Journal of Lie Theory Version of July 14, 1995 Volume 5 (1995) 129–134 C ◯1995 Heldermann Verlag

The density of the image of the exponential function and spacious locally compact groups

Wojciech Jaworski

Communicated by Karl H. Hofmann

A connected locally compact group G is called spacious if there exists a nonempty open set U ⊆ G such that U^{n} ∩ U^{n+1} = Ø for every positive integer n. G is called weakly exponential if the union of its one-parameter subgroups is dense in G. We prove the following conjecture of Hofmann and Mukherjea [Math. Ann. 234, 263-273 (1978)]: G is spacious iff it fails to be weakly exponential.

AMS classification: 22D05, 22E15, 22E46.

Let G be a connected locally compact group. Following Hofmann and Mukherjea [3] we call G spacious if it admits a nonempty open subset U such that U^{n} ∩ U^{n+1} = Ø for every positive integer n ∈ ℕ. We call G weakly exponential if the union of its one-parameter subgroups is dense in G.

When G = SL(n, ℝ) and U ⊆ G is the set of matrices with all entries strictly negative, then obviously U^{n} ∩ U^{n+1} = Ø for all n. SL(n, ℝ) is thus an elementary example of a spacious connected locally compact group. On the other hand, when G = ℝ, U ⊆ ℝ is open, and x ∈ U, there will be a positive integer n such that y = nx/(n + 1) ∈ U. So nx = (n + 1)y ∈ U^{n} ∩ U^{n+1} , and therefore ℝ is not spacious. Using this result one immediately concludes that a weakly exponential locally compact group is not spacious. Hofmann and Mukherjea conjectured that the converse is also true, i.e., G is spacious iff it fails to be weakly exponential [3, Conjecture 2.13]. The purpose of the present paper is to give a proof of this conjecture.

We begin by quoting two lemmas from [3].

Lemma 1. A connected locally compact group G is weakly exponential iff every neighbourhood of the identity e ∈ G contains a compact normal subgroup K such that G/K is a weakly exponential Lie group.

ISSN 0940–2268 / $2.50 ^{_{C} ◯} Heldermann Verlag Of course, a connected Lie group G is weakly exponential iff the image of the exponential map is dense in G.

Lemma 2. Let R denote the radical of a connected locally compact group G. Then G is weakly exponential iff G/R is.

It is easy to see that if N is a closed normal subgroup of G such that G/N is spacious then G is spacious. Therefore it is clear from Lemmas 1 and 2 that it suffices to prove the conjecture in the case that G is a connected semisimple Lie group (as pointed out in [3]).

Let G be a connected semisimple Lie group with Lie algebra 𝔤. Recall that a Cartan subalgebra of 𝔤 is a maximal abelian subalgebra 𝔥 such that adX is semisimple for every X ∈ 𝔥. The Cartan subgroup H(𝔥) associated with 𝔥 is defined as the centralizer of 𝔥 in G: H(𝔥) = Z_{G}(𝔥) = {g ∈ G; Ad(g)↾ 𝔥 = id}. H(𝔥) is a closed Lie subgroup with Lie algebra 𝔥. Thus if H(𝔥) is connected then H(𝔥) = exp(𝔥). Since the union of all Cartan subgroups of G is dense in G [5, Theorem 1.4.1.7], one obtains:

Lemma 3. If all Cartan subgroups of a connected semisimple Lie group G are connected then G is weakly exponential.

As shown in [3, Theorem 2.10] the converse is also true, but we do not need this in our argument. To prove the conjecture it suffices to show that if some Cartan subgroup is disconnected then G is spacious. We will achieve this goal in two steps. First, we will show that spaciousness occurs when the minimal parabolic subgroups of G are disconnected. Then we will show that if the minimal parabolic subgroups are connected then the Cartan subgroups are also connected.

Let G be a real connected semisimple Lie group with Lie algebra 𝔤. Let 𝔤 = 𝔨⊕𝔩 be a Cartan decomposition (𝔨 the subalgebra) and Θ the associated Cartan involution . We shall denote by ⟨·, ·⟩ the bilinear form ⟨X, Y ⟩ = −B(X, ΘY), X, Y ∈ 𝔤, where B(·, ·) is the Killing form. ⟨·, ·⟩ is a scalar product on 𝔤 and we shall write ∥X∥ for the norm

∥X∥ = (⟨X, X⟩)^{1/2}.  (1)

Let 𝔞 be a maximal abelian subspace of 𝔩. For any linear functional λ let,

𝔤_{λ} = {X ∈ 𝔤; (adY)X = λ(Y)X for all Y ∈ 𝔞}.  (2)

When 𝔤_{λ} ≠ {0}, λ is called a root of 𝔤 (relative to 𝔞). One proves that 𝔤 = ⨁_{λ∈Δ} 𝔤_{λ}, where the sum is over the set Δ of roots. Moreover, the 𝔤_{λ} ’s are orthogonal with respect to the scalar product ⟨·, ·⟩. Let ≥ be any total vector order in the dual space 𝔞^{∗} of 𝔞 and denote by Δ_{+} set of positive (nonzero) roots. Set 𝔫 = ⨁_{λ∈Δ_{+}} 𝔤_{λ}. Then 𝔫 is a nilpotent subalgebra of 𝔤 and one obtains an Iwasawa decomposition 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫. Let K,A, and N denote the connected Lie subgroups of G with Lie algebras 𝔨, 𝔞, and 𝔫, respectively. Then the map K ×A×N ∋ (k, a, n) → kan ∈ G is a diffeomorphism onto G and one obtains an Iwasawa decomposition G = KAN. We note that K contains the centre C(G) and is compact iff C(G) is finite.

We shall denote by M the centralizer of A in K. M is either discrete or a closed Lie subgroup with Lie algebra 𝔪 = 𝔤_{0} ∩ 𝔨. The set MAN is a closed cocompact subgroup of G called the minimal parabolic subgroup associated with the Iwasawa decomposition G = KAN. All Iwasawa decompositions of G, and therefore also all minimal parabolic subgroups are conjugate.

Let 𝔫̃ = ⨁_{λ∈−Δ_{+}} 𝔤_{λ}. Then 𝔫̃ is a nilpotent subalgebra. We shall write Ñ for the connected Lie subgroup of G with Lie algebra 𝔫̃. Given a subgroup H ⊆ G we shall denote by H_{0} the connected component of the identity e. We note that (MAN)_{0} = M_{0}AN. The following lemma is a version of Lemma III.6 in [1, p. 66].

Lemma 4. There exists an open semigroup S ⊆ G, an open neighbourhood V of e in G, and an element a ∈ A such that V aM_{0} ⊆ S ⊆ ÑM_{0}AN.

Proof. Choose Z ∈ 𝔞 so that λ(Z) > 0 for all λ ∈ Δ_{+} . Set a = exp(Z) and γ = exp(− min{λ(Z); λ ∈ Δ_{+}}) < 1. Then for every X ∈ 𝔫̃ we have ∥Ad(a)X∥ ≤ γ∥X∥. where ∥·∥ is the norm (1). Let Ω be an open ball in 𝔫 with centre 0. Put E = π(exp(\overline{Ω})) and E′ = π(exp(Ω)) where π: G → G/M_{0}AN is the canonical map. E is clearly a compact subset and E′ ⊆ E. Now, exp(Ω) is open because Ñ is a simply connected nilpotent Lie group. As the map Ñ × MAN ∋ (x, y) → xy is a diffeomorphism onto an open subset of G [2, pp. 406-407], it follows that E′ = π(exp(Ω)) = π(exp(Ω)M_{0}AN) is open. Furthermore, aE = π(exp(Ad(a)\overline{Ω})) ⊆ π(exp(Ω)) = E′.

Define S = {g ∈ G; gE ⊆ E′}. Then S is a semigroup containing a. S is open because E is compact while E′ is open.

Let V be a neighbourhood of e in G such that V a ⊆ S. Let v ∈ V and g ∈ M_{0} . Then vagE = vaπ(g exp(\overline{Ω})) = vaπ(exp(Ad(g)\overline{Ω})). Since Ad(g) ∈ Ad(K) is an isometry, Ad(g)\overline{Ω} = \overline{Ω} and therefore vagE = vaE ⊆ E′. So V aM_{0} ⊆ S.

It remains to show that S ⊆ ÑM_{0}AN. But M_{0}AN = π(e) ∈ E. Hence, π(S) = Sπ(e) ⊆ E′ ⊆ π(Ñ). So S ⊆ ÑM_{0}AN.

Lemma 5. If M is disconnected then G is spacious.

Proof. By Lemma 4 there exists an open semigroup S, an open neighbourhood V of e in G, and a ∈ A such that V aM_{0} ⊆ S ⊆ ÑM_{0}AN. We can assume that V = exp(Ω) where Ω = {X ∈ 𝔤; ∥X∥ < ε} for sufficiently small ε > 0. Let x ∈ M − M_{0} . As Ad(x) preserves the norm ∥ · ∥, we have xV x^{−1} = V . Furthermore, V aM_{0} ⊆ S and xV aM_{0}x^{−1} = V aM_{0} (because x centralizes a and normalizes M_{0} ). Consequently, S_{1} = ⋃_{i=1}^{∞}(V aM_{0})^{i} ⊆ S ⊆ ÑMAN is an open semigroup such that xS_{1}x^{−1} = S_{1} . Hence, setting U = xS_{1} we have U^{n} ∩ U^{n+1} ⊆ x^{n}S_{1} ∩ x^{n+1}S_{1} = x^{n}(S_{1} ∩ xS_{1}) ⊆ x^{n}(ÑM_{0}AN ∩ ÑxM_{0}AN) (because x normalizes Ñ). Using the fact that the map Ñ × M × AN ∋ (g, h, k) → ghk is one-to-one we conclude that U^{n} ∩ U^{n+1} = Ø for all n ∈ ℕ. So G is spacious.

Given a Cartan subalgebra 𝔥 ⊆ 𝔤 we shall write 𝔥_{𝔨} for 𝔥 ∩ 𝔨 and 𝔥_{𝔩} for 𝔥 ∩ 𝔩. If 𝔥 is stable under Θ then 𝔥 = 𝔥_{𝔨} ⊕ 𝔥_{𝔩}. We shall denote by H_{𝔞} the set of all Cartan subalgebras containing 𝔞. H_{𝔞} coincides with the set of maximal abelian subspaces of 𝔤 containing 𝔞. If 𝔥 is a member of H_{𝔞} then 𝔥 is Θ-stable, 𝔥_{𝔩} = 𝔞 and 𝔥_{𝔨} ⊆ 𝔪 = 𝔤_{0} ∩ 𝔨.

Lemma 6. 𝔪 = ⋃_{𝔥∈H_{𝔞}} 𝔥_{𝔨}.

Proof. Let X ∈ 𝔪. If 𝔥 is a maximal abelian subspace containing both X and 𝔞, then 𝔥 ∈ H_{𝔞} and X ∈ 𝔥_{𝔨}.

Lemma 7. There exist:

(i) a compact connected semisimple Lie group G_{#} with Lie algebra 𝔤_{#} and trivial centre;

(ii) a linear isomorphism F : 𝔤 → 𝔤_{#} such that F [X, Y] = [F X, F Y] for

X ∈ 𝔨, Y ∈ 𝔤, and F [X, Y] = −[F X, F Y] for X, Y ∈ 𝔩; (iii) an analytic homomorphism φ: K → G_{#} such that F ↾ 𝔨 is the differential

of φ at e and F Ad(g) = Ad(φ(g))F for all g ∈ K.

Proof. Let 𝔤̃ be the complexification of 𝔤. Then 𝔨 ⊕ i𝔩 is a compact real form of 𝔤̃. Set G_{#} = Int(𝔨 ⊕ i𝔩). Identify 𝔨 ⊕ i𝔩 with the Lie algebra of G_{#} and define F by F (X + Y) = X + iY , X ∈ 𝔨, Y ∈ 𝔩. Given g ∈ K define φ(g) to be the complexification of Ad(g) restricted to 𝔨 ⊕ i𝔩.

Remark 8. Kerφ = C(G). [F 𝔨, F 𝔩] ⊆ F 𝔩, [F 𝔩, F 𝔩] ⊆ F 𝔨.

Lemma 9. Let W ⊆ 𝔤 be a subset. Denote by Z_{K}(W) (resp., Z_{G_{#}}(F W)) the centralizer of W (resp., F W) in K (resp., in G_{#} ). Then Z_{K}(W) = φ^{−1}(Z_{G_{#}}(F W)).

Proof. Obvious by Lemma 7(iii).

We shall write K_{#} for the connected Lie subgroup of G_{#} with Lie algebra F 𝔨. Clearly, K_{#} = φ(K).

Lemma 10. If M is connected then K_{#} ∩ exp(F 𝔞) ⊆ exp(F 𝔪).

Proof. Let g ∈ K_{#} ∩ exp(F 𝔞). Clearly, g ∈ Z_{G_{#}}(F 𝔞). But g = φ(g′) for some g′ ∈ K. By Lemma 9, g′ ∈ Z_{K}(𝔞) = M. As M is connected and has compact Lie algebra 𝔪, M = exp(𝔪). So g = φ(g′) ∈ φ(exp(𝔪)) = exp(F 𝔪) by Lemma 7(iii).

For the next lemma see [5, Proposition 1.4.1.2].

Lemma 11. Let 𝔥 be a Θ-stable Cartan subalgebra. Set H_{K}(𝔥) = H(𝔥) ∩ K. Then H(𝔥) = H_{K}(𝔥) exp(𝔥_{𝔩}) and H(𝔥) is connected iff H_{K}(𝔥) is connected.

Theorem 12. The following conditions are equivalent for a connected semisimple Lie group G:

(a) the minimal parabolic subgroups of G are connected;

(b) the Cartan subgroups of G are connected; (c) G is weakly exponential;

(d) G is not spacious. Proof. (a) ⇒ (b): Every Cartan subalgebra 𝔥 is conjugate in Int(𝔤) to a Θ-stable Cartan subalgebra 𝔥′ such that 𝔥′_{𝔩} ⊆ 𝔞 [5, Proposition 1.3.1.1]. Therefore it suffices to consider Cartan subgroups H = H(𝔥) where 𝔥 is Θ-stable and 𝔥_{𝔩} ⊆ 𝔞. Due to Lemma 11 it suffices to show that H_{K} = exp(𝔥_{𝔨}).

Let g ∈ H_{K}. Then ĝ = φ(g) ∈ Z_{G_{#}}(F 𝔥) (Lemma 9). It is easy to see that F 𝔥 is a maximal abelian subalgebra of 𝔤_{#} . As 𝔤_{#} is compact, this means that F 𝔥 is a Cartan subalgebra [4, Lemma 4.12.1]. As G_{#} is compact and connected, H(F 𝔥) = exp(F 𝔥) [4, Theorems 4.12.3 and 4.12.5]. Therefore ĝ = exp(F X) for some X ∈ 𝔥. Write X = X_{1} + X_{2} , X_{1} ∈ 𝔥_{𝔨}, X_{2} ∈ 𝔥_{𝔩}. Then ĝ = exp(F X_{1}) exp(F X_{2}), and F X_{2} ∈ F 𝔥_{𝔩} ⊆ F 𝔞. As ĝ ∈ K_{#} , ĝ_{2} = exp(F X_{2}) = exp(−F X_{1})ĝ ∈ K_{#} ∩ exp(F 𝔞). By Lemma 10 exp(F X_{2}) = exp(F X′_{2}) where X′_{2} ∈ 𝔪. As 𝔪 = ⋃_{𝔥′∈H_{𝔞}} 𝔥′_{𝔨} (Lemma 6), there exists 𝔥′ ∈ H_{𝔞} with X′_{2} ∈ 𝔥′_{𝔨}. Then 𝔥 and 𝔥′ are two Θ-stable Cartan subalgebras such that 𝔥_{𝔩} ⊆ 𝔥′_{𝔩} = 𝔞. By [5, Proposition 1.3.1.3] there exists k ∈ K such that Ad(k)↾ 𝔥_{𝔩} = id and Ad(k)𝔥′_{𝔨} ⊆ 𝔥_{𝔨}. Now, φ(k)ĝ_{2}φ(k)^{−1} = φ(k) exp(F X_{2})φ(k)^{−1} = exp(Ad(φ(k))F X_{2}) = exp(F Ad(k)X_{2}) = exp(F X_{2}) = ĝ_{2} (by Lemma 7(iii)). Hence, ĝ_{2} = φ(k)ĝ_{2}φ(k)^{−1} = φ(k) exp(F X′_{2})φ(k)^{−1} = exp(Ad(φ(k))F X′_{2}) = exp(F Ad(k)X′_{2}) = exp(F X′′_{2}), where X′′_{2} ∈ 𝔥_{𝔨}. So φ(g) = ĝ = exp(F X_{1}) exp(F X′′_{2}) = exp(F Y) = φ(exp(Y)), where Y ∈ 𝔥_{𝔨}.

Since Kerφ = C(G), g = exp(Y)c where c ∈ C(G). But C(G) ⊆ M = exp(𝔪) because M is connected and 𝔪 is compact. So c = exp(Z) for some Z ∈ 𝔪. Then by Lemma 6 there exists 𝔥′ ∈ H_{𝔞} such that Z ∈ 𝔥′_{𝔨}. Again, 𝔥 and 𝔥′ are two Θ-stable Cartan subalgebras such that 𝔥_{𝔩} ⊆ 𝔥′_{𝔩} = 𝔞. So there exists k ∈ K such that Ad(k)↾ 𝔥_{𝔩} = id and Ad(k)𝔥′_{𝔨} ⊆ 𝔥_{𝔨} [5, Proposition 1.3.1.3]. We then have c = kck^{−1} = exp(Ad(k)Z) = exp(Z′) with Z′ ∈ 𝔥_{𝔨}. Consequently, g = exp(Y + Z′), and Y + Z′ ∈ 𝔥_{𝔨}. Thus H_{K} = exp(𝔥_{𝔨}).

(b) ⇒ (c): Lemma 3.

(c) ⇒ (d): explained at the beginning of the paper.

(d) ⇒ (a): Lemma 5.

Corollary 13. A connected locally compact group fails to be weakly exponential iff it is spacious.

Remark 14. Using Lemma 7 one can prove an analog of Lemma 6 for the group M. Namely, M = ⋃_{𝔥∈H_{𝔞}} H_{K}(𝔥). This, together with Lemma 11 allows to obtain the implication (b) ⇒ (a) of Theorem 12 directly, without involving (c) and (d).

References

[1] Azencott, R., Espaces de Poisson des groupes localement compacts, Lecture Notes in Math., 148, Springer, Berlin etc., 1970.

[2] Helgason, S., “Differential geometry, Lie groups, and symmetric spaces,” Holden-Day, San Francisco, 1965.

[3] Hofmann, K. H., and A. Mukherjea, On the Density of the Image of the Exponential Function, Math. Ann. 234 (1978) 263–273.

[4] Varadarajan, V. S., “Lie groups, Lie algebras, and their representations,” Springer, New York etc., 1984.

[5] Warner, G., “Harmonic Analysis on Semi-Simple Lie Groups I,” Springer, Berlin etc., 1972.

Department of Mathematics, Statistics, and Computing Science Dalhousie University Halifax, Nova Scotia, Canada B3H 3J5

Received March 9, 1995 and in final form May 15, 1995