$
.
The boundary at infinity
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
has a structure of a spherical building, which we now recall. We refer to [Reference Abramenko and BrownAB08] for general definitions and facts about buildings, and to [Reference DuchesneDuc13] for the specific case in which we are interested. The space
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
has a natural structure of a simplicial complex (of dimension
$p-1$
): a simplex (of dimension
$r-1$
) in
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
is defined by a flag
$(V_1\subsetneq \cdots \subsetneq V_r)$
, where all the
$V_i$
are non-zero totally isotropic subspaces of
${\mathcal H}$
. In particular, vertices of
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
correspond to totally isotropic subspaces. A simplex A is contained in a simplex B if all the subspaces appearing in the flag A also appear in the flag B.
Each vertex has a type, which is a number between
$1$
and p given by the dimension of the corresponding isotropic subspace. More generally, the type of a cell is the finite increasing sequence of dimensions of the isotropic subspaces in the associated isotropic flag.
Definition 2.5. Two vertices of
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
, corresponding to isotropic spaces V and W, are opposite if the restriction of Q to
$V+W$
is non-degenerate, which means
$W\cap V^{\bot }=0$
.
Two simplices of the same type, corresponding to two flags
$(V_1\subset \cdots \subset V_r)$
and
$(W_1\subset \cdots \subset W_r)$
of the same type, are opposite if their vertices of the same type are opposite.
Remark 2.6. In terms of CAT(0) geometry, we note that two vertices of
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
of the same type are opposite if and only if they are joined by a geodesic line in
${\mathcal X}_{{\mathbf K}}(p,\infty )$
. For vertices of dimension p, opposition is equivalent to transversality: two vertices
$V,W$
with
$\dim (V)=\dim (W)=p$
are opposite if and only if
$V\cap W=0$
.
2.3 Hermitian symmetric spaces
Let
$(M,g)$
be a Riemannian manifold (possibly of infinite dimension). An almost complex structure is a
$(1,1)$
-tensor J such that for any vector field X,
$J(J(X))=-X$
. A triple
$(M,g,J)$
, where
$(M,g)$
is a Riemannian manifold and J is an almost complex structure, is a Hermitian manifold if for all vector fields
$X,Y$
,
$g(J(X),J(Y))=g(X,Y)$
. If
$(M,g,J)$
is a Hermitian manifold, we define a 2-form
$\omega $
by the formula
$\omega (X,Y)=g(J(X),Y)$
. A Kähler manifold is a Hermitian manifold such that
$d\omega =0$
and
$\omega $
is the Kähler form on M.
Let
$(M,g,J)$
be a Hermitian manifold and
$\nabla $
be the Levi-Civita connection associated to the Riemannian metric g. The almost-complex structure J is parallel if
$\nabla J=0$
, that is, if for all vector fields
$X,Y$
,
$\nabla _X(JY)=J(\nabla _XY)$
. This parallelism condition implies that
$\omega $
is parallel as well, that is, for all vector fields
$X,Y,Z$
,
$(\nabla _X\omega )(Y,Z)=0$
. Because
$d\omega (X,Y,Z)=(\nabla _X\omega )(Y,Z)-(\nabla _Y\omega )(X,Z)+(\nabla _Z\omega )(X,Y)$
, the parallelism condition
$\nabla J=0$
implies that
$\omega $
is closed.
Let N be the Nijenhuis (2,0)-tensor on M, that is, for all vector fields
$X,Y$
,
$$ \begin{align*}N(X,Y)=2([X,Y]-[J(X),J(Y)]-J[J(X),Y]-J[X,J(Y)]).\end{align*} $$
The parallelism of J implies that this tensor vanishes. An almost-complex structure J with vanishing Nijenhuis tensor is called a complex structure. Thus a parallel almost-complex structure is a complex structure.
Definition 2.7. Let
$(M,g)$
be a simply connected Riemannian symmetric space of non-positive sectional curvature. The symmetric space M is said to be a Hermitian symmetric space if it admits a Hermitian almost-complex structure J that is invariant under symmetries. This means that for any
$p,q\in M$
,
$$ \begin{align*}d\sigma_p\circ J_{\sigma_p(q)}\circ d\sigma_p=J_q\end{align*} $$
on the tangent space
$T_qM$
. One also says that the symmetries are holomorphic.
Let us recall some notation in
$\operatorname {\textrm {P}}^2(\mathcal {H})$
. We denote by o the origin in
$\operatorname {\textrm {P}}^2(\mathcal {H})$
, that is, the identity
$\operatorname {\textrm {Id}}$
of
${\mathcal H}$
. The symmetry
$\sigma _o$
at the origin is the map
$x\mapsto x^{-1}$
. The action
$\tau $
of
$\operatorname {\textrm {GL}}^2({\mathcal H})$
on
$\operatorname {\textrm {P}}^2({\mathcal H})$
is given by
$\tau (g)(x)=gx^tg$
. In particular, one has the relation
$$ \begin{align*}\sigma_o\circ\tau(g)=\tau(g^{-1})\circ\sigma_o.\end{align*} $$
The exponential map
$\exp \colon \operatorname {\mathrm{L}}^2({\mathcal H})\to \operatorname {\textrm {GL}}^2({\mathcal H})$
is a local diffeomorphism around the origin and it induces a diffeomorphism
$\exp \colon \operatorname {\textrm {S}}^2({\mathcal H})\to \operatorname {\textrm {P}}^2({\mathcal H})$
. In particular, we identify the tangent space at the origin
$T_o\operatorname {\textrm {P}}^2({\mathcal H})$
with the Hilbert space
$\operatorname {\textrm {S}}^2({\mathcal H})$
. The isotropy group of the origin, that is, the fixator of o, is
$\operatorname {\textrm {O}}^2({\mathcal H})$
. It acts also on
$\operatorname {\textrm {S}}^2({\mathcal H})$
by
$g\cdot v=gv^tg$
and one has
$g\exp (v)^tg=\exp (gv^tg)$
for all
$v\in \operatorname {\textrm {S}}^2({\mathcal H})$
and
$g\in \operatorname {\textrm {O}}^2({\mathcal H})$
. If K is a subgroup of
$\operatorname {\textrm {O}}^2({\mathcal H})$
, we denote by
$K^*$
its image in the isometry group of
$\operatorname {\textrm {S}}^2({\mathcal H})$
.
The following proposition is a mere extension of a classical statement in finite dimension to our infinite-dimensional setting (see for example [Reference HelgasonHel01, Proposition VIII.4.2]). It can be proved with the same methods.
Proposition 2.8. Let
$(M,g)$
be a totally geodesic subspace of the symmetric space
$\operatorname {\textrm {P}}^2(\mathcal {H})$
containing o (the identity element) and corresponding to the Lie triple system
$\mathfrak {p}$
. Let G be the connected component of the stabilizer of M in
$\operatorname {\textrm {GL}}^2(\mathcal {H})$
and let K be the isotropy subgroup of o in G. Assume there is an operator
$J_0\colon \mathfrak {p}\to \mathfrak {p}$
such that:
-
(1)
$J_0^2=-\operatorname {\mathrm{Id}}$
; -
(2)
$J_0$
is an isometry; and -
(3)
$J_0$
commutes with all elements of
$K^*$
.
Then there is a unique G-invariant almost-complex structure J on M which coincides with
$J_0$
on
$T_oM$
. Moreover, J is Hermitian and parallel. Thus,
$(M,g,J)$
is a Hermitian symmetric space and a Kähler manifold.
Remark 2.9. It is well known that a finite-dimensional manifold with a complex structure J is a complex manifold, that is, a manifold modeled on
${\mathbf{C}}^n$
with holomorphic transition maps. The same result does not hold in full generality for infinite-dimensional manifolds but in the case of real analytic Banach manifolds, the result still holds [Reference BeltitaBel05, Theorem 7]. The Hermitian symmetric spaces we consider have a real analytic complex structure and thus are complex manifolds. Nonetheless, we will not need this result.
In the remainder of this section, we exhibit the complex structures J on two classes of Hermitian symmetric spaces we will use later in the paper. Thanks to Proposition 2.8, it suffices to find
$J_0$
with the required properties. The complex structures we describe are all the natural analogs of the corresponding complex structures in finite dimension.
Below, we use orthogonal decompositions
${\mathcal H}=V\oplus W$
and block decompositions for elements of
$\operatorname {\mathrm{L}}({\mathcal H})$
. When we write
$g=\Big [\begin {smallmatrix} A &B\\ C & D \end {smallmatrix}\Big ]$
, this means that
$A=\pi _V\circ g|_V\in \operatorname {\mathrm{L}}(V)$
,
$B=\pi _V\circ g|_W\in \operatorname {\mathrm{L}}(W,V)$
,
$C=\pi _W\circ g|_V\in \operatorname {\mathrm{L}}(V,W)$
, and
$D=\pi _W\circ g|_W\in \operatorname {\mathrm{L}}(W)$
.
2.3.1 The Hermitian symmetric space
${\mathcal X}_{\mathbf{C}}(p,\infty )$
Let
$\mathcal {H}$
be a complex Hilbert space of infinite dimension. We denote by
${\mathcal H}_{\mathbf {R}}$
the underlying real Hilbert space. Let
$V,W$
be closed orthogonal complex subspaces of dimension
$p\in \mathbf{N}\cup \{\infty \}$
and
$\infty $
such that
${\mathcal H}=V\oplus W$
. Let
$I_{p,\infty }=\operatorname {\textrm {Id}}_V\oplus -\operatorname {\textrm {Id}}_W$
. Thus,
$$ \begin{align*}\operatorname{\textrm{O}}^2_{\mathbf{C}}(p,\infty)=\{g\in\operatorname{\textrm{GL}}^2({\mathcal H}),\ g^*I_{p,\infty}g=I_{p,\infty}\}.\end{align*} $$
The symmetric space
${\mathcal X}_{\mathbf{C}}(p,\infty )$
is the
$\operatorname {\textrm {O}}^2_{\mathbf{C}}(p,\infty )$
-orbit of the identity in
$\operatorname {\textrm {P}}^2({\mathcal H}_{\mathbf {R}})$
, that is, the image under the exponential map of the Lie triple system
$$ \begin{align*}\mathfrak p=\left\{\left[\begin{array}{@{}ll}0&A\\A^*&0\end{array}\right],\ A\in\operatorname{\textrm{L}}^2(W,V)\right\}.\end{align*} $$
The complex structure is induced by the endomorphism
$J_0$
of
$\mathfrak p$
defined by
$J_0\Big [\begin {smallmatrix}0&A\\A^*&0\end {smallmatrix}\Big ]=\Big [\begin {smallmatrix}0&iA\\-iA^*&0\end {smallmatrix}\Big ]$
. Because the stabilizer of
$\operatorname {\textrm {Id}}_{{\mathcal H}_{\mathbf {R}}}$
in
$\operatorname {\textrm {O}}_{\mathbf{C}}(p,\infty )$
is given by all the operators that can be expressed as
$\Big [\begin {smallmatrix}P&0\\0&Q\end {smallmatrix}\Big ]$
with
$P\in \operatorname {\textrm {O}}^2_{\mathbf{C}}(V)$
and
$Q\in \operatorname {\textrm {O}}^2_{\mathbf{C}}(W)$
,
$J_0$
satisfies the conditions of Proposition 2.8.
2.3.2 The Hermitian symmetric space
${\mathcal X}_{\mathbf {R}}(2,\infty )$
Let
${\mathcal H}$
be a real Hilbert space of infinite dimension. Let
$V,W$
be closed orthogonal subspaces of dimension two and
$\infty $
such that
${\mathcal H}=V\oplus W$
. Let
$I_{2,\infty }=\operatorname {\textrm {Id}}_V\oplus -\operatorname {\textrm {Id}}_W$
. Thus,
$$ \begin{align*}\operatorname{\textrm{O}}^2_{\mathbf{R}}(2,\infty)=\{g\in\operatorname{\textrm{GL}}^2({\mathcal H}),\ ^tgI_{2,\infty}g=I_{2,\infty}\}.\end{align*} $$
The symmetric space
${\mathcal X}_{\mathbf {R}}(2,\infty )$
is the
$\operatorname {\textrm {O}}^2_{\mathbf {R}}(2,\infty )$
-orbit of the identity in
$\operatorname {\textrm {P}}^2({\mathcal H})$
, that is, the image under the exponential map of the Lie triple system
$$ \begin{align*}\mathfrak p=\left\{\left[\begin{array}{@{}ll}0&A\\{}^tA&0\end{array}\right],\ A\in\operatorname{\textrm{L}}(W,V)\right\}.\end{align*} $$
Fix some orthonormal basis
$(e_1,e_2)$
of V and let
$I=\Big [\begin {smallmatrix}0&-1\\1&0\end {smallmatrix}\Big ]$
. This element belongs to the group
$\operatorname {\textrm {SO}}_{\mathbf {R}}(2)$
, which is commutative. The complex structure is defined by
$J_0\Big [\begin {smallmatrix}0 &A \\ {}^t A &0 \end {smallmatrix}\Big ]=\Big [\begin {smallmatrix}0 &IA \\-^{t}(IA)&0\end {smallmatrix}\Big ]$
. Because the stabilizer of
$\operatorname {\textrm {Id}}_{\mathcal H}$
in the identity component of
$\operatorname {\textrm {O}}^2_{\mathbf {R}}(2,\infty )$
is given by operators of the form
$\Big [\begin {smallmatrix}P &0\\0 &Q\end {smallmatrix}\Big ]$
, with
$P\in \operatorname {\textrm {SO}}_{\mathbf {R}}(2)$
and
$Q\in \operatorname {\textrm {O}}^2_{\mathbf {R}}(\infty )$
, let us denote by
$\operatorname {\textrm {O}}^{+}_{\mathbf {R}}(2,\infty )$
the set of elements in
$\operatorname {\textrm {O}}_{\mathbf {R}}(2,\infty )$
of the form
$\Big [\begin {smallmatrix} A &B\\ C & D \end {smallmatrix}\Big ]$
, where A has positive determinant. So there is a
$\operatorname {\textrm {O}}^{+}_{\mathbf {R}}(2,\infty )$
-invariant complex structure on
${\mathcal X}_{\mathbf {R}}(2,\infty )$
. Let us denote by
$\operatorname {\mathrm{PO}}_{\mathbf {R}}^+(2,\infty )$
the image of
$\operatorname {\textrm {O}}^{+}_{\mathbf {R}}(2,\infty )$
under the quotient map
$\operatorname {\textrm {O}}_{\mathbf {R}}(2,\infty )\to \operatorname {\mathrm{PO}}_{\mathbf {R}}(2,\infty )$
.
2.4 Tube-type Hermitian symmetric spaces
In finite dimension, the class of Hermitian symmetric spaces splits into two classes: those of tube type and those that are not of tube type. This distinction is important for the approach we use to understand maximal representations. For a definition in finite dimension, we refer to [Reference Burger, Iozzi and WienhardBIW09]. Let us briefly recall that if
${\mathcal X}$
is a finite-dimensional Hermitian symmetric space, a chain is the boundary (as a subset of the Shilov boundary of
${\mathcal X}$
) of a maximal tube-type subspace. By definition, if
${\mathcal X}$
is of tube type, there is a unique maximal tube-type subspace:
${\mathcal X}$
itself. However, if
${\mathcal X}$
is not of tube type, chains lie in a unique
$\operatorname {\mathrm{Isom}}({\mathcal X})$
-orbit and it yields a new incidence geometry: the chain geometry (see [Reference PozzettiPoz15, §3]). Let us give an ad hoc definition of tube-type Hermitian symmetric spaces in infinite dimension.
Definition 2.10. An irreducible Hermitian symmetric space is of tube type if there is a dense increasing union of tube-type finite-dimensional totally geodesic holomorphic Hermitian symmetric subspaces.
Lemma 2.11. The Hermitian symmetric spaces
${\mathcal X}_{\mathbf{C}}(\infty ,\infty )$
,
${\mathcal X}_{\mathbf {R}}(2,\infty )$
,
$\operatorname {\mathrm{Sp}}^2({\mathcal H})/\operatorname {\mathrm{U}}^2({\mathcal H})$
, and the space
$\operatorname {\mathrm{O}}^{*2}(\infty )/\operatorname {\mathrm{U}}^2(\infty )$
are of tube type.
The Hermitian symmetric space
${\mathcal X}_{\mathbf{C}}(p,\infty )$
with
$p<\infty $
is not of tube type.
Proof. The four first cases are simply the closure of an increasing union of Hermitian totally geodesic holomorphic subspaces isomorphic to respectively
${\mathcal X}_{\mathbf{C}}(n,n),{\mathcal X}_{\mathbf {R}}(2,n), \operatorname {\textrm {Sp}}(2n)/\operatorname {\mathrm{U}}(n)$
, and
$\text {SO}^{*}(4n)/\operatorname {\mathrm{U}}(2n)$
. All those spaces are of tube type.
For
${\mathcal X}_{\mathbf{C}}(p,\infty )$
with
$p<\infty $
, we know that any finite-dimensional totally geodesic and holomorphic Hermitian symmetric subspace Y is contained in some standard copy of
${\mathcal X}_{\mathbf{C}}(p,q)$
for
$q>p$
large enough. In particular, if Y is of tube type, then it lies in some standard copy of
${\mathcal X}_{\mathbf{C}}(p,p)$
and thus standard copies of
${\mathcal X}_{\mathbf{C}}(p,p)$
are maximal finite-dimensional Hermitian symmetric subspaces of tube type.
Remark 2.12. Among the infinite-dimensional Hermitian symmetric spaces of tube type,
${\mathcal X}_{\mathbf {R}}(2,\infty )$
is remarkable. This is the only one to be of tube type and of finite rank.
Remark 2.13. A theory of tube-type domains and Jordan algebras in infinite dimension has been developed. We refer to [Reference Kaup and UpmeierKU77] and references for an entrance to this subject. We do not rely on this theory.
3 Algebraic groups in infinite dimension
3.1 Algebraic subgroups of bounded operators of Hilbert spaces
Algebraic subgroups of finite-dimensional linear Lie groups are well understood and equipped with a useful topology: the Zariski topology. In infinite dimension, some new and baffling phenomena may appear. For example, one-parameter subgroups may be non-continuous. In [Reference Harris and KaupHK77], Harris and Kaup introduced the notion of linear algebraic groups and showed that they behave nicely with respect to the exponential map. In particular, the exponential map is a local homeomorphism and any point sufficiently close to the origin lies in some continuous one-parameter subgroup.
Let
$A,B$
be two real Banach algebras and let
$G(A)$
be the set of all invertible elements of A. A map
$f\colon A\to B$
is a homogeneous polynomial map of degree n if there is a continuous n-linear map
$f_0\in \operatorname {\mathrm{L}}^n(A,B)$
such that for any
$a\in A$
,
$f(a)=f_0(a,\ldots ,a)$
. Now, a map
$f\colon A\to B$
is polynomial if it is a finite sum of homogeneous polynomial maps. Its degree is the maximum of the degrees that appear in the sum.
Let
${\mathcal H}$
be a real Hilbert space. The Banach algebras we will use are
$\operatorname {\mathrm{L}}({\mathcal H})$
endowed with the operator norm and the field of real numbers
$\textbf {R}$
. The group of invertible elements in
$\operatorname {\mathrm{L}}({\mathcal H})$
is
$\operatorname {\textrm {GL}}({\mathcal H})$
.
Definition 3.1. A subgroup G of
$G(A)$
is an algebraic subgroup if there is a constant n and a set
$\mathcal {P}$
of polynomial maps of degrees at most n on
$A\times A$
such that
$$ \begin{align*}G=\{g\in G(A);\ P(g,g^{-1})=0,\text{ for all } P\in \mathcal{P}\}.\end{align*} $$
Observe that
$\mathcal {P}$
may be infinite but the degrees of its elements are uniformly bounded. The main result of [Reference Harris and KaupHK77] is the fact that an algebraic subgroup is a Banach Lie group (with respect to the norm topology) and that the exponential map gives a homeomorphism in a neighborhood of the identity.
Remark 3.2. In this context, one could define a generalized Zariski topology by choosing the smallest topology such that zeros of polynomial maps (or standard polynomial maps, see below) are closed. This topology behaves differently from the finite-dimensional case because the Noetherian property does not hold. We will not use any such topology.
Moreover, the intersection of an infinite number of algebraic subgroups has no reason to be an algebraic group. Degrees of the defining polynomials may be unbounded.
Examples 3.3.
-
(1) Let
${\mathcal H}$
be a Hilbert space of infinite dimension over
${\mathbf{C}}$
and let
${\mathcal H}_{\mathbf {R}}$
be the underlying real Hilbert space. Let I be the multiplication by the complex number i. Then I is an isometry of
${\mathcal H}_{\mathbf {R}}$
of order two and Because the map
$$ \begin{align*}\operatorname{\textrm{GL}}({\mathcal H})=\{g\in\operatorname{\textrm{GL}}({\mathcal H}_{\mathbf{R}}), gI=Ig\}.\end{align*} $$
$M\mapsto MI-IM$
is linear on
$\operatorname {\mathrm{L}}({\mathcal H})$
,
$\operatorname {\textrm {GL}}({\mathcal H})$
is an algebraic subgroup of
$\operatorname {\textrm {GL}}({\mathcal H}_{\mathbf {R}})$
. Similarly, if
$\textbf {H}$
is the field of quaternions and
${\mathcal H}$
is a Hilbert space over
$\textbf {H}$
(with underlying real Hilbert space
${\mathcal H}_{\mathbf {R}}$
), then
$\operatorname {\textrm {GL}}({\mathcal H})$
is an algebraic subgroup of
$\operatorname {\textrm {GL}}({\mathcal H}_{\mathbf {R}})$
.
-
(2) Let
${\mathcal H}$
be a Hilbert space of infinite dimension over
${\textbf K}$
and
${\mathcal H}=V\oplus W$
be an orthogonal splitting where V has dimension
$p\in \mathbf{N}$
. Let
$I_{p,\infty }$
be the linear map
$\operatorname {\textrm {Id}}_V\oplus -\operatorname {\textrm {Id}}_W$
. By definition, the group
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )$
is and because the map
$$ \begin{align*}\operatorname{\textrm{O}}_{{\mathbf K}}(p,\infty)=\{g\in\operatorname{\textrm{GL}}({\mathcal H}),g^*I_{p,\infty}g=I_{p,\infty}\}\end{align*} $$
$(L,M)\mapsto L^*I_{p,\infty }M$
is bilinear on
$\operatorname {\mathrm{L}}({\mathcal H}_{\mathbf {R}})\times \operatorname {\mathrm{L}}({\mathcal H}_{\mathbf {R}})$
, the group
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )$
is a (real) algebraic subgroup of
$\operatorname {\textrm {GL}}({\mathcal H}_{\mathbf {R}})$
. This is a particular case of [Reference Harris and KaupHK77, Example 4].
In finite dimension, linear algebraic groups of
$\operatorname {\textrm {GL}}_n(\textbf {R})$
are given by polynomial equations in matrix coefficients. We generalize this notion to subgroups of
$\operatorname {\textrm {GL}}(\mathcal {H})$
.
Definition 3.4. Let
${\mathcal H}$
be a real Hilbert space. A matrix coefficient is a linear map
$f\colon \operatorname {\mathrm{L}}({\mathcal H})\to \textbf {R}$
such that there are
$x,y\in {\mathcal H}$
such that
$f(L)=\langle Lx,y\rangle $
for any
$L\in \operatorname {\mathrm{L}}({\mathcal H})$
. A homogeneous polynomial map P on
$\operatorname {\mathrm{L}}({\mathcal H})\times \operatorname {\mathrm{L}}({\mathcal H})$
of degree d is standard if there is an orthonormal basis
$(e_n)_{n\in \mathbf{N}}$
of
${\mathcal H}$
, non-negative integers
$m,l$
such that
$d=m+l$
, and families of real coefficients
$(\unicode{x3bb} _i)_{i\in \mathbf{N}^{2m}}$
,
$(\mu _j)_{j\in \mathbf{N}^{2l}}$
such that for all
$(M,N)\in \operatorname {\mathrm{L}}({\mathcal H})\times \operatorname {\mathrm{L}}({\mathcal H})$
,
$P(M,N)$
can be written as an absolutely convergent series
where
$P_i(M)\!=\!\prod _{k\!=\!0}^{m-1}\langle Me_{i_{2k}},e_{i_{2k+1}}\rangle $
for
$i\in \mathbf{N}^{2m}$
and similarly
$P_j(N)\!=\!\prod _{k\!=\!0}^{l-1}\langle Ne_{i_{2k}},e_{i_{2k+1}}\rangle $
for
$j\in \mathbf{N}^{2l}$
.
A polynomial map is standard if it is a finite sum of standard homogeneous polynomial maps. A subgroup of
$\operatorname {\textrm {GL}}({\mathcal H})$
is a standard algebraic subgroup if it is an algebraic subgroup defined by a family of standard polynomials.
Examples 3.5.
-
(1) Any matrix coefficient is a standard homogeneous polynomial map of degree one. For
$x,y\in {\mathcal H}$
and any orthonormal basis
$(e_n)_{n\in \mathbf{N}}$
, we define
$x_n=\langle x,e_n\rangle $
and similarly
$y_n=\langle y,e_n\rangle $
for any
$n\in \mathbf{N}$
. Then the matrix coefficient
$P(M)=\langle Mx,y\rangle $
is given by the series For any finite subset
$$ \begin{align*}\sum_{i,j\in\textbf{N}}x_iy_j\langle Me_i,e_j\rangle.\end{align*} $$
$K\subset \mathbf{N}^2$
finite containing
$\{1,\ldots ,n\}^2$
, where
$$ \begin{align*}\bigg|\sum_{i,j\leq n}x_iy_j\langle Me_i,e_j\rangle-\langle Mx,y\rangle\bigg|\leq \|M\| (\Vert x\Vert \Vert \pi_n(y)-y\Vert+\Vert y\Vert \Vert \pi_n(x)-x\Vert ),\end{align*} $$
$\pi _n(x)$
is the projection on the space spanned by the n first vectors of the basis. This implies that the series is absolutely convergent (see [Reference ChoquetCho69, VII-3-§8]).
-
(2) The group
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )$
is not only an algebraic group, it is also a standard algebraic group. Let
$(e_i)_{i\in \mathbf{N}}$
be an orthonormal basis of
${\mathcal H}$
adapted to the decomposition
${\mathcal H}=V\oplus W$
as in Example 3.3. An element
$g\in \operatorname {\textrm {GL}}({\mathcal H})$
is in
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )$
if and only if
$\langle I_{p,\infty }ge_i,ge_j\rangle =\langle I_{p,\infty }e_i,e_j\rangle $
for any
$i,j\in \mathbf{N}$
. Because
$\langle I_{p,\infty }ge_i,ge_j\rangle =\sum _{k\in \mathbf{N}}\langle ge_i,I_{p,\infty }e_k\rangle \langle g e_j,e_k\rangle $
and the coefficient
$\langle ge_i,I_{p,\infty }e_k\rangle $
is
$\varepsilon _k\langle ge_i,e_k\rangle $
with
$\varepsilon _k=-1$
for
$m\leq p$
and
$\varepsilon _k=1$
for
$m\geq p$
, we see that
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )$
is a standard algebraic group. -
(3) Let
${\mathcal H}$
be a real Hilbert space and
$V<{\mathcal H}$
be a closed subspace, then
$H=\operatorname {\textrm {Stab}}(V)$
is a standard algebraic subgroup of
$\operatorname {\textrm {GL}}({\mathcal H})$
. Actually, and thus is a standard algebraic subgroup of
$$ \begin{align*}H=\{g\in\operatorname{\textrm{GL}}({\mathcal H}),\ \langle gx,y\rangle=0,\text{ for all } x\in V,y\in V^{\bot}\},\end{align*} $$
$\operatorname {\textrm {GL}}({\mathcal H})$
.It follows that stabilizers of simplices of the building at infinity of
${\mathcal X}_{{\mathbf K}}(p,\infty )$
are standard algebraic subgroups of
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )$
. Moreover, if
$\xi $
is a point at infinity, its stabilizer coincides with the stabilizer of the minimal simplex that contains it. See [Reference DuchesneDuc13, Proposition 6.1] for details in the real case. The same argument works as well over
${\mathbf{C}}$
and
$\textbf {H}$
. In particular, stabilizers of points at infinity are standard algebraic subgroups.
Let H be a standard algebraic subgroup of
$\operatorname {\textrm {GL}}({\mathcal H})$
. If E is a finite-dimensional subspace of
${\mathcal H}$
, we denote by
$H_E$
the subgroup of elements
$g\in H$
such that
$g(E)=E$
,
$g|_{E^{\bot }}=\operatorname {\textrm {Id}}$
. We identify
$H_E$
with an algebraic subgroup of
$\operatorname {\textrm {GL}}(E)$
. By a strict algebraic group H, we mean that H is algebraic and
$ H\neq \operatorname {\textrm {GL}}({\mathcal H})$
.
Lemma 3.6. If H is a strict standard algebraic subgroup, then there is a finite-dimensional subspace
$E\subset {\mathcal H}$
such that
$H_E$
is a strict algebraic subgroup of
$\operatorname {\mathrm{GL}}(E)$
.
Proof. Let
$\mathcal {P}$
be the family of standard polynomials defining the algebraic subgroup H. Let
$P\in \mathcal {P}$
be a non-constant standard polynomial. Choose an orthonormal basis
$(e_n)$
such that P can be written as an absolutely convergent series. For
$n\in \mathbf{N}$
, let us set
$E_n$
to be the space spanned by
$(e_1,\ldots ,e_n)$
.
In particular for n large enough, the restriction of P to pairs
$(g,g^{-1})$
is a non-constant polynomial map on
$\operatorname {\textrm {GL}}(E_n)$
and thus defines a strict algebraic subset of
$\operatorname {\textrm {GL}}(E_n)$
.
Remark 3.7. Not all polynomial maps are standard. The set of compact operators
$\operatorname {\mathrm{L}}_c({\mathcal H})$
is closed in
$\operatorname {\mathrm{L}}({\mathcal H})$
(it is the closure of the set of finite-rank operators) and by Hahn–Banach theorem, there is a non-trivial bounded linear form that vanishes on
$\operatorname {\mathrm{L}}_c({\mathcal H})$
. This linear form is not standard because it vanishes on all finite-rank operators.
Remark 3.8. By Proposition 1.5 (proved in §3.2), Zariski density (Definition 1.4) implies geometric density and we do not know if the converse holds. Lemma 3.6 shows that algebraic subgroups can be tracked by considering finite-dimensional subspaces. So, one can think that phenomena similar to those in the finite-dimensional case happen and this is maybe a clue that the converse implication between geometric density and Zariski density holds. In particular, one can show that if H is a strict algebraic subgroup of
$\operatorname {\textrm {O}}_{{\mathbf K}}(p,\infty )$
such that there is some finite-dimensional subspace E with
$H_E\neq \{\operatorname {\textrm {Id}}\}$
, then H is not geometrically dense.
3.2 Exterior products
Let
${\mathcal H}_{{\mathbf K}}$
be a Hilbert space over
${\textbf K}$
with Hermitian form Q of signature
$(p,\infty )$
. We denote by
${\mathcal H}$
the underlying real Hilbert space and by
$(\cdot ,\cdot )$
the real quadratic form
$\Re (Q)$
.
The exterior product
${\textstyle \bigwedge }^k\mathcal {H}$
has a natural structure of pre-Hilbert space and there is a continuous representation
$\pi _k\colon \operatorname {\textrm {GL}}(\mathcal {H})\to \operatorname {\textrm {GL}}({\textstyle \bigwedge }^k\mathcal {H})$
given by the formula
$\pi _k(g)(x_1\wedge \cdots \wedge x_k)=gx_1\wedge \cdots \wedge gx_k$
. An orthonormal basis of
${\textstyle \bigwedge }^k\mathcal {H}$
is given by
$(e_{i_1}\wedge \cdots \wedge e_{i_k})_{i\in \mathcal {I}}$
, where
$\mathcal {I}$
is the set of elements
$i\in \mathbf{N}^k$
such that
$i_1<\cdots <i_k$
and
$(e_i)$
is an orthonormal basis of
$\mathcal {H}$
. In other words, if
$\langle \cdot ,\cdot \rangle $
is the scalar product on
${\mathcal H}$
, then the bilinear form applied to two vectors
$x_1\wedge \cdots \wedge x_k$
and
$y_1\wedge \cdots \wedge y_k$
is given by the Gram determinant
$\det ( \langle x_i,y_j\rangle _{i,j=1..k})$
. As usual, the completion of
${\textstyle \bigwedge }^k\mathcal {H}$
is denoted
$\overline {{\textstyle \bigwedge }^k\mathcal {H}}$
.
The space
${\textstyle \bigwedge }^k\mathcal {H}$
is also endowed with a quadratic form built from Q. One defines
$(x_1\wedge \cdots \wedge x_k,y_1\wedge \cdots \wedge y_k)$
to be
$\det ((x_i,y_j))$
. As soon as
$k\geq 2$
, this quadratic form is non-degenerate of signature
$(\infty ,\infty )$
and extends continuously to
$\overline {{\textstyle \bigwedge }^k\mathcal {H}}$
. Moreover,
$\pi _k(\operatorname {{\textrm O_{\mathbf {K}}}}(Q))$
preserves this quadratic form.
Lemma 3.9. Let
$(e_i)_{i\in \mathcal I}$
be an orthonormal basis of
$\mathcal {H}$
and let
$v,w$
be vectors of
${\textstyle \bigwedge }^k \mathcal {H}$
. There are families
$(\unicode{x3bb} _i)$
and
$(\mu _i)$
such that for any
$g\in \operatorname {\mathrm{GL}}(\mathcal {H})$
,
is a standard polynomial in g.
Proof. If suffices to write
$v=\sum _{i\in \mathcal {I}}\bigwedge _{l=1}^k e_{i_l}\unicode{x3bb} _i$
and
$w=\sum _{j\in \mathcal {I}}\bigwedge _{l=1}^k e_{j_l}\mu _j$
and express the scalar product of
${\textstyle \bigwedge }^k\mathcal {H}$
in the basis
$(e_{i_1}\wedge \cdots \wedge e_{i_k})_{i\in \mathcal {I}}$
. The sum is absolutely convergent because
$(\unicode{x3bb} _i)$
and
$(\mu _j)$
are Hilbert coordinates. Finally,
$(ge_{i_l},e_{j_{\sigma (l)}})=\langle ge_{i_l}, I_{p,\infty }e_{j_{\sigma (l)}}\rangle $
and writing
$I_{p,\infty }e_{j_{\sigma (l)}}$
in the Hilbert base, one recovers an absolutely convergent series of matrix coefficients in the Hilbert base
$(e_i)$
.
Lemma 3.10. Let V be a non-trivial subspace of
${\textstyle \bigwedge }^k\mathcal {H}$
. The stabilizer of
$\overline {V}$
in
$\operatorname {{\mathrm{O}_{\mathbf {K}}}}(Q)$
is a standard algebraic subgroup.
Proof. If V is a non-trivial subspace of
${\textstyle \bigwedge }^k\mathcal {H}$
, one can choose an orthonormal basis
$(v_i)_{i\in I}$
of
${\textstyle \bigwedge }^k\mathcal {H}$
such that the closure
$\overline {V}$
in the Hilbert completion
$\overline {{\textstyle \bigwedge }^k\mathcal {H}}$
is the closed span of
$(v_i)_{i\in I_0}$
for some
$I_0\subset I$
.
Let H be the subgroup of
$\operatorname {{\textrm O_{\mathbf {K}}}}(Q)$
stabilizing
$\overline {V}$
. Thus, by Lemma 3.9, g belongs to H if and only if, for all
$ i\in I_0$
and
$j\in I\setminus I_0$
, we have
$(\pi _k(g)v_i,v_j)=0$
. Thus, H is the algebraic subgroup of
$\operatorname {{\textrm O_{\mathbf {K}}}}(Q)$
defined by the family of polynomials
$\mathcal {P}=\{P_{ij}\}$
where
$P_{ij}(g)=(\pi _k(g)v_i,v_j)$
.
Proof of Proposition 1.5
We have seen in Example 3.5 that stabilizers of points at infinity are standard algebraic subgroups. Assume
$\mathcal Y$
is a strict totally geodesic subspace of
${\mathcal X}$
. Without loss of generality, we assume that
$o\in \mathcal Y$
and thus
$\mathcal Y$
corresponds to some Lie triple system
$\mathfrak {p}< \operatorname {\textrm {S}}^2({\mathcal H})$
. Let
$\mathfrak {k}=\overline {[\mathfrak {p},\mathfrak {p}]}$
and
$\mathfrak m$
be the Lie algebra
$\mathfrak {k}\oplus \mathfrak {p}\leq \operatorname {\mathrm{L}}^2({\mathcal H})$
. Because
$\mathfrak m$
is a Lie algebra,
$G=\exp (\mathfrak m)$
is a subgroup of
$\operatorname {\textrm {GL}}^2({\mathcal H})$
that is generated by transvections along geodesics in
$\mathcal Y$
. In particular, for any
$h\in \operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )$
, h normalizes G if and only if h preserves
$\mathcal Y$
. Because
$G=\exp (\mathfrak m)$
, h normalizes G if and only if it stabilizes
$\mathfrak m$
under the adjoint action (that is,
$\operatorname {\textrm {Ad}}(h)(\mathfrak m)=\mathfrak m$
, which means that for any
$X\in \mathfrak m$
,
$hXh^{-1}\in \mathfrak m$
). Because
$\mathfrak m$
is a closed subspace of
$\operatorname {\mathrm{L}}^2({\mathcal H})$
, we have the splitting
$\operatorname {\mathrm{L}}^2({\mathcal H})=\mathfrak m\oplus \mathfrak m^{\bot }$
. So, h stabilizes
$\mathfrak m$
if and only if for any
$X\in \mathfrak m$
and
$Y\in \mathfrak m^{\bot }$
,
$\langle hXh^{-1},Y\rangle =0$
, where
$\langle \ ,\ \rangle $
is the Hilbert–Schmidt scalar product. Finally, because the map
$(M,N)\mapsto \langle MXN,Y\rangle $
is bilinear on
$\operatorname {\mathrm{L}}({\mathcal H})\times \operatorname {\mathrm{L}}({\mathcal H})$
, H is an algebraic subgroup of
$\operatorname {{\textrm O_{\mathbf {K}}}}(p,\infty )$
. It remains to show that these bilinear maps are standard. Let
$(e_n)$
be an orthonormal basis of
${\mathcal H}$
and let
$E_{i,j}=e_i\otimes e_j^*$
be the associated orthonormal basis of
$\operatorname {\mathrm{L}}^2({\mathcal H})$
, that is,
$E_{i,j}(x)=\langle x,e_j\rangle e_i$
. Thus, let us write
$X=\sum _{i,j}X_{i,j}E_{i,j}$
and
$Y=\sum _{i,j}X_{i,j}E_{i,j}$
to obtain
$$ \begin{align*}\langle MXN,Y\rangle=\sum_{i,j,k,l}X_{i,j}Y_{k,l}\langle ME_{i,j}N,E_{k,l}\rangle,\end{align*} $$
where
$$ \begin{align*} \langle ME_{i,j}N,E_{k,l}\rangle&=\operatorname{\textrm{Trace}}((ME_{i,j}N)^*E_{k,l})\\ &=\sum_{n\in\textbf{N}}\langle N^*E_{i,j}M^*E_{k,l}(e_n),e_n\rangle\\ &=\langle N^*E_{j,i}M^*(e_k),e_l\rangle\\ &=\langle E_{j,i}M^*(e_k),N(e_l)\rangle\\ &=\langle e_i,M^*(e_k)\rangle\langle e_j,N(e_l)\rangle\\ &=\langle M(e_i),e_k\rangle\langle e_j,N(e_l)\rangle. \end{align*} $$
The absolute convergence of the series can be proven with the same arguments as in Example 3.5.(1).
Let
${\mathcal H}$
be a Hilbert space over
${\textbf K}$
with a non-degenerate Hermitian form Q of signature
$(p,\infty )$
with
$p\in \mathbf{N}$
. For a finite-dimensional non-degenerate subspace
$E\subset {\mathcal H}$
of Witt index p, we denote by
${\mathcal X}_E\subset {\mathcal X}_{{\mathbf K}}(p,\infty )$
the subset of isotropic subspaces of E of dimension p. This corresponds to a standard embedding of
${\mathcal X}_{{\mathbf K}}(p,q)\hookrightarrow {\mathcal X}_{{\mathbf K}}(p,\infty )$
, where
$(p,q)$
is the signature of the restriction of Q to E. Let
$\mathcal {E}$
be the collection of all such finite-dimensional subspaces. We conclude this section with a lemma that shows that the family
$({\mathcal X}_E)_{E\in \mathcal {E}}$
is cofinal among finite-dimensional totally geodesic subspaces.
Lemma 3.11. For any finite-dimensional totally geodesic subspace
$\mathcal {Y}\subset {\mathcal X}_{{\mathbf K}}(p,\infty )$
, there is
$E\in \mathcal {E}$
such that
$\mathcal {Y}\subset {\mathcal X}_E$
.
Proof. We claim that one can find finitely many points
$x_1,\ldots ,x_n\in \mathcal {Y}$
such that
$\mathcal {Y}$
is the smallest totally geodesic subspace of
${\mathcal X}$
that contains
$\{x_1,\ldots ,x_n\}$
. We define by induction points
$x_1,\ldots ,x_k\in \mathcal {Y}$
and
$\mathcal {Y}_k$
, that is, the smallest totally geodesic subspace containing
$\{x_1,\ldots ,x_k\}$
. Observe that
$\mathcal {Y}_k$
has finite dimension because
$\{x_1,\ldots ,x_k\}\subset \mathcal {Y}$
and
$\mathcal {Y}$
has finite dimension. For
$x_1$
, choose any point in
$\mathcal {Y}$
and let
$\mathcal {Y}_1$
be
$\{x_1\}$
. Assume
$x_1,\ldots ,x_k$
have been defined. If
$\mathcal {Y}_k\neq \mathcal {Y}$
, choose
$x_{k+1}\in \mathcal {Y}\setminus \mathcal {Y}_k$
. One has
$\mathcal {Y}_{k+1}\varsupsetneq \mathcal {Y}_{k}$
and thus
$\dim (\mathcal {Y}_{k+1})>\dim (\mathcal {Y}_{k})$
. So, in finitely many steps, one gets that there is
$n\in \mathbf{N}$
such that
$\mathcal {Y}_n=\mathcal {Y}$
.
The points
$x_1,\ldots ,x_n$
are positive definite subspaces (with respect to Q) of
$\mathcal {H}$
. So, let E be the span of these subspaces. Observe that this space has finite dimension and
$x_1,\ldots ,x_n\in {\mathcal X}_E$
. Up to adding finitely many vectors to E, we may moreover ensure that E is non-degenerate with Witt index p.
4 Boundary theory
4.1 Maps from strong boundaries
Let G be a locally compact, second countable group acting continuously by isometries on
${\mathcal X}_{{\mathbf K}}(p,\infty )$
, where p is finite. This section is dedicated to the analysis of Furstenberg maps also known as boundary maps from a measurable boundary of G to the geometric boundary
$\partial {\mathcal X}_{{\mathbf K}}(p,\infty )$
, or more precisely, to some specific part of this boundary. We use a suitable notion of a measurable boundary of a group introduced in [Reference Bader and FurmanBF14, §2]. This definition (see Definition 4.7) is a strengthening of previous versions introduced by Furstenberg [Reference FurstenbergFur73] and Burger and Monod [Reference Burger and MonodBM02].
Let us recall that a Polish space is a topological space which is separable and completely metrizable. By a Lebesgue G-space, we mean a standard Borel space (that is, a space and a
$\sigma $
-algebra given by some Polish space and its Borel
$\sigma $
-algebra), equipped with a Borel probability measure and an action of G which is measurable and preserves the class of the measure. We denote by
$\textbf {P}(\Omega )$
the space of probability measures on a standard Borel space
$\Omega $
. This is a Polish space for the topology of weak convergence.
Definition 4.1. Let
$\Omega $
be a standard Borel space,
$\unicode{x3bb} \in \textbf {P}(\Omega )$
. Assume that G acts on
$\Omega $
with a measure-class preserving action. The action of G on
$\Omega $
is isometrically ergodic if for every separable metric space Z equipped with an isometric action of G, every G-equivariant measurable map
$\Omega \to Z$
is essentially constant.
Remark 4.2. If the action of G on
$\Omega $
is isometrically ergodic, then it is ergodic (take
$Z=\{0,1\}$
and the trivial action). Furthermore, if the action of G on
$\Omega \times \Omega $
is isometrically ergodic, then it is also the case for the action on
$\Omega $
.
Definition 4.3. Let Y and Z be two Borel G-spaces and
$p:Y\to Z$
be a Borel G-equivariant map. We denote by
$Y\times _p Y$
the fiber product over p, that is, the subset
$\{(x,y)\in Y^2,\ p(x)=p(y)\}$
with its Borel structure coming from
$Y^2$
.
We say that p (or Y) admits a fiberwise isometric action if there exists a Borel, G-invariant map
$d:Y\times _p Y\to \textbf {R}$
such that any fiber
$Y'\subset Y$
of p endowed with
$d|_{Y'\times Y'}$
is a separable metric space.
Before going on, let us give a few examples of fiberwise isometric actions. These examples are closed to measurable fields of metric spaces that appear in [Reference Bader, Duchesne and LécureuxBDL16] and are simpler versions of fiberwise isometric actions that will appear in the proof of Theorem 1.7.
Example 4.4. Let
$(M,d)$
be a metric space. The Wisjman hyperspace
$2^M$
is the set of closed subspaces in M. This space can be embedded in the space
$C(M)$
of continuous functions on M: to any close subspace A, one associates the distance function
$x\mapsto d(A,x)$
. The topology of pointwise convergence on
$C(M)$
induces the so-called Wisjman topology on
$2^M$
. If
$(M,d)$
is complete and separable, then the Wisjman hyperspace is a Polish space. Actually, when M is separable, the topology is the same as the topology of pointwise convergence on a countable dense subset.
Let
${\mathcal X}$
be a complete separable CAT(0) space. We denote by
$\mathcal {F}_k$
the space of flat subspaces of dimension k in
${\mathcal X}$
(that is, isometric copies of
$\textbf {R}^k$
). One can check that
$\mathcal {F}_k$
is closed in
$2^{\cal X}$
: flatness is encoded in three conditions (equality in the CAT(0) inequality, convexity, and geodesic completeness), the dimension is encoded in the Jung inequality (see e.g. [Reference Lang and SchroederLS97]), and all these conditions are closed. The visual boundary
$\partial {\mathcal X}$
with the cone topology is a closed subspace of
$\overline {{\mathcal X}}$
, which is an inverse limit of a countable family of closed balls [Reference Bridson and HaefligerBH99]. Thus,
$\partial {\mathcal X}$
is a Polish space.
Let G act by isometries on
${\mathcal X}$
and let
$k>0$
be such that there exists a k-dimensional flat in
${\mathcal X}$
. Let
$\partial \mathcal F_k=\{(F,\xi )\mid F\in \mathcal F_k\text { and } \xi \in \partial F\}$
. This is a closed subspace of
$\mathcal {F}_k\times \partial {\mathcal X}$
. Then the continuous projection
$\partial \mathcal F_k\to \mathcal F_k$
admits a fiberwise isometric action of
$\operatorname {\mathrm{Isom}}(\cal X)$
, each
$\partial F$
being endowed with the Tits metric.
Definition 4.5. Let A and B be Lebesgue G-spaces. Let
$\pi :A\to B$
be a measurable G-equivariant map. We say that
$\pi $
is relatively isometrically ergodic if each time we have a G-equivariant Borel map
$p:Y\to Z$
of standard Borel G-spaces, which admits a fiberwise isometric action, and measurable G-maps
$A\to Y$
and
$B\to Z$
such that the following diagram commutes:
then there exists a measurable G-map
$\phi \colon B\to Y$
which makes the following diagram commutative.
Remark 4.6. If a G-Lebesgue space B is such that the first projection
$\pi _1\colon B\times B\to B$
is relatively isometrically ergodic, then it is isometrically ergodic. Indeed, if Y is a separable metric G-space and
$f:B\to Y$
is G-equivariant, then it suffices to apply relatively isometric ergodicity to the map
$\tilde {f}\colon (b,b')\mapsto f(b')$
and the trivial fibration
$Y\to \{\ast \}$
.
Actually, relative isometric ergodicity yields a measurable map
$\phi \colon B\to Y$
such that for almost all
$(b,b')$
,
$\phi (b)=f(b')$
, and thus f, is essentially constant.
Let B be a Lebesgue G-space. We use the definition of amenability for actions introduced by Zimmer, see [Reference ZimmerZim84, §4.3]. The action
$G\curvearrowright B$
is amenable if for any compact metrizable space M on which G acts continuously by homeomorphisms, there is a measurable G-equivariant map
$\phi \colon B\to \textbf {P}(M)$
.
Definition 4.7. The Lebesgue G-space B is a strong boundary of G if:
-
• the action of G on
$(B,\nu )$
is amenable (in the sense of Zimmer); and -
• the first projection
$\pi _1\colon B\times B\to B$
is relatively isometrically ergodic.
Example 4.8. The most important example for us is the following [Reference Bader and FurmanBF14, Theorem 2.5]. Let G be a connected semisimple Lie group and P a minimal parabolic subgroup. Then
$G/P$
, with the Lebesgue measure class, is a strong boundary for the action of G. If
$\Gamma <G$
is a lattice, then
$G/P$
is also a strong boundary for the action of
$\Gamma $
. More generally, this is also true if G is a semisimple algebraic group over a local field.
The next example shows that every countable group admits a strong boundary.
Example 4.9. Let