Journal of Lie Theory Volume 9 (1999) 285–291 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag Berlin

Characterizing weakly symmetric spaces as Gelfand pairs

Hiêú Nguyêñ

Communicated by J. Faraut

Abstract. Let (M, G, μ) be a Riemannian weakly symmetric space. Fix a base point x_{0} ∈ M and denote by H to be the compact isotropy subgroup of G at x_{0} . It is proven that L^{1}(H \G/H) is commutative, i.e. (G, H) is a Gelfand pair. This extends É. Cartan’s result for Riemannian symmetric spaces. Conversely, if (G, H) is a Riemannian weakly symmetric pair, then M = G/H can be made to be Riemannian weakly symmetric. An application of this result is presented.

1. Introduction

In this paper, we characterize Riemannian weakly symmetric spaces as a special class of Gelfand pairs. The notion of a weakly symmetric space was introduced by A. Selberg in [8] as a generalization to that of a symmetric space. He also constructed examples of weakly symmetric (but not symmetric) spaces by endowing the total space of principal S^{1}-bundles over the Siegel half-space with a twisted action by the group Sp(n, ℝ) × S^{1} (cf. [6]). The topic has recently gained much interest with discoveries of many new examples, especially by J. Berndt and L. Vanhecke in [1], who found an alternative characterization of weak symmetry based on the reversal of geodesics, and by W. Ziller in [12], who replaced geodesics with tangent vectors. Recently, the author in [7] was able to extend Selberg’s results to hermitian symmetric spaces based on Ziller’s infinitesimal characterization.

Let M be a Riemannian manifold, G a locally compact transitive Lie group of isometries of M, and μ a fixed isometry of M (not necessarily in G) such that (M, G, μ) is a Riemannian weakly symmetric space. Fix a base point x_{0} ∈ M and let H be the compact isotropy subgroup of G at x_{0}. Selberg was able to prove in [8] that D(G/H), the space of G-invariant differential operators on M = G/H, is commutative. If we further assume that G is connected, then a result of E.G.F. Thomas [9] says that commutativity of D(G/H) is equivalent to commutativity of L^{1}(H \G/H), the space of integrable functions on G bi-invariant under H. Therefore, under this additional assumption, it follows that if (M = G/H, G, μ) is weakly symmetric, then (G, H) is a Gelfand pair. In this paper, we remove the restriction that G must be connected and give a simple direct proof of this

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag result as stated in the following theorem, which extends É. Cartan’s result for Riemannian symmetric spaces (cf. [10], Cor. 2.7).

Theorem 1.1. If (M, G, μ) is a Riemannian weakly symmetric space and H is the compact isotropy subgroup of G at a base point of M, then L^{1}(H\G/H) is commutative, i.e. (G, H) is a Gelfand pair.

Our argument proceeds as follows. We first define a Riemannian weakly symmetric space (M, G, μ) and demonstrate that it is better characterized as a homogeneous weakly symmetric space G/H. We then establish an equivalence between G/H with weakly symmetric pairs (G, H). Our theorem will follow by showing that when H is compact, these weakly symmetric pairs (G, H) are in fact Gelfand pairs. We also prove as a converse that a Riemannian weakly symmetric pair (G, H) produces a Riemannian weakly symmetric space M = G/H and go on to apply this result in an example.

2. Proof of Theorem

Let us begin with some preliminaries. We shall always assume in this paper that G is a locally compact Lie group, H a closed subgroup of G and e the identity element in G. We also denote by Inn_{H}(G) to be the subgroup of Aut(G) consisting of inner automorphisms of the form I_{h} : g ↦ hgh^{−1}, where h ∈ H and g ∈ G.

Definition 2.1. (Selberg [8]) The triple (M, G, μ) is called a Riemannian weakly symmetric space if the following properties are satisfied:

(i) M is a Riemannian manifold, G is a locally compact transitive Lie group of isometries of M, and μ is a fixed isometry of M (not necessarily in G) satisfying μGμ^{−1} = G and μ^{2} ∈ G, (ii) Given any two points x and y in M, there exists an element g ∈ G such that

gx = μy and gy = μx.  (1)

Unfortunately, this definition of Selberg does not make any reference to the isotropy subgroups of G with respect to its action on M. Since such a subgroup of G must be specified in order to make the connection with Gelfand pairs, we revise his definition to better fit this framework.

Definition 2.2. G/H is called a homogeneous weakly symmetric space if there exists an analytic diffeomorphism μ of the homogeneous manifold M = G/H such that

(i) μGμ^{−1} = G, μ(eH) = eH and μ^{2} ∈ H, (ii) Given any two points x and y in M, there exists an element g ∈ G such that

gx = μy and gy = μx.  (2)

The following lemma shows that indeed Selberg’s definition is compatible with ours.

Lemma 2.3. Let (M, G, μ) be a Riemannian weakly symmetric space. Fix any base point x_{0} ∈ M and let H be the compact isotropy subgroup of G at x_{0}. Then there exists an isometry μ̃ of M = G/H which makes G/H a homogeneous weakly symmetric space.

Proof. Set y_{0} = μx_{0}. By transitivity of G, there exists an element g̃ ∈ G such that g̃y_{0} = x_{0}. Define μ̃ = g̃μ. Then μ̃ fixes x_{0} and satisfies μ̃Gμ̃^{−1} = G. We next identify M with the homogeneous space G/H and the point x_{0} with the coset eH so that μ̃(eH) = eH and μ̃^{2} ∈ H. By assumption, given any two points x and y of M, there exists an element g ∈ G such that gx = μy and gy = μx. It follows that g̃gx = μ̃y and g̃gy = μ̃x. Hence, G/H is a homogeneous weakly symmetric space with respect to μ̃.

We next establish an equivalence between homogeneous weakly symmetric spaces and weakly symmetric pairs. These pairs possess special symmetry conditions that will ensure that they are Gelfand pairs.

Definition 2.4. (G, H) is called a weakly symmetric pair if there exists an automorphism θ of G such that

(i) θ(H) ⊆ H and θ^{2} ∈ Inn_{H}(G), (ii) Hθ(g)H = Hg^{−1}H for all g ∈ G.

Furthermore, (G, H) is called a Riemannian weakly symmetric pair if Ad_{G}(H) is compact.

Lemma 2.5. (G, H) is a weakly symmetric pair if and only if G/H is a homogeneous weakly symmetric space.

Proof. Let (G, H) be a weakly symmetric pair and θ the corresponding automorphism of G. We define a map μ of the homogeneous manifold M = G/H by

μ(gH) = θ(g)H, gH ∈ G/H.

Then μ is an analytic diffeomorphism since θ is an automorphism. It is also clear that μ(eH) = θ(e)H = eH and μ^{2} ∈ H since θ^{2} ∈ Inn_{H}(G). This proves property (i) of a homogeneous weakly symmetric space.

To prove property (ii), we will first show that it holds for any point x_{0} ∈ M and the origin eH. Since G is transitive on M, there exists an element g_{1} ∈ G such that g_{1}x_{0} = g_{1}g_{0}H = eH. Also, by assumption, we have Hθ(g_{0})H = Hg_{0}^{−1}H. This allows us to write h_{0}θ(g_{0})H = g_{0}^{−1}H for some element h_{0} ∈ H. Setting g̃ = h_{0}^{−1}g_{0}^{−1}, it follows that

g̃(eH) = h_{0}^{−1}(g_{0}^{−1}H) = h_{0}^{−1}h_{0}θ(g_{0})H = μx_{0},  (3)

and

g̃x_{0} = h_{0}^{−1}g_{0}^{−1}(g_{0}H) = eH = μ(eH).  (4)

For the arbitrary case, fix x and y to be any two points of M and let g_{0} be the element of G which maps y to eH. Denote by x_{0} the image of x under g_{0}. By our previous work, there exists an element g̃ which satisfies (3) and (4) for the two points x_{0} and eH. Setting g = θ(g_{0}^{−1})g̃g_{0}, it follows that

gx = μy and gy = μx.

Hence, G/H is a homogeneous weakly symmetric space with respect to μ.

Conversely, let G/H be a homogeneous weakly symmetric space and μ the corresponding analytic diffeomorphism of G/H. We define a homomorphism θ of G by

θ(g) = μgμ^{−1}, g ∈ G.

Then θ is an automorphism of G since μ is an analytic diffeomorphism and μGμ^{−1} = G. It is clear that θ(H) ⊆ H and θ^{2} ∈ Inn_{H}(G) since μ fixes the point x_{0} = eH and μ^{2} ∈ H. This proves property (i) of a weakly symmetric pair. Furthermore, we have

μ(gH) = μgμ^{−1}(eH) = θ(g)(eH) = θ(g)H, gH ∈ G/H.

To prove property (ii), fix any element g_{0} ∈ G and denote x_{0} = g_{0}H. By assumption, there exists an element g ∈ G such that g(eH) = μx_{0} = θ(g_{0})H and gx_{0} = gg_{0}H = θ(e)H = eH. It follows that HgH = Hθ(g_{0})H and Hgg_{0}H = HeH. This allows us to write h_{1}gg_{0}h_{2} = e, for some h_{1}, h_{2} ∈ H. In other words, h_{1}g = h_{2}^{−1}g_{0}^{−1} or HgH = Hg_{0}^{−1}H. Hence, Hg_{0}^{−1}H = Hθ(g_{0})H and (G, H) is a weakly symmetric pair with respect to θ.

We are now ready to make the connection between Riemannian weakly symmetric spaces with Gelfand pairs.

Definition 2.6. Let H be a compact subgroup of G. Then (G, H) is called a Gelfand pair if the algebra L^{1}(H \G/H) is commutative under the convolution product.

Lemma 2.7. If (G, H) is a weakly symmetric pair and H is compact, then (G, H) is a Gelfand pair.

Remark 2.8. We mention that this lemma is a minor extension of a result due to J. Faraut [2], Prop. I.2., which in turn is an extension of the following classical result due to I.M. Gelfand [4]:

Theorem 2.9. ([4]) Let θ be an involutive automorphism of G = KP where K is compact, θ(k) = k for all k ∈ K and θ(p) = p^{−1} for all p ∈ P. Then (G, K) is a Gelfand pair.

Proof. (of Lemma 2.7) We follow the argument used in [2], Prop. I.2. Let f ∈ C_{c}(G), the space of continuous functions on G with compact support, and define

f^{θ}(g) = f(θ(g)) and f^{♯}(g) = f(g^{−1}).

Let dg be the Haar measure on G. The maps

f ↦ ∫_{G} f(g)dg and f ↦ ∫_{G} f^{θ}(g)dg

are left-invariant positive Radon measures on G. But it is well known that such a Radon measure is unique up to a constant factor. It follows that there exists a positive constant c such that

∫_{G} f^{θ}(g)dg = c ∫_{G} f(g)dg.  (5)

Now, choose any f ∈ C_{c}(H \G/H), the space of functions in C_{c}(G) bi-invariant under H, such that ∫_{G} f(g)dg ≠ 0. Since θ(H) ⊆ H and θ^{2} ∈ Inn_{H}(G) for a weakly symmetric pair, we must have (f^{θ})^{θ} = f. This implies c^{2} = 1 or c = 1 because of (5). It follows that (f_{1} ∗ f_{2})^{θ} = f_{1}^{θ} ∗ f_{2}^{θ} for any two functions f_{1}, f_{2} ∈ C_{c}(G).

Next, we notice that property (ii) of a weakly symmetric pair implies f^{θ} = f^{♯} for any f ∈ C_{c}(H \G/H). It follows that

∫_{G} f(g^{−1})dg = ∫_{G} f(θ(g))dg = ∫_{G} f(g)dg

and shows G is unimodular (cf. [2], Prop. I.1). Since ♯ is an anti-automorphism, it follows that (f_{1} ∗ f_{2})^{♯} = f_{2}^{♯} ∗ f_{1}^{♯} for any two functions f_{1}, f_{2} ∈ C_{c}(H \G/H). Hence, f_{1} ∗ f_{2} = f_{2} ∗ f_{1} and C_{c}(H\G/H) is commutative. According to [9], Prop. 2, this is equivalent to commutativity of L^{1}(H \G/H).

Proof. (of Theorem 1.1) Apply Lemmas 2.3, 2.5 and 2.7 in the order listed and use the fact that H is compact.

3. Criterion for Weak Symmetry

Notice that Lemma 2.5 also lets us state conditions as to when a pair (G, H) produces a Riemannian weakly symmetric space M = G/H.

Lemma 3.1. If (G, H) is Riemannian weakly symmetric pair, then M = G/H can be made to be Riemannian weakly symmetric.

Proof. Because of Lemma 2.5, G/H is a homogeneous weakly symmetric space with respect to the analytic diffeomorphism μ induced from θ. The argument now copies that used in the classical situation where (G, H) is a Riemannian symmetric pair (cf. [5], Ch. IV, §3, Prop. 3.4). Since Ad_{G}(H) is compact, there exists a Riemannian structure Q on the homogeneous manifold M = G/H such that it is invariant under the action of G by left-translation of cosets. Let N consist of those elements n ∈ G such that n acts as the identity mapping on M. Then there exists a closed Lie subgroup G̃ ⊂ I(M), the full isometry group of M, such that G̃ is isomorphic to G/N. Furthermore, it is clear that μ is an isometry of M since θ is an automorphism of G. Hence, (M, G̃, μ) is Riemannian weakly symmetric.

As an application, we shall use Lemma 3.1 to find examples of Riemannian weakly symmetric spaces. Following [3] and [7], we let (𝔤, σ) be an orthogonal symmetric Lie algebra of hermitian compact or non-compact type. Then 𝔤 is real semisimple. Decompose 𝔤 = 𝔨 + 𝔭 into its ±1-eigenspaces. Then 𝔨 is a compact Lie algebra and 𝔨 = 𝔨_{s} + 𝔷_{𝔨}, where 𝔨_{s} = [𝔨, 𝔨] is the semisimple part and 𝔷 is the center of 𝔨. Let 𝔞 be a maximal abelian subalgebra of 𝔭. Let G be a connected real Lie group with Lie algebra 𝔤 and denote by K, K_{s}, Z_{K} and A be the analytic subgroups of G corresponding to 𝔨, 𝔨_{s}, 𝔷_{𝔨} and 𝔞. Here, K_{s} is compact and K and Z_{K} are compact if and only if G has finite center.

According to [3], Prop. 2.1, there exists an involutive automorphism γ of 𝔤, which we shall assume lifts to G, such that γ(K_{s}) = K_{s}, γ(a) = a^{−1} for all a ∈ A and γ(k) = k^{−1} for all k ∈ Z_{K}. This gives the following result, which is implied in [3], Theorem 3.1, and independently proven by J.A. Wolf and A. Korányi in an unpublished work [11] using a similar argument. We mention that both of these papers also go on to prove that (G, K_{s}) is a Gelfand pair.

Lemma 3.2. If the hermitian symmetric space G/K contains no irreducible factors of tube type, then (G, K_{s}) is a Riemannian weakly symmetric pair with respect to θ = γ.

Proof. It is clear that (G, K_{s}) satisfies property (i) of a weakly symmetric pair. As for property (ii), Theorem 3.3 of [3] says that if G/K contains no irreducible factors of tube type, then it is equivalent to the following property:

K_{s}kgk^{−1}K_{s} = K_{s}gK_{s} for all k ∈ Z_{K} and g ∈ G.  (6)

Now, G has the well-known decomposition G = KAK. Also, K = K_{s}Z_{K}. It is now easy to check using these two decompositions and (6) that

K_{s}θ(g)K_{s} = K_{s}g^{−1}K_{s} for all g ∈ G.  (7)

This proves property (ii).

As a result, it follows that there exists an isometry group G̃ ≅ G/N such that (M = G/K_{s}, G̃, μ) is Riemannian weakly symmetric because of Lemma 3.1, where μ is the isometry of M induced from θ.

Remark 3.3. The author has already proven in [7] that M = G/K_{s} is weakly symmetric, but with respect to an isometry group larger than G, namely G^{1} := G×Z_{K}. More precisely, it was proven that (M, G̃^{1}, μ) is Riemannian weakly symmetric and proven without the restriction that G/K not contain any irreducible factors of tube type. It is in this sense that our result here that M is weakly symmetric with respect to G̃ is new. Furthermore, it was also proven in [7] that M is not Riemannian symmetric with respect to a special one-parameter family of Riemannian metrics in the case where G/K is a classical irreducible bounded symmetric domain.

References

[1] Berndt, J. and L. Vanhecke, Geometry of weakly symmetric spaces, J. Math. Soc. Japan 48 (1996), 745–760.

[2] Faraut, J., Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, “Analyse Harmonique,” 315–446, Les Cours de CIMPA, 1982.

[3] Flensted-Jensen, M., Spherical functions on a simply connected semisimple Lie group, Amer. J. Math. 99 (1977), 341–361.

[4] Gelfand, I.M., Spherical functions on a symmetric space, AMS Transl. 37 (1964), no. 2, 39–44.

[5] Helgason, S., “Differential Geometry, Lie Groups, and Symmetric Spaces”, Pure and Applied Mathematics 80, Academic Press, 1978.

[6] Maaß, H., “Siegel’s Modular Forms and Dirichlet Series,”Lecture Notes in Mathematics 216, Springer-Verlag, 1971.

[7] Nguyêñ, H., Weakly symmetric spaces and bounded symmetric domains, Transformation Groups 2 (1997) no. 4, 351-374.

[8] Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47–87.

[9] Thomas, E. G. F., An infinitesimal characterization of Gelfand pairs, Contemporary Mathematics 26 (1984) 379–385.

[10] Wolf, J. A., The uncertainty principle for Gelfand pairs, Nova J. Algebra Geom. 1 (1992), no. 4, 383–396.

[11] Wolf, J. A., and A. Korányi, Working notes, August, 1983.

[12] Ziller, W., Weakly symmetric spaces, “Topics in Geometry: In Memory of Joseph D’ Atri,” Progress in Nonlinear Diff. Eqs. 20, Birkhäuser, 1996, 355–368.

Department of Mathematics Rowan University Glassboro, NJ 08028, USA nguyen@rowan.edu

Received October 7, 1997 and in final form August 25, 1997