Karpelevich’s compactification and extensions of Poisson integrals

In this short blog post, we study to what extent the Poisson integral of continuous functions on the Furstenberg boundary ${G/P_0}$ of a symmetric space of non-compact type can be extended continuously to the whole Karpelevich boundary or what parts thereof it can be extended. The interest in such integrals comes from the fact that the Poisson integrals of functions in ${L^{\infty}(G/P_0)}$ exhaust all bounded harmonic functions on ${X}$ , a deep result of Furstenberg. The study of such extensions hence has its origins in classical harmonic analysis and solutions to Dirichlet problems on the Poincare disk.

1. Definitions and notation

I refer to and for the basic definitions of the Karpelevich compactification, though some parts of the definition will be recalled in the next section.

In what follows ${X}$ will denote a (Riemannian) symmetric space of non-compact type, ${G}$ the identity component of isometry group of ${X}$ , ${K\subset G}$ the stabilizer of a point ${x_0\in X}$ and ${\Gamma \subset G}$ a torsion free lattice in ${G}$ .

${P\subset G}$ will denote a parabolic subgroup, while ${P_0\subset G}$ will be a fixed minimal parabolic subgroup. The Langlands decomposition of ${P}$ will be denoted by ${P = M_P A_P N_P}$ . ${\Delta}$ denotes the collection of simple (restricted) roots of ${G}$ , with respect to a fixed ${\R}$ -split torus ${A\subset G}$ . For a subset ${I\subset \Delta}$ it is customary to denote the associated parabolic subgroup by ${P^I}$ and its Langlands decomposition by ${P^I = M_IA_I N_I}$ and I will occasionally adhere to this convention. The ${P^I}$ are called standard parabolic subgroups, they all contain ${P_0 = P^{\empty}}$ and they form a complete set of representatives of the ${G}$ (or equivalently ${K}$ ) conjugacy classes of parabolic subgroups of ${G}$ . Lastly, if ${z\in \partial X}$ is a point on the geodesic boundary, then the parabolic subgroup ${Stab_G(z)}$ will be denoted ${P_z}$ .

We denote by ${\partial X}$ the geodesic boundary and ${\overline{X} = X\cup \partial X}$ . ${\partial X^K}$ and ${\overline{X}^K}$ will denote the Karpelevich boundary and compactification respectively.

For a parabolic subgroup ${P}$ the space ${X_P := M_P/(M_P\cap K)}$ is a symmetric space of rank strictly less than ${X}$ , called the called the boundary symmetric space.

2. The functions ${\pi_{v, w}: X_v \to \overline{X_w}^K}$

Every point in ${\partial X^K}$ can be written as a pair ${(v, w)}$ where ${v\in \partial X}$ and ${w\in \overline{X_{P_v}}^K}$ . If ${X}$ is rank 1,then ${\overline{X}^K:= X\cup \partial X}$ by definition. The boundary is thus definied inductively. If ${v}$ is regular ${X_{P_{v}} }$ is a point, so ${w = \star}$ is uniquely determined by ${v}$ in this case and we write ${(v,\star)}$ which shows that one can think of the regular boundary ${\partial X_{reg}\subset \partial X}$ as sitting inside the Karpelevich boundary. If ${v}$ is an irregular boundary point of a rank 2 symmetric space, then ${X_{P_z}:= X_z}$ is a rank 1 symmetric space hence ${w\in \overline{X}_{z}^K = X_z\cup \partial X_z}$ . If it is of higher rank, the process is continued by induction till one reaches the familiar geodesic comapctification. The main difficulty in defining the topology on the boundary of the Karpelevich compactification is the definition of the maps ${\pi_{v, w}: X_v\to \overline{X_w}^K}$ for “close enough” points ${v, w \in \partial X}$ (see ). The authors definie these maps first for points ${v,w}$ where ${P_v\subset P_w}$ , then extend it to a small neighborhood of ${v}$ by perturbing it with a small neighborhood of ${e\in G}$ . After defining these maps, one can define the topology as the weak topology with respect to these family of maps, together with the quotient map ${\partial X^K\to \partial X}$ . More explicitly, one says that a sequence ${(v_i, w_i)}$ converges to ${(v, w)}$ in ${\partial X^K}$ if there is a sequence ${g_i \to e}$ in ${G}$ such that

${v_i \to v }$ in ${\partial X}$ ,
${P_{gv_i}\subset P_z}$ (for large enough ${i}$ )
${\pi_{gv_i, z}(w_i) \in \overline{X_z}^K}$ converges to ${w}$

We need the perturbation by ${g_i}$ for ${\pi_{gv_i, z}}$ to be well defined, as they are only defined for nested parabolic subgroups.

To simplify the exposition significantly, the we look at the case where ${rk(X) = 2}$ . In this case ${\overline{X}^K}$ is equal to the Martin compactification ${\overline{X}^M}$ . Let ${Q\subset P}$ be nested parabolic subgroups, then ${Q}$ determines a parabolic subgroup ${\tilde{Q}}$ of ${X_P}$ , hence a point on the geodesic boundary ${\partial X_P}$ . The parabolic subgroup is (eventually) defined in (see I.1.21 and onward) to be simply given by

$\displaystyle \tilde{Q}= Q \cap M_P$

We have the following consequence of the discussion in the above reference

Lemma 1 Let ${Q, Q'\subset P}$ be parabolic subgroups of a parabolic subgroup ${P}$ , then

${\tilde{Q} = Q\cap M_P = Q'\cap M_P = \tilde{Q}'} \quad \text{if and only if} {Q = Q'}.$

Proof

The “if” part is trivial. To show the converse implication, we use the expression for the Langlands decomposition of ${\tilde{Q} = Q\cap M_P}$ given in I.1.21. Namely,

$\displaystyle N_Q = N_PN_{\tilde{Q}} \qquad A_Q = A_PA_{\tilde{Q}} \qquad M_Q = M_{\tilde{Q}}$

Hence if ${\tilde{Q} = \tilde{Q}'}$ then ${N_{\tilde{Q}} = N_{\tilde{Q'}} }$ , ${A_{\tilde{Q}} = A_{\tilde{Q'}} }$ and ${M_{\tilde{Q}} = M_{\tilde{Q'}}}$ . This implies that ${N_Q = N_{Q'}}$ , ${A_Q = A_{Q'}}$ and ${M_Q = M_{Q'}}$ and thus ${Q = Q'}$

Using the above lemma one can show that sequences in ${\partial X_{reg}}$ converging to irregular points lift to convergent sequences in ${\partial X^K}$ whose limit now depends on what Weyl chamber the sequence is in:

Lemma 2 Assume that ${X}$ has rank 2 and let ${\partial X_{reg}}$ be the regular boundary, treated as a subset of the Karpelevich boundary. Let ${A, A'\subset \partial X_{reg}}$ be two distinct positive Weyl chambers at infinity. If ${v_i \in A}$ and ${v_i'\in A'}$ are two sequences converging to an irregular boundary point ${z \in \partial X_{irr} \subset \partial X}$ , then the corresponding sequences ${(v_i, \star)}$ and ${(v_i', \star)}$ converge to distinct points in ${\partial X^K}$ , namely

$\displaystyle \lim (v_i, \star) = (z, \tilde{P_{v_0}}) \qquad \lim(v_i', \star) = (z, \tilde{P_{v'_0}} )$

where ${v_0}$ is the first (or any other) element of the sequence ${v_i}$ and ${\tilde{P_{v_0}}}$ is the associated parabolic subgroup of ${X_{P_z}}$ treated as a point in ${\partial X_{P_z}^K = \partial X_{P_z}}$ .

Proof

Sine we are in rank 2, there is a neighborhood ${U}$ of the irregular boundary point ${z\in \partial X}$ such that ${P_x\subset P_z}$ for all ${x\in U}$ , hence we can ignore the perturbation sequence ${g_i}$ in the definition of convergence above. Since ${v_i}$ is regular, ${\pi_{v_i, z}}$ maps to the boundary ${\partial X^K = \partial X}$ (see the remark on p.77 of ). The corresponding point on this boundary is ${\tilde{P}_{v_i} = P_{v_i}\cap M_z}$ . Since all ${P_{v_i}}$ are isomorphic as ${v_i}$ lie in the same positive Weyl chamber, the maps ${\pi_{v_i, z}}$ are constant in ${i}$ , hence clearly convergent and using the previous lemma and the definition of convergence the claim follows.

3. Extending the Poisson integral on the boundary

Using the lemmas in the previous section we can show the following:

Proposition 3 Let ${f\in C(G/P_0)}$ be a continuous function on the Furstenberg boundary ${G/P_0 \subset \partial X^K}$ and ${(\mu_x)_{x\in X}}$ the Patterson-Sullivan densities. Let ${F}$ be the (Poisson) integral

$\displaystyle F(x) : = \int_{G/P_0} f(v) d\mu_x(v)$

and ${F' = F|_{\partial X_{reg}}}$ the restriction of ${F}$ to the regular boundary. Then ${F'}$ extends to the closure of the regular boundary in ${\partial X^K}$ .

Proof

The function ${F}$ is known to extend continuously to ${X\cup \partial X_{reg}}$ (see ) and since we have an equivariant imbedding ${\iota: X\cup \partial X_{reg} \to \overline{X}^K}$ extending the identity on ${X}$ , the function ${F}$ also extends to this image in ${\overline{X}^K}$ .

Denote by ${F' = F|_{\partial X_{reg}}}$ . Then ${F'}$ is known to be constant on positive Weyl chambers with value equal to ${f(v)}$ where ${v}$ is the unique point in ${G/P_0}$ that intersects the Weyl chamber (recall that ${G/P_0}$ parameterizes the positive Weyl chambers). It follows that ${F'}$ rarely extends to ${\partial X}$ as any irregular boundary point will be in the closure of (uncountably) many positive Weyl chambers and ${F'}$ . However, by the previous lemma, the limit of a sequences of regular boundary points in ${\partial X^K}$ depend on the positive Weyl chamber, hence the function ${F'}$ can be extended continuously to the closure of ${\partial X_{reg}}$ in ${\partial X^K}$ , and it is easy to check that this extension is also continuous.

It may be tempting to hope that the above Poisson integral can be extended to the closure of ${X\cup \partial X_{reg}}$ inside ${\overline{X}^K}$ , however the proof of the next lemma shows why this is not the case:

Lemma 4 Let ${\overline{X}^T}$ be any compactification of ${X}$ for which there is a surjective map

$\displaystyle p: \overline{X}^T\to X\cup \partial X$

sending ${X}$ to ${X}$ . Then there are functions ${f\in C(G/P_0)}$ for which ${F}$ does not extend to ${\overline{X}^T}$ .

Proof

Let ${x_i\in X}$ be a sequence with ${\lim x_i = z\in \partial X}$ . Then ${\lim F(x_i)}$ is well defined (even if ${z}$ is irregular). This implies a possible extension ${F'}$ of ${F}$ to ${\overline{X}^T}$ would have to be constant on each fiber ${\phi^{-1}(z)}$ ( ${z\in \partial X}$ ).

We have seen that ${F}$ in general does not extend to a continuous function on ${X\cup \partial X}$ but has jump discontinuities when passing between cells of the closed Weyl chambers. We denote also by ${F}$ this discontinuous extension. There are thus sequences ${v_i, w_i \in \partial X}$ converging to a common point ${z}$ for which ${\lim F(v_i) \neq \lim F(w_i)}$ . Let ${\tilde{v_i}, \tilde{w}_i}$ be lifts of the above sequences to ${\overline{X}^T}$ assumed wlog to converge. Again, let ${F'}$ denote a candidate for our extension of ${F}$ to ${\overline{X}^T}$ . Since ${F'}$ is constant on each fiber of ${\phi}$ , ${\lim F'(\tilde{w_i}) = \lim F(w_i) \neq \lim F(v_i) = \lim F'(\tilde{v_i})}$ . Since both sequences converge to points in ${\phi^{-1}(z)}$ this shows ${F'}$ is not constant on the fiber of ${z}$ , which is a contradiction.

The above Lemma shows that for all the classical compactifications (Martin, Satake, Furstenberg, Karpelevich etc.) the Poisson integral does not extend continuously to the boundary in general. The reason is they all “dominate” the geodesic compactification, meaning the identity map on $X$ extends to a surjective map onto the geodesic compactification, hence the above Lemma applies.

In a nutshell, the use of Poisson integrals to solve Dirichlet type problems for the Poincare disc can be extended to rank 1 symmetric spaces but no further even if we augment the boundary of the space. See for the situation in rank 1.

References

Szabo, G., Wu, J., & Zacharias, J. (2017). Rokhlin dimension for actions of residually finite groups. ArXiv:1408.6096 [Math]. http://arxiv.org/abs/1408.6096

Guentner, E., Willett, R., & Yu, G. (2015). Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and $C^*$-algebras. ArXiv:1510.07769 [Math]. http://arxiv.org/abs/1510.07769

Conley, C., Jackson, S., Marks, A., Seward, B., & Tucker-Drob, R. (2020). Borel asymptotic dimension and hyperfinite equivalence relations. ArXiv:2009.06721 [Math]. http://arxiv.org/abs/2009.06721

Bell, G., & Dranishnikov, A. (2007). Asymptotic Dimension. ArXiv:Math/0703766. http://arxiv.org/abs/math/0703766

doi:10.1016/j.topol.2008.02.011 | Elsevier Enhanced Reader. (n.d.). https://doi.org/10.1016/j.topol.2008.02.011

Serre, J.-P. (1980). Trees and Amalgams. In J.-P. Serre, Trees (pp. 1–68). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-61856-7_1

Hill, T. P. (2021). An Evolutionary Theory for the Variability Hypothesis. ArXiv:1703.04184 [q-Bio]. http://arxiv.org/abs/1703.04184

Albuquerque, P. (1999). Patterson-Sullivan Theory in Higher Rank Symmetric Spaces. Geometric & Functional Analysis GAFA, 9(1), 1–28. https://doi.org/10.1007/s000390050079

Kadison, R. V., & Ringrose, J. R. (1972). Cohomology of operator algebras. III : reduction to normal cohomology. Bulletin de la Société Mathématique de France, 100, 73–96. https://doi.org/10.24033/bsmf.1731

Kadison, R. V., & Ringrose, J. R. (1971). Cohomology of operator algebras II. Extended cobounding and the hyperfinite case. Arkiv För Matematik, 9(1–2), 55–63. https://doi.org/10.1007/BF02383637

Kadison, R. V., & Ringrose, J. R. (1971). Cohomology of operator algebras: I. Type I von Neumann algebras. Acta Mathematica, 126(none), 227–243. https://doi.org/10.1007/BF02392032

Pop, F., & Smith, R. R. (2010). Vanishing of second cohomology for tensor products of type II1 von Neumann algebras. Journal of Functional Analysis, 258(8), 2695–2707. https://doi.org/10.1016/j.jfa.2010.01.013

Christensen, E., Pop, F., Sinclair, A., & Smith, R. (2003). Hochschild Cohomology Of Factors With Property Gamma. Annals of Mathematics, 158, 635–659.

Hochschild cohomology of factors with property $\Gamma$ | Annals of Mathematics. (n.d.). Retrieved September 4, 2021, from https://annals.math.princeton.edu/2003/158-2/p07

Cameron, J. (2009). Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras. Proceedings of the Edinburgh Mathematical Society, 52, 287–295. https://doi.org/10.1017/S0013091507000053

Cameron, J. (2009). Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras. Proceedings of the Edinburgh Mathematical Society. https://doi.org/10.1017/S0013091507000053

Johnson, B. E., Kadison, R. V., & Ringrose, J. R. (1972). Cohomology of operator algebras. III : reduction to normal cohomology. Bulletin de La SociéTé MathéMatique de France, 79, 73–96. https://doi.org/10.24033/bsmf.1731

Sinclair, A. M., & Smith, R. R. (1998). Hochschild Cohomology for Von Neumann Algebras with Cartan Subalgebras. American Journal of Mathematics, 120(5), 1043–1057. https://www.jstor.org/stable/25098635

Qian, W., & Shen, J. (2014). Hochschild cohomology of type II$_1$ von Neumann algebras with Property $\Gamma$. ArXiv:1407.0664 [Math]. http://arxiv.org/abs/1407.0664

Sinclair, A. M., & Smith, R. R. (1995). Hochschild Cohomology of Von Neumann Algebras. Cambridge University Press.

Hochschild Cohomology of Von Neumann Algebras | Abstract analysis. (n.d.). Cambridge University Press. Retrieved September 4, 2021, from https://www.cambridge.org/it/academic/subjects/mathematics/abstract-analysis/hochschild-cohomology-von-neumann-algebras, https://www.cambridge.org/it/academic/subjects/mathematics/abstract-analysis

Eynard, B. (2014). A short overview of the “Topological recursion.” ArXiv:1412.3286 [Hep-Th, Physics:Math-Ph]. http://arxiv.org/abs/1412.3286

Pimsner, M. V. (1986). KK-groups of crossed products by groups acting on trees. Inventiones Mathematicae, 86(3), 603–634. https://doi.org/10.1007/BF01389271

Pimsner, M. (1986). KK-groups of crossed products by groups acting on trees. https://doi.org/10.1007/BF01389271

Rieffel, M. (1982). Applications of strong Morita equivalence for transofrmation C*-algebras. 38(1). https://math.berkeley.edu/~rieffel/papers/applications.pdf

Kloeckner, B. (2010). Symmetric spaces of higher rank do not admit differentiable compactifications. Mathematische Annalen, 347(4), 951–961. https://doi.org/10.1007/s00208-009-0464-z

Paradan, P.-E. (2009). Symmetric spaces of the non-compact type: Lie groups. Séminaires et Congrès, Géométries à courbure négative ou nulle, groupes discrets et rigidités(18), 39–76. https://hal.archives-ouvertes.fr/hal-00773255

Connes, A., & Skandalis, G. (1984). The Longitudinal Index Theorem for Foliations. Publications of the Research Institute for Mathematical Sciences, 20(6), 1139–1183. https://doi.org/10.2977/prims/1195180375

Schroeder, M. (2010). Patterson-Sullivan distributions for symmetric spaces of the noncompact type. ArXiv:1012.1113 [Math]. http://arxiv.org/abs/1012.1113

Helgason, S. (1979). Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press.

Emerson, H., & Meyer, R. (2006). Euler characteristics and Gysin sequences for group actions on boundaries. Mathematische Annalen, 334(4), 853–904. https://doi.org/10.1007/s00208-005-0747-y

Blackadar, B. (1986). K-Theory for Operator Algebras. Springer-Verlag. https://doi.org/10.1007/978-1-4613-9572-0

Ji, L., & Macpherson, R. (2002). Geometry of compactifications of locally symmetric spaces. Annales de l’institut Fourier, 52(2), 457–559. https://doi.org/10.5802/aif.1893

Ulsnaes, T. (n.d.). Note series.

Ulsnaes, T. (2021). The Karpelevic compactification.

(N.d.).

Jin, X. (2021). Homological Mirror Symmetry for the universal centralizers I: The adjoint group case. ArXiv:2107.13395 [Math-Ph]. http://arxiv.org/abs/2107.13395

Soule, C. (2004). An introduction to arithmetic groups. https://hal.archives-ouvertes.fr/hal-00001348

Rosenberg, J. M., & Schochet, C. (1986). The Künneth theorem and the universal coefficient theorem for equivariant K-theory and KK-theory. American Mathematical Society.

Rieffel, M. A. (2002). Group C∗-Algebras as Compact Quantum Metric Spaces. Documenta Mathematica, 48.

Ulsnaes, T. (n.d.). Compactifications of symmetric spaces and their applications. X ., 52.

Ulsnaes, T. (n.d.). An example of the index of a Rockland operator on a contact manifold. 68.

Ulsnaes, T. (n.d.). A simple proof of the Hard Lefschetz theorem. 37.

Ulsnaes, T. (n.d.). Pensum MATH4450 Videreg˚aende Funksjonal analyse V˚ar 2017. 22.

Videregående Funksjonalanalyse. (n.d.). Retrieved June 22, 2021, from https://www.overleaf.com/project/59920ace6080cd3eb133870e

Erratum:

Lemma 4 is not true! The reason it fails it that it relies on the fact that for any convergent seuqnece $x_i \in X$ with $x_i\to x_\infty \in X(\infty)$ the corresponding measures $\mu_{x_i}$ converges weakly. If this was true, the Lemma would be correct. Unfortunately, this result, which Theorem 2.4 of this article, is not true. The Poisson integral to the boundary for any compactification dominating the maximal Furstenberg compactification, like the Karpelevich compactification.

1. Definitions and notation

2. The functions

3. Extending the Poisson integral on the boundary

References

Erratum:

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2. The functions ${\pi_{v, w}: X_v \to \overline{X_w}^K}$