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Compactifications of symmetric spaces. (English) Zbl 1053.31006
Progress in Mathematics (Boston, Mass.) 156. Boston, MA: Birkhäuser (ISBN 0-8176-3899-7/hbk). x, 284 p. (1998).
In the 60’s, the necessity for understanding “boundaries at infinity” of Riemannian symmetric spaces of higher rank led to landmark papers by Satake (1960), Furstenberg (1963) and Karpelevič (1965). Compactifications of symmetric spaces were constructed, in different ways and from various points of views, ranging from Representation Theory to Geometry, via Probability Theory. The book under review is the first systematic presentation of all these constructions. It includes a new proof of Moore’s isomorphism between the Satake and the Furstenberg compactifications together with some new characterizations thereof. But the main novelty lies in the complete description of the Martin compactifications of a symmetric space. These compactifications arise naturally in Potential Theory and Probability Theory, but they are also interesting from a geometric point of view: For a negatively curved Hadamard manifold, the determination of the Martin compactification(s) was obtained by Anderson and Schoen and by Ancona in the 1980’s but it remains an open problem for non-positively curved manifolds. For symmetric spaces associated to complex groups, where an explicit expression for the Green function is available, this was done by Dynkin in the 1960’s and it is only recently that the authors could deal with a general symmetric space of higher rank. This is presented for the first time in the present book.
Let \(X=G/K\) be a noncompact Riemannian symmetric space, with \(G\) semisimple (satisfying the usual standard assumptions) and \(K\) maximal compact. A compactification of \(X\) is a compact \(G\)-space \(\overline X\) together with a \(G\)-equivariant topological embedding \(i:X\to\overline X\) such that \(i(X)\) is dense in \(\overline X\). A compactification dominates another one if there is a continuous \(G\)-equivariant map from the first onto the second which is compatible with the embeddings. One denotes by \(\overline X{}^1\vee\overline X{}^2\) the smallest compactification which dominates both compactifications \(\overline X{}^1\) and \(\overline X{}^2\). Let us now introduce the compactifications under investigation.
First, the geometry of \(X\) leads to the visibility compactification \(\overline X(\infty)\) and to the more refined Karpelevič compactification \(\overline X{}^K\). Representation theory of \(G\) yields the family of Furstenberg-Satake compactifications \(\overline X{}^{SF}_I\) parametrized by the various subsets \(I\) of the set \(\Delta \) of simple roots; when \(I=\Delta\), this is just the one-point compactification of \(X\) , whereas, for \(I=\emptyset\), it is called the maximal Satake-Furstenberg compactification \(\overline X_{\max}^{SF}\) of \(X\). The third family belongs to Potential Theory and consists of the Martin compactifications \(\overline X(\lambda)\), where \(\lambda \) runs through the positive spectrum \(]-\infty, \lambda_0]\) of the Laplace operator on \(X\). Of particular interest is the Martin compactification \(\overline X(\lambda_0)\) attached to the ground eigenvalue \(\lambda _0\).
The first two thirds of the book (Chapters II–X) are devoted to a detailed presentation of the following diagram, where an arrow between two compactifications indicates that the first one dominates the second one: \[ \begin{alignedat}{2}4 && &&\quad\overline X(\lambda_0)\leftrightarrow\overline X{}^{SF}_{\max} \quad\rightarrow\quad\overline X{}^{SF}_I\\ && \nearrow && && \searrow \\ \overline X{}^K\quad\rightarrow\quad\overline X(\lambda)\leftrightarrow \overline X{}^{SF}_{\max}\vee\overline X(\infty)\quad && && && & \quad\overline X{}^{SF}_\Delta\\ && \searrow && && \nearrow \\ && && \overline X(\infty)\mkern90mu && & \end{alignedat} \] The way the Martin compactifications fit into this picture is developed in the central part of the book (Chapters V–X).
The last part of the book (Chapters XI–XV) deals with analogous results in the discrete setting: Given a semisimple Lie group or algebraic group \(G\) defined over a local field and a \(K\)-biinvariant probability measure \(p\) on \(G\), one can define a Martin compactification \(\overline X(p,r)\) of the corresponding symmetric space or building \(X=G/K\) for any \(r\geq r_0\), where \(r_0\) is the ground eigenvalue of \(p\). In the case \(r=r_0\), it is still isomorphic to the maximal Furstenberg-Satake compactification (which can be also defined for buildings). Let us be more specific. Denoting by \(\mathcal G\) and \(\mathcal K\) the Lie algebras of \(G\) and \(K\), let \(\mathcal A\) be a Cartan subspace (whose dimension is the rank of \(X\)) in the tangent space \(\mathcal P=\mathcal G\ominus\mathcal K\), \(A=\exp\mathcal A\) the corresponding subgroup of \(G\), and \(\Sigma\subset\mathcal A^*\) the root system of \((\mathcal G,\mathcal A)\). Then \(\mathcal A-\bigcup_{\alpha\in\Sigma}\ker\alpha\) is the disjoint union of open convex cones, one of which being selected as the positive Weyl chamber \(\mathcal A^+\). Its closure is denoted \(\overline{\mathcal A^+}\), so that the polar decomposition of \(X\) reads \(X=K\exp\overline{\mathcal A^+}.0\). The positive roots are those roots \(\alpha\in\Sigma\) which are positive on \(\mathcal A^+\) and \(\rho\) is their half-sum with multiplicities. Recall that \(\Delta\) denotes the set of simple positive roots, which constitute a basis of \(\mathcal A^*\).
Part I: Geometric realizations of Furstenberg-Satake and Karpelevič compactifications (Chapters II–V). Parabolic subgroups of \(G\) occur basically in most compactifications of \(X\). Their definition and properties are recalled in Chapter II: To any subset \(I\) of \(\Delta\) is associated a standard parabolic subgroup \(P^I\) in such a way that \(I\subset J\iff P^I\subset P^J\), the minimal one being \(P=P^\emptyset\). A subgroup of \(G\) is parabolic if it is conjugate under \(G\) (or equivalently under \(K\)) to some \(P^I\). The set of parabolic subgroups is given the structure of a spherical building, the Borel-Tits building of \(X\), by declaring that \(P\) is incident to (or a face of) \(Q\) if \(P\supset Q\).
In the next chapter, this building is realized as a simplicial complex \(\Delta(X)\) on the sphere at infinity: This complex is induced by the decomposition of \(\mathcal P\) (or equivalently \(X=\exp P.0\)) into convex cones \(\text{Ad}\,k.C_I\), where \(k\) runs through \(K\), \(I\) through the proper subsets of \(\Delta\), and the face \(C_I\) is defined by \[ C_I=\{\,H\in\mathcal A\mid \alpha(H)=0\;\forall\,\alpha\in I\text{ and }\alpha(H)>0\;\forall\,\alpha\in\Delta\setminus I\}. \] Then the parabolic subgroup \(gP^Ig^{-1}\) is the stabilizer of the face at infinity \(g.C_I(\infty)\).
Using this decomposition, Chapter III proceeds with the construction of the following geometric compactification \(\overline X{}^*=X\sqcup\Delta^*(X)\) of the symmetric space \(X\): A sequence \(x_n:=\exp H_n.0\) in the flat \(A.0\) is \(I\)-canonical if \(H_n\) belongs to the face \(C_I\) and \(\alpha(H_n)\to+\infty\) for every \(\alpha\in\Delta\setminus I\). A sequence \((y_n)\) in \(X\) is \(I\)-fundamental if there is a convergent sequence \(g_n\to g\) in \(G\) such that \(g_n^{-1}y_n\) is \(I\)-canonical in \(A.0\). The pair \((g.C_I(\infty),g.0)\) is the formal limit of \((y_n)\) and two fundamental sequences are equivalent if they have the same formal limit. The set \(\Delta^*(X)\) of formal limits is shown to have the structure of a cellular complex dual to \(\Delta (X)\), each cell being identified with a symmetric space of smaller rank. The space \(\overline X{}^*\) is topologized in Appendix A (Theorem A.5).
The main result (Theorem 4.43) in Chapter IV is the identification of this geometric compactification \(\overline X{}^*\) with Satake’s maximal compactification \(\overline X{}^S_{\max}\). To this end Satake’s construction is recalled thoroughly. Furstenberg’s boundary theory is also introduced. First of all, \(G/P^I\) is shown to be a \(G\)-boundary, i.e. a compact space on which \(G\) acts minimally and proximally (see Definitions 4.45 or 9.29). Next, denoting by \(M^1(G/P^I)\) the (compact convex) set of probability measures on this boundary and by \(\overline m\) the unique \(K\)-invariant measure therein, which is denoted by \(m_{P^I}\) in Appendix B and should not be confused with the measure \(m_I\in M^1(G/P)\) in Theorem 7.22), one gets Furstenberg’s compactification \(\overline X{}^F_I\) by considering the map \(\phi^I:g.0\mapsto g.\overline m\) of \(X\) into \(M^1(G/P^I)\) and by taking the closure of its image, provided it is one-to-one, which has been investigated by Moore (see Proposition 4.49 and Lemma 9.11). It is clear by construction that \(\overline X{}^F_I\) dominates \(\overline X{}^F_J\) whenever \(I\subset J\). Chapter IV concludes with the partial result that the geometrical compactification \(\overline X{}^*\) dominates Furstenberg’s maximal compactification \(\overline X{}^F_{\max}:=\overline X{}^F_\emptyset\).
Chapter V is devoted to a geometric presentation of the Karpelevič compactification \(\overline X{}^K\), which is completed in Appendix B. It differs slightly from the original construction of Karpelevič , which proceeded by induction over the rank of the boundary symmetric spaces.
Part II: Martin compactifications (Chapters VI–X). Chapter VI consists of a brief introduction to the classical Martin compactification of a manifold \(X\) with respect to a Laplacian \(L\): The set of \(\lambda\in\mathbb R\) for which the equation \[ Lf+\lambda f=0 \tag{\({(\partial_\lambda)}\)} \] has a positive solution \(f\) is called the positive spectrum of \(L\) and consists of the interval (\(-\infty,\lambda_0]\), where \(\lambda_0\) is the bottom of the \(L^2\)-spectrum of \(-L\). For \(\lambda<\lambda_0\), one defines the Green function \(G^\lambda(x,y)\) as the integral kernel of \((\lambda I+L)^{-1}\) and the Martin kernel as the renormalized Green function \[ K^\lambda(x,y):={G^\lambda(x,y)\over G^\lambda(x,0)}. \] In some cases, this can be defined also for \(\lambda= \lambda_0\). By Harnack’s inequality, the family \(\{\,K^\lambda (x,.)\mid x\in X\,\}\) is pre-compact for the topology of uniform convergence on compact subsets. The \(\lambda\)-Martin compactification \(\overline X(\lambda)\) of \(X\) is obtained by considering the map \(x\longmapsto K^\lambda (x,.)\) and by taking the closure of its image. For a symmetric space, equipped with the standard Riemannian structure, the ground eigenvalue is \(\lambda_0=\|\rho\|^2\), and the extremal solutions of \((\partial_\lambda)\) were already determined by Karpelevič.
In Chapter VII (Theorem 7.33) the authors manage to identify the ground Martin compactification \(\overline X(\lambda_0)\) with the compactification \(\overline X{}^*\), relying solely on invariance properties of limit functions, which is quite remarkable. Let us sketch their argument, which is based on the invariance of \(K^{\lambda_0}(g_n.0,.)\) under \(g_nKg_n^{-1}\). If the subgroups \(g_nKg_n^{-1}\) happen to “converge” to a subgroup \(D\), for some sequence \(g_n\in G\) tending to infinity, then any limit function \(h\) of \(K^{\lambda_0}(g_n.0,.)\) will be \(D\)-invariant. On the other hand, as a limit of renormalized Green functions whose singularities are removed to infinity, \(h\) is a solution of \((\partial_{\lambda_0})\). Thus, if \((\partial_{\lambda_0})\) has a unique normalised \(D\)-invariant solution \(h^D\), then \(K^{\lambda_0}(g_n.0,.)\) has to converge to \(h^D\). The authors show that all these assumptions are fulfilled whenever \((g_n)\) is a fundamental sequence (Propositions 7.20, 9.14 and Theorem 7.22). As a consequence, \(\overline X{}^*\) dominates \(\overline X(\lambda _0)\). The actual identification \(\overline X{}^*\equiv\overline X(\lambda_0)\) is the content of Proposition 7.31.
Chapter VIII is devoted to the Martin compactification \(\overline X(\lambda)\) below the ground value \(\lambda_0\) and contains one of the main achievement of the book (Theorem 8.21), namely the characterization of \(\overline X(\lambda)\) as the smallest compactification which dominates both the visibility compactification \(\overline X(\infty)\) and the compactifications \(\overline X{}^*\equiv\overline X{}^S_{\max}\equiv\overline X(\lambda_0)\). The first part of the argument is the same as for \(\overline X(\lambda_0)\): The limit of \(K^\lambda(g_n.0,.)\) along any fundamental sequence \((g_n)\) is invariant under some subgroup \(D\). But now \((\partial_\lambda)\) has more than one (normalized) \(D\)-invariant solution and the authors need two additional ingredients: (a) the classification of extremal positive solutions of \((\partial_\lambda)\), which goes back to Karpelevič and which is nicely reproved in Chapter XIII, and (b) the behavior at infinity of the Green function \(G^\lambda\), which has been recently determined by Anker and Ji.
The convergence alluded to above of \(g_nKg_n^{-1}\) towards a limit subgroup \(D\) is made precise in Chapter IX: it is observed that the set \(\mathcal S\) of all closed subgroups of \(G\) endowed with the topology of local Hausdorff convergence is a compact space, and that the map \(i:g.0\mapsto gKg^{-1}\) embeds \(X\) into \(\S\). The closure of \(i(X)\) in \(\S\) is again isomorphic to \(\overline X^*\)(Theorem 9.18), and consists exactly of all {maximal distal subgroups of \(G\), i.e. those subgroups which act distally on the Lie algebra \(\mathcal G\) and are maximal with this property (Theorem 9.27). This beautiful realization serves in particular to complete the identification \(\overline X{}^F_{\max}\equiv\overline X{}^S_{\max}\) between the maximal compactifications of Fustenberg and Satake (Theorem 9.47). The other nonmaximal compactifications are dealt with in Appendix B. Finally another possible realization of all these isomorphic compactifications is presented in Chapter X, using the ground spherical function \(\Phi^0\).
Part III: Martin compactifications for random walks and the fixed line property (Chapters XI–XV). The Laplace-Beltrami operator \(L\) on \(X\) is the generator of the heat semigroup \[ e^{\,t\,L}f(g.0)=\int_Gf(h.0)\,p_{t}(g^{-1}h)\,dh, \] which is a right convolution operator defined by a \(K\)-biinvariant probability measure on \(G\) with smooth density, namely the heat kernel \(p_t\). This leads to the natural question whether the previous Martin theory can be extended to the class of convolution operators, which may indicate to which extent the above results relied upon the Riemannian structure of the symmetric space or on the sole structure of \(G\). Actually the investigation of such Martin compactifications makes sense in a much wider setting, namely for general locally compact groups \(G\) and probability measures \(p\) (satisfying some regularity assumptions) with associated convolution operator \(*p\): \[ *pf(g)= \int _G f(gh) p(dh). \] As explained in Chapter XI, the positive spectrum of \(*p\) still consists of an interval \([r(p),+\infty)\), where \(r(p)=\limsup_{n\to+\infty}p^{*n}(e)^{1/n}\). This number is smaller or equal to the \(L^2\) spectral radius \(r_0(p)\) of \(*p\) and the case of equality is investigated in Chapters XI–XII. For \(r\geq r(p)\) the set of Radon measures \(\mu\) such that \(\mu*p\leq r\mu\) is shown to be a convex cone with compact base, on which \(G\) acts by convolution on the right side. The \(r\)-Martin compactification of \(G\) with respect to \(p\) is the closure of the \(G\)-orbit of the potential measure \(\mu^r=\sum_nr^{-n}p^{*n}\) for the projective action on the base of this cone. Then, particular emphasis is put on Gelfand pairs \((G,K)\) and in particular on the following two cases: (a) Riemannian symmetric spaces associated to semisimple Lie groups and (b) Bruhat-Tits buildings associated to semisimple algebraic groups defined over local fields. In both cases \(G=KP\), where \(P\) satisfies a strong version of amenability, namely Furstenberg’s fixed line property, which means that every affine action of \(P\) on a convex cone with compact basis has a fixed line (or equivalently that the projective action on the basis has a fixed point). In most of Chapters XI-XV, it is assumed that \(G\) has such a decomposition and that \(p\) is a \(K\)-biinvariant probability measure.}

MSC:
31C35 Martin boundary theory
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
58J65 Diffusion processes and stochastic analysis on manifolds
60J45 Probabilistic potential theory
60J50 Boundary theory for Markov processes
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