Gelfand pairs admit an Iwasawa decomposition

TL;DR

This paper proves that all Gelfand pairs can be decomposed into a product involving an amenable subgroup, leading to new classifications and a canonical family of spherical functions.

Contribution

It establishes that every Gelfand pair admits an Iwasawa decomposition with an amenable subgroup, providing new insights into their structure and applications.

Findings

Gelfand pairs admit an Iwasawa decomposition G=KP.

Complete classification of non-positively curved Gelfand pairs.

Introduction of a canonical family of spherical functions.

Abstract

Every Gelfand pair (G,K) admits a decomposition G=KP, where P<G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous. Applications include the complete classification of non-positively curved Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand--Godement for general Gelfand pairs.

Equations

$$\phi (x)\phi (y)={\int }_K\phi (xky)dk\forall x,y\in G$$

$${\phi }_s(g)={\int }_K\varrho (g^{-1}k{)}^{\frac{1}{2}+is}dk.$$

$$({\pi }_s(g)f)(xP)=f(g^{-1}xP){\left(\frac{\varrho (g^{-1}x)}{\varrho (x)}\right)}^{\frac{1}{2}+is}.$$

Full text

Applications include the complete classification of non-positively curved Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand–Godement for general Gelfand pairs.

Let $G$ be a locally compact group. The space $\mathscr{M}^{\mathrm{b}}(G)$ of bounded measures on $G$ is an algebra for convolution, which is simply the push-forward of the multiplication map $G\times G\to G$.

This definition, rooted in Gelfand’s 1950 work [14], is often given in terms of algebras of functions [12]. This is equivalent, by an approximation argument in the narrow topology, but has the inelegance of requiring the choice (and existence) of a Haar measure on $G$.

Examples of Gelfand pairs include notably all connected semi-simple Lie groups $G$ with finite center, where $K$ is a maximal compact subgroup. Other examples are provided by their analogues over local fields [18], and non-linear examples include automorphism groups of trees [23],[2].

All these “classical” examples also have in common another very useful property: they admit a co-compact amenable subgroup $P<G$. In the semi-simple case, $P$ is a minimal parabolic subgroup. Moreover, the Iwasawa decomposition implies that $G$ can be written as $G=KP$. This note shows that this situation is not a coincidence:

The mere existence of $P$ has a number of strong consequences discussed below. Most immediate is that $G$ belongs to the exclusive club whose members boast a homogeneous Furstenberg boundary:

Then the Furstenberg boundary of $G$ is the homogeneous space $\partial G=G/P\cong K/(K\cap P)$.

In particular, $P$ is unique up to conjugacy.

Another general consequence is that $G$ is exact in the sense of C*-algebras [20, §7.1].

We next derive a more geometric illustration of how consequential the existence of $P$ is. The above classical examples of Gelfand pairs are all CAT(0) groups in the sense that they occur as cocompact isometry groups of non-positively curved spaces: either Riemannian symmetric spaces or Euclidean buildings. General CAT(0) groups constitute a much more cosmopolitan category populated by all sorts of exotic spaces hailing from combinatorial group theory, Kac–Moody theory, etc. Using the “indiscrete Bieberbach theorem” established with P.-E. Caprace [9], the Theorem of this note leads to a complete classification of CAT(0) Gelfand pairs:

Then $X$ is a product of Euclidean spaces, Riemannian symmetric spaces of non-compact type, Bruhat–Tits buildings and biregular trees.

In particular, $G$ lies in a product of Gelfand pairs belonging to the classical sets of examples above.

This statement contains for instance a result by Caprace–Ciobotaru [5], namely: let $X$ be an irreducible locally finite thick Euclidean building. If $G={\rm Aut}(X)$ (or any co-compact subgroup $G<{\rm Isom}(X)$) is a Gelfand pair for some compact $K<G$, then $X$ is Bruhat–Tits.

Similarly, the statement contains some cases of results by Abramenko–Parkinson–Van Maldeghem [1] and Lécureux [21, §7],[22] establishing the non-commutativity of Hecke algebras associated to certain Coxeter groups. Namely, when Kac–Moody theory associates to them a locally finite thick building, Corollary 2 implies that the Hecke algebra can only be commutative in the affine case.

The “Iwasawa decomposition” $G=KP$ is stronger yet than the existence of $P$. For instance, it is a key ingredient for results of Furman [13, Thm. 10] and it could shed some light on the spherical dual of $G$, see below. It should also impose further restrictions on the centraliser lattice in case $G$ is a compactly generated simple group, see [10]. Already the existence of $P$ implies that this lattice is at most countable: see [10, pp. 11–12] and use that $G/P$ is metrisable in this setting.

We now contemplate some of the analytic legacy that the decomposition $G=KP$ bestows upon a general Gelfand pair $(G,K)$. Following Gelfand and Godement [17], the fundamental building block of non-commutative Fourier–Plancherel theory is given by positive definite spherical functions on Gelfand pairs, namely continuous $\varphi\colon G\to\mathbf{C}$ satisfying

[TABLE]

where the integration is with respect to the unique Haar probability measure on $K$; see also [11] and [26]. This is the abstract generalisation of addition formulas for special functions such as Legendre functions [25].

Here is how $P$ enters the picture:

Let $\nabla_{P}$ be the modular function of $P$, which is non-trivial unless $G$ itself is amenable and $G=P$. Then $\varrho(kp)=\nabla_{P}(p)$ gives a well-defined continuous function $\varrho\colon G\to\mathbf{R}_{>0}$ when $k\in K$, $p\in P$ because $\nabla_{P}$ vanishes on $K\cap P$. For every parameter $s\in\mathbf{C}$, define

[TABLE]

In view of Corollary 1, $\varphi_{s}$ is actually canonically attached to the pair $(G,K)$ up to conjugation. On the other hand, $\varphi_{s}$ is the matrix coefficient of the (projectively) unique $K$-fixed vector in a parabolically induced representation from $P$. In particular, $\varphi_{s}$ is a pure positive definite spherical function on $G$ for each real $s$.

This is classical for semi-simple groups, where $\varphi_{s}$ above is the Harish-Chandra formula; the Theorem makes it available for general Gelfand pairs, as desired by Godement [16, §16]. Of course this only gives a principal series and suggests to investigate fully the characters of $P$.

Proof of the Theorem and of Corollary 1. We recall that an affine $G$-flow is a non-empty compact convex set $C$ in some locally convex topological vector space over $\mathbf{R}$, endowed with a jointly continuous $G$-action preserving the affine structure of $C$. An affine flow is called irreducible if it does not contain any proper affine subflow. An argument due to Furstenberg implies that $G$ admits an irreducible flow $\Delta G$ which is universal in the sense that it maps onto every irreducible flow. Moreover, $\Delta G$ is unique up to unique isomorphisms. It turns out that $\Delta G$ is the simplex of probability measures $\mathscr{P}(\partial G)$ over the Furstenberg boundary $\partial G$ of $G$, and that this is actually one of the possible definitions of $\partial G$. For all this, we refer to [15].

We shall be more interested in the convex subset $\mathscr{P}(G)$ of $\mathscr{M}^{\mathrm{b}}(G)$ consisting of the probability measures, as well as in the corresponding subset $\mathscr{P}(G)^{K,K}$. We note the following straightforward facts:

• •

$\mathscr{P}(G)$ is closed under the multiplication given by convolution.

• •

The monoid $\mathscr{P}(G)$ contains $G$ via the identification of points with Dirac masses.

• •

The normalised Haar measure $\kappa$ of $K$ is an idempotent belonging to $\mathscr{P}(G)^{K,K}$.

• •

$\mathscr{P}(G)^{K,K}=\kappa\mathscr{P}(G)\kappa$; it is a monoid with $\kappa$ as identity.

By generalised vector-valued integration [4, IV§7.1], any affine $G$-flow $C$ is endowed with an action of the monoid $\mathscr{P}(G)$ which is affine in both variables. It will be crucial below that this action is moreover continuous for the variable in $C$. One way to see this is to check first that any $\mu\in\mathscr{P}(G)$ induces a continuous map $C\to\mathscr{P}(C)$ by push-forward on orbits, using that the $G$-action on $C$ is equicontinuous over compact subsets of $G$. Then observe that the action of $\mu$ is obtained by composing this map $C\to\mathscr{P}(C)$ with the continuous barycenter map $\mathscr{P}(C)\to C$.

Since $K$ is compact, it has a non-empty fixed-point set $C^{K}$; better yet, the idempotent $\kappa$ provides a continuous projection $\kappa\colon C\to C^{K}$. In particular, the monoid $\mathscr{A}=\kappa\mathscr{P}(G)\kappa$ preserves the convex compact set $C^{K}$.

Only now do we use the assumption that we have a Gelfand pair: the monoid $\mathscr{A}$ is commutative. Since $\mathscr{A}$ acts by continuous operators, the Markov–Kakutani theorem therefore implies that $\mathscr{A}$ fixes a point $p$ in $C^{K}$. From now on, we assume that $C$ is irreducible. The convex set $\mathscr{P}(G)p$ is $G$-invariant and hence must be dense. It follows that $\kappa\mathscr{P}(G)p$ is dense in $C^{K}$, but $\kappa\mathscr{P}(G)p$ is $\mathscr{A}p$ which is reduced to $p$. In conclusion, we have shown that $K$ has a unique fixed point in $C$.

We now apply this to the case where $C=\Delta G$ is the simplex of probability measures on $\partial G$ and deduce that $K$ fixes a unique such measure on $\partial G$. Since $K$ is compact, every $K$-orbit supports an invariant measure: the push-forward of $\kappa$. This implies that $K$ has a single orbit in $\partial G$. In particular, $\partial G=G/P$ for some co-compact subgroup $P<G$ and moreover $G=KP$.

Next, we observe that $P$ is relatively amenable in $G$, which means by definition that every affine $G$-flow has a $P$-fixed point. Indeed, this property characterises the subgroups that fix a point in $\Delta G$: this follows from the universal property of $\Delta G$. This characterisation also implies that this $P$ is already maximal relatively amenable. Indeed, if $P^{\prime}<G$ is relatively amenable and contains $P$, it also fixes a point in $\Delta G$; this induces an affine $G$-map $\mathscr{P}(G/P^{\prime})\to\mathscr{P}(G/P)$, which must be the identity by universality of $\mathscr{P}(G/P)=\Delta G$.

We recall that relative amenability is equivalent to amenability in a wide class of ambient locally compact groups $G$ including all exact groups, but it is only a posteriori that the Theorem implies that $G$ is exact, see [20, §7.1]. In the locally compact setting, it is still an open question to exhibit an example where the weaker relative notion does not coincide with amenability [8]. In the co-compact case, however, we can settle the question with the Proposition below and conclude that $P$ is amenable. Thus the Proposition will complete the proof.

The following statement is a very basic case of much more general results by Andy Zucker [27, Thm. 7.5]; the elementary proof below is inspired by reading his preprint.

We now justify our claims concerning the functions $\varphi_{s}$ on $G$. Since $G$ is unimodular (reference above), Weil’s integration formula [3, VII§2.5] implies that the push-forward of $\kappa$ on $G/P$ has a Radon–Nikodým cocycle given at $(g,xP)$ by $\varrho(g^{-1}x)/\varrho(x)$. Therefore, the unitary induction $\pi_{s}$ of the character $\nabla_{P}^{is}$ is given on various spaces of functions $f$ on $G/P$ by

[TABLE]

The only $K$-invariant vectors $v$ are constant functions on $G/P$ and hence the associated matrix coefficient $\varphi_{s}(g)=\langle\pi_{s}(g)v,v\rangle$ is uniquely defined once $v$ has unit norm. The fact that $\varphi_{s}$ is pure and spherical (for $s\in\mathbf{R}$) now follows from the general theory of Gelfand pairs, specifically I.II.6 and I.III.2 in [12].

Acknowledgements. I am grateful to Andy Zucker for sending me his preprint and to Pierre-Emmanuel Caprace for several insightful comments on a preliminary version.