Journal of Lie Theory Volume 9 (1999) 355–360 C ◯1999 Heldermann Verlag

Homogeneous spaces of compact connected Lie groups which admit nontrivial invariant algebras

Ilyas A. Latypov

Communicated by E. B. Vinberg

Abstract. In 1965 Wolf [10] and Gangolli [1] proved that compact semisimple groups are distinguished in the class of all compact connected Lie groups by the following property: every uniformly closed function algebra which is invariant with respect to left and right translations is also invariant with respect to the complex conjugation. In this article we extend this result to the class of homogeneous spaces of compact connected Lie groups with connected stable subgroups: a homogeneous space admits only self-conjugated invariant function algebras if and only if the isotropy representation has no nonzero fixed vectors.

1. Introduction

Let M be a homogeneous space of a compact connected Lie group G acting on M by left. Denote by H the stable subgroup of some point x_{0} ∈ M, by N_{G}(H) the normalizer of H in G, and by N the group N_{G}(H)/H. Let C(M) be the commutative Banach algebra of all complex-valued continuous functions on M. Then G acts on C(M) by the formula

(L_{g}f)(x) = f(g^{−1}x), g ∈ G, x ∈ M.

We shall say that A is an invariant algebra on M if A is an invariant under this action of G uniformly closed subalgebra of C(M). A function algebra A will be called self-conjugated if it contains the complex conjugated function f̄ for any f ∈ A. The Stone-Weierstrass theorem implies that any self-conjugated uniformly closed function algebra on a compact Q may be identified with C(Q̃), where Q̃ is the factor of Q by the equivalence x ∼ y ⟺ f(x) = f(y) for all f ∈ A. If A is an invariant algebra then the equivalence is invariant. Hence the factor space is a homogeneous space M′ of G and the factorization is a continuous equivariant mapping π : M → M′ such that A = C(M′) ∘ π. Therefore, self-conjugated algebras on M are in one-to-one correspondence with closed subgroups of G which include H.

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag

The main result of this paper is

Theorem. Let G be a compact connected Lie group, H is its closed connected subgroup. Then the following conditions are equivalent:

(1) every uniformly closed invariant algebra on M = G/H is self-conjugated;

(2) the group N_{G}(H)/H is finite;

(3) the isotropy group of any point x_{0} ∈ M has no fixed nonzero vectors in the tangent space T_{x_{0}}M.

Bi-invariant uniform algebras on a compact group G are G×G-invariant algebras on G, where G × G acts on G by left anf right translations. The condition that the group N_{G×G}(G)/G is finite is equivalent to the condition that the group G is semisimple. Hence the theorem generalizes results of Wolf and Gangolli.

The case of invariant algebras on spheres is most investigated one among homogeneous spaces which are not of the type G×G/G. de Leeuw and Mirkil [6] showed that every invariant algebra on the real sphere S^{n} = SO(n + 1)/ SO(n) is self-conjugated. Nagel and Rudin [7] showed that there are invariant algebras on the complex speres S^{2n−1} = U(n)/ U(n − 1) which are not self-conjugated. The other transitive actions on S^{n} which admit only self-conjugated invariant algebras were described in [5]. In [11] Wolf proved that almost all compact homogeneous symmetric spaces admit only self-conjugated invariant algebras.

The way of the proof of the theorem was suggested by Gichev [2].

2. Finitely generated invariant algebras

Definition. An invariant algebra A will be called finitely generated if A is generated as a Banach algebra by a finite-dimensional invariant vector space V ⊂ C(M).

Let A be a finitely generated invariant algebra on M, V be a generating finite-dimensional invariant vector space.

The group G acts on the complex vector space V^{∗} of all linear functionals on V . This action extends to the action of the complexification G^{C} of G in V^{∗}.

Let δ be the evaluating functional for the point x_{0} on V : δ(f) = f(x_{0}) for all f ∈ V . Let O_{δ,V} be the G-orbit of the vector δ in V^{∗} and Oδ,V^{C} be the G^{C} -orbit of δ in V^{∗}. Denote by Stab δ and Stab_{C} δ the stabilizers of δ in G and G^{C} respectively; note that H ⊆ Stab δ. We obtain a continuous equivariant mapping M → O_{δ,V} . Since linear functions on V^{∗} are in one-to-one correspondence with functions in V , we may identify A with the uniform closure of the algebra of all holomorphic polynomials on O_{δ,V} .

Lemma 1. Let A be an invariant algebra on M generated by a finite dimensional invariant subspace V . If Oδ,V^{C} is closed and

(Stab δ)^{C} = Stab_{C} δ

then A is self-conjugated. Proof. Let B be the algebra of all holomorphic polynomials on Oδ,V^{C} . Then A is the uniform closure of B|_{M}. Denote by ^{#} the involutive automorphism of G^{C} whose set of fixed points is the maximal compact subgroup G of G^{C} . The condition of the lemma implies that ^{#} induces an involution in Oδ,V^{C} with O_{δ,V} as the set of fixed points. For any holomorphic on Oδ,V^{C} function f the function f^{#}(z) = \overline{f(z^{#})} is also holomorphic. Hence the restriction to O_{δ,V} of the algebra of all holomorphic on Oδ,V^{C} functions is self-conjugated. By the Stone-Weierstrass theorem, the uniform closure of the algebra of all holomorphic on Oδ,V^{C} functions restricted to O_{δ,V} is C(O_{δ,V}).

Since Oδ,V^{C} is closed, any holomorphic on Oδ,V^{C} function can be extended to an entire function by H.Cartan extension theorem (see, for example, [Chapter VIII A, Theorem 18, 4]. Hence any holomorphic on Oδ,V^{C} function can be approximated by polynomials uniformly on compact sets (Taylor series), in particular on O_{δ,V} . Therefore, polynomials are dense in C(O_{δ,V}). This easily implies that the restriction of the algebra of polynomials on O_{δ,V} is self-conjugated.

Lemma 2. If every finitely generated invariant subalgebra of an invariant algebra A on M is self-conjugated then A is self-conjugated. Proof. Let A_{0} be the subset of A consisting of functions which generate finite dimensional invariant subspaces. If f ∈ A_{0} then f̄ ∈ A by the assumption of the lemma. Clearly, f̄ ∈ A_{0} . Hence A_{0} is self-conjugated. It remains to note that A_{0} is dense in A because G is compact.

3. Proof of the theorem

Lemma 3. The conditions (2) and (3) of the theorem are equivalent. Proof. The Lie algebra of N_{G}(H) coincides with the normalizer 𝔫𝔤(𝔥) of 𝔥 in 𝔤 because H is connected. There is an Ad H -invariant decomposition of 𝔫𝔤(𝔥): 𝔫𝔤(𝔥) = 𝔥 ⊕ 𝔫. Since 𝔫 is an ideal in 𝔫𝔤(𝔥) the Lie algebra 𝔫 can be identified with the set of all Ad H -invariant vectors in the tangent space T_{x_{0}}M. Therefore the group N is finite if and only if 𝔫 = 0.

Lemma 4. If the group N is infinite then there exists an invariant algebra on M which is not self-conjugated. Proof. Suppose the group N is infinite. Since gHn = gnH for all g ∈ G and n ∈ N, N acts on G/H by right translations. Hence there is an action of the group U(1) = {e^{iφ}} commuting with the action of G.

For every integer m set

S_{m} = {f ∈ C(M)|ρ(e^{iφ})f = e^{imφ}f}.

The spaces S_{m} are G-invariant since the actions of the groups G and U(1) commute. Then the algebra

A = ∑_{m∈ℤ_{+}} S_{m}

of all functions extending on φ holomorphically to the unit disc is an invariant algebra on M which is not self-conjugated.

Let N_{C} be the factor of the normalizer of H^{C} in G^{C} by H^{C} . Denote by τ the antilinear involutive automorphism which distinguishes the compact real form 𝔤 of the Lie algebra 𝔤^{C} .

Lemma 5. If N if finite then N_{C} is finite.

Proof. Denote by 𝔤 and 𝔥 the Lie algebras of G and H respectively. Since N is finite, the Lie algebra of N_{G}(H) coincides with 𝔥. Let 𝔫_{C} be the Lie algebra of N_{G^{C}}(H^{C}); 𝔫_{C} coincides with the normalizer of 𝔥^{C} in 𝔤^{C} because H^{C} is connected. Since 𝔥^{C} is τ -invariant 𝔫_{C} is also τ -invariant. Hence 𝔫_{C} = 𝔥^{C} . Therefore the identity component of N_{G^{C}}(H^{C}) coincides with H^{C} . The group N_{C} is finite because N_{G^{C}}(H^{C}) is an algebraic group.

Lemma 6. Let V be a finite-dimensional complex vector space, G^{C} ⊆ GL(V) and the stabilizer of v ∈ V contains H. If the group N is finite then G^{C} -orbit of v is closed.

Proof. By the Luna criterion (see [Theorem 6.17, 9], it is sufficient to prove that the orbit of normalizer of H^{C} in G^{C} is closed. This is an easy consequence of Lemma 5.

Lemma 7. Let V be a finite-dimensional complex vector space, G^{C} ⊆ GL(V) and the stabilizer of v ∈ V contains H. If the group N is finite then the stabilizer of v in G^{C} is the complexification of the stabilizer of v in G.

Proof. By Lemma 6 G^{C} -orbit of v is closed. Then the stabilizer Stab_{C} v of v in G^{C} is a complex reductive Lie group (see [Theorem 4.17, 9]). Hence Stab_{C} v is the complexification of its maximal compact subgroup K_{1} . We may assume that H ⊆ K_{1} .

Let K be a maximal compact subgroup of G^{C} containing K_{1} . Denote by 𝔨 the Lie algebra of K. Then 𝔨 is a real compact form of 𝔤^{C} . Since N is finite the center of 𝔤 is contained in 𝔥. Let 𝔤0^{C} , 𝔨_{0} and 𝔤_{0} be the semisimple parts of 𝔤^{C} , 𝔨 and 𝔤 respectively. Denote by μ the antilinear involutive automorphism defining the compact real form 𝔨_{0} of the Lie algebra 𝔤0^{C} .

Consider the automorphism θ = μτ_{0} where τ_{0} is the restriction of τ to 𝔤0^{C} . The standard proof of the fact that any two compact real forms of the complex semisimple Lie algebra are conjugated shows that there is an element z ∈ 𝔤0^{C} , such that θ^{2} = exp(ad(4z)), exp(ad z)𝔤_{0} = 𝔨_{0} , and adz commutes with all linear transformations of 𝔤0^{C} commuting with θ^{2} (for details see [8] or [3].

Since θ^{2}|_{𝔥^{C}∩𝔤_{0}^{C}} = id, z is contained in the centralizer of 𝔥^{C} ∩ 𝔤0^{C} in 𝔤0^{C} . Hence vectors τz +z and i(τz −z) are elements of the centralizer of 𝔥∩𝔤_{0} in 𝔤_{0} .

By Lemma 3 z is contained in the center of 𝔥^{C} . Since 𝔥 includes the center of 𝔤, exp(ad z)𝔤 = 𝔨. Then exp(− ad z)K ⊆ G because K is connected. Therefore

exp(− ad z)K_{1} ⊆ Stab_{C} v ∩ G.

The group exp(− ad z)K_{1} is a maximal compact subgroup of Stab_{C} v. Thus Stab_{C} v = (Stab_{C} v ∩ G)^{C}.

Proof of Theorem. The equivalence (2) ⇔ (3) is proved in Lemma 3. The implication (1) ⇒ (2) follows from Lemma 4.

Suppose that the group N is finite. Lemmas 1, 6 and 7 imply that every invariant finitely generated algebra on M is self-conjugated. By Lemma 2 every invariant algebra on M is self-conjugated. Therefore (2) implies (1).

Corollary 1. Let G be a compact connected Lie group, H is its closed connected subgroup. If the group N_{G}(H)/H is finite then invariant function algebras on M are in one-to-one correspondence with the closed subgroups of G which include H.

The class of homogeneous spaces satisfying the condition of Corollary 1 is rather wide. It contains, for example, all compact homogeneous symmetric spaces.

Corollary 2. Let V be a finite-dimensional complex vector space, G be a compact connected Lie subgroup of GL(V). Suppose that H is a closed connected subgroup of G such that the group N_{G}(H)/H is finite. If H is contained in the stabilizer of an element v ∈ V then G-orbit Gv of v is polynomially convex and holomorphically convex.

Proof. Denote by P the algebra of all polynomials on V . Let A be the uniform closure of the restriction of P to Gv. By the theorem A is selfconjugated. Clearly, P separates the points of Gv. By the Stone-Weierstrass theorem A = C(Gv). Hence the set of all multiplicative linear functionals on C(Gv) coincides with Gv. Since every point x of the polynomially convex hull of Gv defines the multiplicative linear functional on A (the evaluating functional at x) Gv is polynomially convex. Since the holomorphically convex hull of Gv is contained in the polynomially convex hull, Gv is holomorphically convex.

References

[1] Gangolli R., Invariant function algebras on compact semisimple Lie groups, Bull. Amer. Math. Soc., 71 (1965), 634–637.

[2] Gichev V. M., Domains with homogeneous sceletons and invariant algebras, Proc. Tambov Summer School-Seminar “Harmonic analysis on homogeneous spaces,” August 26-31, 1996. Vestnik Tambovskogo Universiteta, 1998, 3 (1998), 38–44.

[3] Goto M., and F. D. Grosshans, “Semisimple Lie algebras,” Marcel Dekker, New York and Basel, 1978.

[4] Gunning R. C., and H. Rossi, “Analytic Functions of Several Complex Variables,” Prentice-Hall, Englewood Cliffs, N. J., 1965.

[5] Latypov I. A., Invariant algebras of functions on spheres, Proc. Tambov Summer School-Seminar “Harmonic analysis on homogeneous spaces,” August 26-31, 1996. Vestnik Tambovskogo Universiteta, 1998, 3 (1998), 53–61.

[6] de Leeuw, K., and H. Mirkil, Rotation-invariant algebras on the n-sphere, Duke Math. J., 30 (1961), 667-672.

[7] Nagel A., and W. Rudin, Moebius-invariant function spaces on balls and spheres, Duke Math. J., 43 (1976), 841–865.

[8] Vinberg E. B., and A. L. Onishchik, “Seminar on Lie groups and algebraic groups,” Nauka, Moscow, 1988 [Russian].

[9] Vinberg E. B., and V. L. Popov, Invariant Theory, Encyclopaedia Math., VINITI, 55 (1990), 137-309 [Russian].

[10] Wolf J., Translation-invariant function algebras on compact groups, Pacif. J. Math., 15 (1965), 1093–1099.

[11] —, Self-adjoint functional spaces on Riemannian symmetric manifolds, Trans. Amer. Math. Soc., 113 (1965), 299–315.

Omsk State University prosp. Mira 55a 644077, Omsk, Russia latypov@univer.omsk.su

Received April 24, 1998 and in final form June 30, 1998