Notes on polar coordinates for symmetric spaces

In this short blog post, we are going to introduce a “coordinate system” on Riemannian symmetric spaces called polar coordinates, that arise as an immediate consequence of Cartan’s decomposition theorem. The quotation marks are due to the fact that we have some choice in how to express a point in these coordinates.

Let ${G}$ be a non-compact reductive Lie group with maximal compact subgroup ${K\subset G}$ fixed by the Cartan involution. By the definition of reductive Lie groups such a ${K}$ always exist (see p. 384 of ). Let ${\mathfrak{g} = \mathfrak{k}\oplus \mathfrak{p}}$ be the Lie algebra of ${G}$ with ${\mathfrak{k}}$ the Lie algebra of ${K}$ . Let ${\mathfrak{a}\subset \mathfrak{p}}$ be a maximal abelian subalgebra. Let ${A = Exp(\mathfrak{a}) \subset G}$ . We call ${\mathfrak{a}}$ and ${A}$ a Cartan subalgebra and subgroup respectively. Let ${\Sigma}$ be the collection of restricted roots of ${G}$ relative to ${A}$ , that is,

$\Sigma = \{ \lambda\in \mathfrak{a}^*~|~ \exists X\neq 0, ad_H(X) = [H, X] = \lambda(H) X ~\forall H\in \mathfrak{a}\}.$

Denote by ${W(G, A)}$ the associated Weyl group of the root system, that is the (Coxeter) group generated by reflections in ${\mathfrak{a}}$ about the hyperplanes ${ker(\lambda)}$ for ${\lambda\in \Sigma}$ . By identifying ${A = exp(\mathfrak{a})}$ , ${W(G, A)}$ also acts on ${A}$ .

The connected components of

$\left(\mathfrak{a} - \bigcup_{\lambda \in \Sigma} ker\lambda \right)= \{ H\in \mathfrak{a} ~|~ \lambda(H) \neq 0 ~ \forall \lambda\in \Sigma\}$

are called the positive Weyl chambers of ${\mathfrak{a}}$ . We pick one of these components and denote its image in ${A}$ under ${Exp}$ by ${A^+}$ and call it “the” positive Weyl chamber of ${A}$ . Then the closure ${\overline{A^+}}$ is a fundamental domain for the action of ${W(G, A)}$ on ${A}$ , meaning every element in ${A}$ is in the ${W(G, A)}$ orbit of a unique point in ${\overline{A^+}}$ .

Assuming now ${(G, K)}$ is a symmetric pair for which ${G/K}$ is a non-compact symmetric space, then the limit points of ${A^+\subset G/K}$ on the geodesic boundary is denoted ${A^+(\infty)}$ and called the positive Weyl chamber at infinity. We have a homeomorphism

$A^+(\infty ) = A^+\times \R$

The following theorem of Cartan can be found in the book of Knapp

Theorem 1 () Theorem 7.39 } Every element in ${G}$ has a decomposition as ${k_1 a k_2}$ with ${k_1, k_2 \in K}$ and ${a\in A}$ . In this decomposition, ${a}$ is uniquely determined up to conjugation by a member of the Weyl group ${W(G, A)}$ . If ${a = exp(H)}$ for ${H\in \mathfrak{a}}$ such that ${\lambda(H) \neq 0 }$ for all ${\lambda \in \Sigma}$ (i.e. ${a}$ is in a positive Weyl chamber), then ${k_1}$ is unique up to right multiplication by a member of ${M = Z_K(A)}$ (the centralizer in ${K}$ of ${A}$ ).

We have the following corollary

Corollary 2 (Polar coordinates on ${G/K}$ ) There is a well defined surjective continuous, ${K}$ -equivariant map ${\phi: K/M \times \overline{A}^+ \to X = G/K}$ given by

$\phi(kM, a) = kaK$

with following properties

${\phi}$ restricts to an imbedding ${K/M \times A^+ \to X}$ with open dense image.
${\phi(kM,a) = \phi(k'M, a')}$ implies ${a = a'}$ .

It follows that every element ${x\in X}$ can be written as ${x = ka}$ , where ${a \in \overline{A^+}}$ is unique, and if ${a}$ is not a on the boundary of ${\overline{A}^+}$ , ${k}$ is unique, modulo ${M = Z_K(A)}$ .

Proof

The previous theorem and the fact that ${\overline{A}^+}$ is a fundamental domain for ${W(G,A)}$ gives us that ${G = K\overline{A^+}K}$ and so

$X = G/K = K\overline{A^+}.$

The other claims follow more or less readily from the Theorem.

Remarks

If ${x = ka}$ with ${e\neq a \in \partial \overline{A^+}}$ , then ${k}$ is unique up multiplication by an element in the subgroup ${S \subset K}$ generated by
$S = \langle M_i ~|~ M_i = Z_K(A_i)\rangle$
where ${A_i}$ runs over all Cartan subgroups containing the geodesic ray associated with ${a}$ , that is, if ${a = exp(H)}$ for some ${H\in \mathfrak{a}}$ , then ${t\mapsto exp(tH)}$ is in ${A_i}$ for all ${t\geq 0}$ . Note that if ${a}$ was regular it is contained in a unique Cartan subgroup ${A}$ of ${G}$ through ${x_0}$ , hence ${S = M = Z_K(A)}$ for a unique ${A}$ .
The projection $\overline{k}: G\to K$ given by the Iwasawa decomposition $G = KAN$ of $G$ yields an action of $G$ on $K/M$ given by $g\cdot kM = \overline{k}(gk)M$ . With respect to this action the natural identification $K/M = G/P$ given by $kM \mapsto kP$ is $G$ -equivariant. To see that this action is well definied not that for any $g\in G$ and any $m\in M$
$\overline{k}(gm) = \overline{k}(kan m) = \overline{k}(km a n') = k(g)m$
as $M$ normalizes $N$ and centralizes $A$ . With respect to this $G$ -action the polar coordinate map is $G$ -equivariant on the dense open subset $K/M\times A^+$ , however for points $x \in X$ with $x = kaK$ and $a\in \partial A^+$ , we have $kaK = kmaK$ for $m\in S$ (defined above), and $S$ need not normalize $N$ nor centralize $A$ . We conclude that this $G$ -action is not well defined on outside this open dense region.

Example

Let us look at an example in 3 dimensions, which we can actually visualize. The drawing below illustrates the situation. Let $G = SL_2(\R)\rtimes \R$ . Any maximal compact subgroup of $G$ is conjugate to $K = SO_2(\R)\rtimes \{0\}$ .

The space $G/K$ is homeomorphic to a 3-dimensional ball, its geodesic boundary is thus the 2-sphere. There are two irregular boundary points which we can depict as the north and the south pole.

A choice of positive Weyl chamber $A^+$ is, in this picture, a half disk inside ball $G/K$ containing the axis that connects the north and the south pole. The space $K/M$ is called the Furstenberg boundary and sits inside the sphere as a circle perpendicular to $A^+(\infty)$ .

The polar coordinates at a point $x\in G/K$ expresses $x$ as an element in this half disk, together with an ‘angle of rotation’ about the north-south axis determined by an element in $K/M$ . If $x$ is on the axis of rotation, then the geodesic corresponding to $x$ is precisely this axis. Any Cartan subgroup of $G$ containing the origin in $G/K$ also contains this geodesic ray. One can then show this implies

$S = \langle M_i~|~ M_i = Z_K(A_i) \rangle = K.$

By the previous remark $x$ can thus be written uniquely as an product of an element $a\in \partial A^+$ and $k\in K/S = K/K = \{e\}$ .

The open dense subset ${K/M \times A^+ \subset X}$ is simply ${X}$ removed all geodesic rays centered at ${x_0 }$ in the direction of irregular boundary points. For the example of say $X = (SL_2(\R)\times \R) / SO_2(\R)\times \{0\}$ this is just the ball removed a straight line connecting the north and south pole.

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

Example

Leave a Reply Cancel reply