Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 1, 37-46.

On the Exponential Function of Real Splittable and Real Semisimple Lie Groups

Michael Wüstner

Fachbereich Mathematik, Technische Hochschule Darmstadt, Schloßgartenstr. 7, D-64289 Darmstadt, Germany e-mail: wuestner@mathematik.th-darmstadt.de

Abstract. A Lie group G is called exponential if its exponential function is surjective. This article contains a characterization of real analytic splittable groups and real semisimple Lie groups with surjective exponential function. It is shown that they are exponential if and only if for each nilpotent element x of the Lie algebra L(G) the centralizer Z_{G}(x) = {g ∈ G| Ad(g)(x) = x} is weakly exponential, i.e., has dense exponential image.

Until today the problem of the surjectivity of the exponential function of Lie groups is not solved in general. J. Dixmier gave a characterization for simply connected exponential Lie groups [3], H. L. Lai characterized complex simple Lie groups [11], [12], [13] while D. Ž. Djoković characterized complex algebraic reductive exponential Lie groups [5] and together with T. Q. Nguyen reductive real exponential Lie groups [6]. They gave also a table of all real linear simple exponential Lie groups. The author characterized real solvable exponential Lie groups [17],[19] and complex splittable Lie groups [17],[18]. This note generalizes the results of [6] and [18]. If there is no danger of confusion we denote the Lie algebra of a Lie group G with 𝔤. A Lie group G is called exponential if G = exp 𝔤. It is called weakly exponential if G = \overline{exp 𝔤}. We shall prove that a real analytic splittable group is exponential if and only if the centralizers Z_{G}(x) of all nilpotent x ∈ 𝔤 are weakly exponential. Furthermore, we show that this characterization is even true for exponential covering groups of real analytic splittable groups. This is a modification of the analogous statement for real reductive Lie groups [6, Theorem 2.2]. Our present result also generalizes our earlier one in [18, Theorem 4.22], where it is shown that for complex splittable Lie groups the connectedness of

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

M. Wüstner: On the exponential function of real splittable and real semisimple Lie groups 38

the centralizers Z_{G}(x) = {g ∈ G| Ad(g)(x) = x} and the surjectivity of the exponential map are equivalent conditions.

1. Preliminaries

In this section we summarize some material on splittable Lie groups and on regular elements. In the whole article V denotes a real finite-dimensional vector space. Let End(V) be the space of all endomorphisms on V. An element A ⊆ End(V) is called semisimple, if for each invariant subspace U ⊆ V there is an invariant subspace W ⊆ V such that V = U ⊕W. This is equivalent to the condition that A is diagonalizable with respect to the complexification V_{ℂ}. The endomorphism A is called nilpotent if A^{n} = 0. It is called unipotent, if A − id is nilpotent. If λ is an eigenvalue of A then V_{λ}(A) denotes the eigenspace with respect to λ. Let G be a Lie group. An element g ∈ G is called Ad-unipotent if Ad(g) is unipotent. It is called Ad-semisimple if Ad(g) is semisimple. If we consider Gl(V), there are elements which are semisimple and unipotent, respectively, in Gl(V). We call them semisimple and unipotent, respectively. A semisimple element is Ad-semisimple and a unipotent element is Ad-unipotent but, in general, the converse is wrong. So we have to distinguish between both definitions. Consider the multiplicative Jordan decomposition g = g_{s}g_{u} = g_{u}g_{s} with semisimple g_{s} and unipotent g_{u}, both in Gl(V). Note that the Jordan decomposition is unique. A subgroup G ⊆ Gl(V) is called splittable if for all g ∈ G we have g_{s}, g_{u} ∈ G. If, in addition, G is analytic, it is called a splittable analytic group. Analogously, let 𝔤 be a Lie algebra. An element x ∈ 𝔤 is called ad-semisimple if ad x is semisimple, and ad-nilpotent if ad x is nilpotent. Now assume that 𝔤 ⊆ gl(V) and x ∈ 𝔤. We call x semisimple if it is semisimple in gl(V) and nilpotent if it is nilpotent in gl(V). If x is semisimple (nilpotent) then it is ad-semisimple (ad-nilpotent) but the converse is wrong in general. But note that in the case of semisimple Lie algebras, ad-semisimple (ad-nilpotent) elements are semisimple (nilpotent) (c.f. [10, Theorem 6.4]). Consider the additive Jordan decomposition x = x_{s} + x_{n} with semisimple x_{s} and nilpotent x_{n}, both in gl(V), with [x_{s}, x_{n}] = 0. This decomposition is also unique. A Lie algebra 𝔤 is called splittable if for all x ∈ 𝔤 we have x_{s}, x_{n} ∈ 𝔤. An analytic subgroup G ⊆ Gl(V) is splittable if and only if 𝔤 is splittable. This is proved in [18, Proposition 4.17]. Furthermore, in [1, VII, §5, n^{o}5, Corollaire 3] it is shown that a Lie algebra 𝔤 is splittable if and only if the complexification 𝔤_{ℂ} is splittable. In particular, an analytic group G is splittable if and only if its complexification G_{ℂ} is splittable. By the splittability of complex semisimple Lie groups this implies that real semisimple linear Lie groups are splittable. Let G be a subgroup of Gl(V). The splittable hull E(G) of G is defined as the smallest splittable subgroup in Gl(V) which contains G. If, in addition, G is analytic, its Lie algebra is denoted by 𝔢(𝔤). Note that this is the smallest splittable Lie algebra which contains 𝔤. So it is called the splittable hull of 𝔤. The algebraic hull A(G) of G is defined as the smallest algebraic subgroup in Gl(V) which contains G.

Now we review some standard facts on regular elements of Lie groups and Lie algebras. An element g of a Lie group G is called regular if and only if for each element g′ of a neighborhood of g the nilspace of Ad(g′) − id has the same dimension as the nilspace of Ad(g) − id. If, in addition, G is connected, g is regular if and only if the dimension of the nilspace is minimal. This dimension is called the rank of 𝔤. We will denote by Reg(G) the set of all regular elements of a Lie group G, by reg 𝔤 the set of the regular elements of 𝔤 and by reg exp the set of the elements of 𝔤 where the exponential function is regular. By [9, Lemma 1.4], we know that exp^{−1}(Reg(G)) = reg 𝔤 ∩ reg exp holds. Moreover, a regular element g of a connected Lie group G is contained in one and only one Cartan subgroup. The next proposition is a corollary of [9, Lemma 1.4].

Proposition 1.1. If g ∈ Reg(G) ∩ exp 𝔤, i.e, if there is a y ∈ reg 𝔤 ∩ reg exp with g = exp y, then e^{ad y}x = x implies [y, x] = 0.

We note that, in general, this y is not unique (cf. the circle group). Next we investigate the preimage of unipotent and of semisimple elements of Gl(V). For this, we need a standard proposition. A proof can be found in [7, Proposition I.2.7].

Proposition 1.2. Consider the linear Lie group Gl(V) and its Lie algebra gl(V). We define 𝔫_{p} = {x ∈ gl(V)|x^{p} = 0} and N_{p} = {g ∈ Gl(V)|g = 1 + x, x ∈ 𝔫_{p}}. Then exp |_{𝔫_{p}}: 𝔫_{p} → N_{p} is a diffeomorphism with the inverse function

log: g ↦ ∑_{k=1}^{∞}(−1)^{k+1} \frac{(g − 1)^{k}}{k} .

In the whole article we make use of the following statement. It ensures that a unipotent g in a splittable analytic group G ⊆ Gl(V) has a preimage in 𝔤.

Lemma 1.3. Let G ⊆ Gl(V) be an analytic group and assume that for nilpotent x, y ∈ 𝔤 we have exp x = exp y. Then we get x = y. In particular, 0 is the only nilpotent element of 𝔤 which is in exp^{−1}(1). If, in addition, G is splittable, then for each unipotent g ∈ G there is a nilpotent x ∈ 𝔤 with g = exp x.

Proof. The first part is a consequence of Lemma 1.2 and the second part is shown in [18, Lemma 4.16]. □

Now we turn to semisimple elements. For this, we denote by S the set of all semisimple elements of G and by 𝔰 the set of all semisimple elements of 𝔤.

Proposition 1.4. Let G ⊆ Gl(V) be an analytic group. Then we have exp^{−1}(S) = 𝔰.

Proof. Since the exponential map of a semisimple endomorphism is semisimple itself, we have exp 𝔰 ⊆ S, hence 𝔰 ⊆ exp^{−1}(S). Now assume that y ∈ exp^{−1}(S) is not semisimple. We consider the splittable hull E(G) of G in Gl(𝔤). In 𝔢(𝔤), we have the Jordan decomposition y = y_{s} + y_{n} with [y_{s}, y_{n}] = 0 where y_{s} ∈ 𝔢(𝔤) is semisimple and y_{n} ∈ 𝔢(𝔤) is nilpotent. Then

exp_{E(G)} y = exp_{E(G)} y_{s} exp_{E(G)} y_{n}

is a semisimple element. By the uniqueness of the Jordan decomposition we get exp_{E(G)} y_{n} = 1. By Lemma 1.3, this implies y_{n} = 0. Thus y is semisimple. □

Corollary 1.5. Let G be a real connected Lie group. Then 𝔰^{ad} = exp^{−1}(S^{Ad}).

2. Cartan subgroups

Cartan subgroups are a very useful tool in Lie theory. We shall need them, too. Let 𝔥 be a Cartan algebra and N_{G}(𝔥) the normalizer of 𝔥 in G. Furthermore, Λ_{𝔥} ⊆ 𝔥_{ℂ}^{∗} denotes the set of the roots belonging to 𝔥. Let 𝔤_{ℂ}^{λ} be the root space with respect to λ. We define

C(𝔥) = {g ∈ N_{G}(𝔥)|λ ∘ Ad(g)|_{𝔥_{ℂ}} = λ for all λ ∈ Λ}.

A subgroup H of G is called Cartan subgroup if and only if L(H) is a Cartan subalgebra and H = C(L(H)). Moreover, since Ad(g^{−1}) · 𝔤_{ℂ}^{λ} = 𝔤_{ℂ}^{λ∘Ad(g)} we have C(𝔥) = {g ∈ G| Ad(g) · 𝔤_{ℂ}^{λ} = 𝔤_{ℂ}^{λ} for all λ ∈ Λ ∪ {0}}. In [2, IV.5.1], C. Chevalley has given another definition of Cartan subgroups. In [14, Theorem A.4], K. H. Neeb has proven that these definitions are equivalent for connected Lie groups. We observe that the center of G is contained in each Cartan subgroup of G. We need the following characterization of weakly exponential Lie groups ([14, Theorem 12] and [8, Corollary 18]).

Theorem 2.1. Let G be a connected Lie group and 𝔤 its Lie algebra. Then the following conditions are equivalent:

(i) G is weakly exponential. (ii) All Cartan subgroups of G are connected. (iii) Each regular element of G is contained in exp 𝔤.

Proposition 2.2. If G ⊆ Gl(n, ℂ) is a connected complex analytic subgroup, then each semisimple element g is contained in the center of a Cartan subgroup.

Proof. We consider the algebraic hull A(G). By [16, p. 69], each semisimple element g is contained in a ℂ-torus of A(G). Hence it is contained in the center of a Cartan subgroup C of A(G). Thus, g is contained in the center of C ∩ G. By [2, IV.5.1. Proposition 1], C ∩ G is a Cartan subgroup of G. □

Lemma 2.3. Let G ⊆ Gl(n, ℝ) an analytic subgroup and G_{ℂ} its complexification, i.e., the smallest complex analytic subgroup of Gl(𝔤_{ℂ}) which contains G cf. [15, p. 35]. Let g ∈ G and denote by Z_{G}(g) the centralizer of g in G. Then L(Z_{G_{ℂ}}(g)) = L(Z_{G}(g))_{ℂ}.

Proof. We note that L(Z_{G_{ℂ}}(g)) ⊆ 𝔤_{ℂ}. Thus we have L(Z_{G_{ℂ}}(g)) = {x + iy | x, y ∈ 𝔤 and exp ℂ(x + iy) ⊆ Z_{G_{ℂ}}(g)}. So, x + iy ∈ L(Z_{G_{ℂ}}(g)) is equivalent to exp c(x + iy) = g exp(c(x + iy))g^{−1} = exp Ad(g)(c(x + iy)) for all c ∈ ℂ. Since exp maps a neighborhood of 0 homeomorphically onto a neighborhood of 1, this is equivalent to Ad(g)(x + iy) = x + iy. Since Ad(g) acts linear on 𝔤_{ℂ}, we have Ad(g)x + i Ad(g)y = Ad(g)(x + iy) = x + iy. Considering the real and the imaginary part, this is equivalent to Ad(g)x = x and Ad(g)y = y, i.e., x, y ∈ L(Z_{G}(g)). This implies the claim. □

Proposition 2.4. If G ⊆ Gl(n, ℝ) is an analytic subgroup, then each semisimple element g is contained in a Cartan subgroup.

Proof. Denote by Z_{G}(g) the centralizer of g. We set 𝔴 := L(Z_{G}(g)). Let 𝔥 be a Cartan subalgebra of 𝔴. Thus 𝔥_{ℂ} is a Cartan subalgebra of 𝔴_{ℂ}. By Proposition 2.2, g is contained in the center of a Cartan subgroup C of G_{ℂ}. Thus, C is contained in Z_{G_{ℂ}}(g), hence 𝔠 ⊆ L(Z_{G_{ℂ}}(g)) = 𝔴_{ℂ} by Lemma 2.3. But 𝔠 is maximal nilpotent and the normalizer 𝔫_{𝔴_{ℂ}}(𝔠) of 𝔠 in 𝔴_{ℂ} is contained in 𝔫_{𝔤_{ℂ}}(𝔠) ∩ 𝔴_{ℂ} = 𝔠 ∩ 𝔴_{ℂ} = 𝔠 ⊆ 𝔫_{𝔴_{ℂ}}(𝔠). Thus, 𝔠 is a Cartan subalgebra of 𝔴_{ℂ}. Since in a complex Lie algebra all Cartan subalgebras are conjugate, 𝔥_{ℂ} is conjugate to 𝔠, hence a Cartan subalgebra of 𝔤_{ℂ}. Since a subalgebra of 𝔤 is a Cartan subalgebra of 𝔤 if and only if its complexification is a Cartan subalgebra of 𝔤_{ℂ}, 𝔥 is a Cartan subalgebra of 𝔤 and g ∈ C(𝔥). □

Proposition 2.5. Let G be a connected real Lie group. Then each Ad-semisimple element of G is contained in some Cartan subgroup.

Proof. We note that Ad(G) is an analytic subgroup of Gl(𝔤). Consider the Lie group topology of Ad(G), i.e., the topology which makes Ad(G) to a Lie group. (Note that this, in general, is not the induced topology.) Together with this topology Ad(G) is isomorphic to G/Z by the canonical decomposition of functions. Since the center of G is contained in each Cartan subgroup H of G, by [9, Proposition 1.13] the Cartan subgroups of G/Z are exactly of the form H/Z with Cartan subgroup H in G. Thus the Cartan subgroups of Ad(G) are exactly of the form Ad(H) with Cartan subgroup H in G. Thus, there is a Cartan subgroup H of G with g ∈ H if and only if there is a Cartan subgroup W ⊆ Ad(G) with Ad(g) ∈ W. Thus, we have reduced our problem to the linear case. Proposition 2.4 implies the claim. □

We get the following result:

Proposition 2.6. Let G be a real Lie group and x ∈ 𝔤 nilpotent. Assume that g ∈ Z_{G}(x) is Ad-semisimple in G. Then it is Ad-semisimple in Z_{G}(x), too. If, in addition, Z_{G}(x) is connected, then g is contained in a Cartan subgroup of Z_{G}(x).

Proof. We observe that Ad(g) and ad x commute with each other. Thus they leave their eigenspaces invariant. In particular, we have Ad(g).𝔷_{𝔤_{ℂ}}(x) ⊆ 𝔷_{𝔤_{ℂ}}(x) and hence

Ad(g).((𝔤_{ℂ})_{λ}(Ad(g)) ∩ 𝔷_{𝔤_{ℂ}}(x)) ⊆ (Ad(g).(𝔤_{ℂ})_{λ}(Ad(g))) ∩ (Ad(g).𝔷_{𝔤_{ℂ}}(x))
⊆ ((𝔤_{ℂ})_{λ}(Ad(g))) ∩ 𝔷_{𝔤_{ℂ}}(x).

Thus Ad(g)|_{𝔷_{𝔤_{ℂ}}} is diagonalizable. Proposition 2.5 implies the last statement. □

3. Real analytic exponential splittable groups

We develop the main result of this article. For the following lemma we do not need that the group is splittable.

Lemma 3.1. Let G ⊆ Gl(V) be a real analytic exponential group. Then for each nilpotent x ∈ 𝔤 and each semisimple g_{s} ∈ Z_{G}(x) there is a semisimple x_{s} ∈ 𝔷_{𝔤}(x) such that g_{s} = exp x_{s}.

Proof. Set g = g_{s} exp x. Since G is exponential, there is a y ∈ 𝔤 with g = exp y. Now we turn to E(G). We find a semisimple x_{s} ∈ 𝔢(𝔤) and a nilpotent x_{n} ∈ 𝔢(𝔤) with y = x_{s} + x_{n} and [x_{s}, x_{n}] = 0. Thus we get g_{s} exp_{E(G)} x = exp_{E(G)} x_{s} exp_{E(G)} x_{n}. Moreover, x is also nilpotent and g_{s} is semisimple. By the uniqueness of the multiplicative Jordan decomposition we have g_{s} = exp_{E(G)} x_{s} and exp_{E(G)} x = exp_{E(G)} x_{n}. Lemma 1.3 implies x = x_{n} ∈ 𝔤. This means that x_{s} = y − x_{n} ∈ 𝔤 ∩ 𝔷_{𝔢(𝔤)}(x) = 𝔷_{𝔤}(x) and hence g_{s} = exp x_{s} ∈ exp 𝔷_{𝔤}(x). □

Proposition 3.2. If G ⊆ Gl(V) is a splittable analytic subgroup then the centralizer Z_{G}(y) of each element y ∈ 𝔤 is splittable, too.

Proof. Suppose that g ∈ Z_{G}(y). Then we find a semisimple g_{s} ∈ G and a unipotent g_{u} ∈ G with g = g_{s}g_{u} = g_{u}g_{s}. Note that the semisimple Jordan component is a polynomial in g. Thus g_{s}, g_{u} ∈ Z_{G}(y). □

Lemma 3.3. Let G ⊆ Gl(V) be a real splittable analytic group and assume that for each nilpotent x ∈ 𝔤 and each semisimple g_{s} ∈ Z_{G}(x) there is a semisimple x_{s} ∈ 𝔷_{𝔤}(x) such that g_{s} = exp x_{s}. Then the centralizer Z_{G}(x) is weakly exponential.

Proof. Let x ∈ 𝔤 be nilpotent and g ∈ Z_{G}(x). Since Z_{G}(x) is splittable, we find a semisimple g_{s} ∈ Z_{G}(x) and a unipotent g_{u} ∈ Z_{G}(x) with g = g_{s}g_{u} = g_{u}g_{s}. By Lemma 1.3, there is a unique nilpotent x_{n} ∈ 𝔤 with g_{u} = exp x_{n}. Furthermore, since x_{n} is exp-regular we have x_{n} ∈ 𝔷_{𝔤}(x) because of Proposition 1.1. By assumption, there is a semisimple x_{s} ∈ 𝔷_{𝔤}(x) with g_{s} = exp x_{s}. Thus g ∈ Z_{G}(x)_{0}, i.e., Z_{G}(x) is connected.

Now assume that, in addition, g is regular in Z_{G}(x). We note that g_{s}g_{u} = g ∈ Z_{G}(x) is regular in Z_{G}(x) if and only if g_{s} is regular in Z_{G}(x). So g_{s} is regular in Z_{G}(x), hence x_{s} is exp-regular in 𝔷_{𝔤}(x). But then e^{ad x_{s}}x_{n} = Ad(g_{s})x_{n} = x_{n} and Proposition 1.1 implies [x_{s}, x_{n}] = 0. Thus [x_{s} + x_{n}, x] = 0 and g = exp x_{s} exp x_{n} = exp(x_{s} + x_{n}). This means that g ∈ exp 𝔷_{𝔤}(x). By Theorem 2.1, the centralizer is weakly exponential. □

Lemma 3.4. If G ⊆ Gl(V) is a splittable analytic group and if for each nilpotent x ∈ 𝔤 the centralizer Z_{G}(x) is weakly exponential then G is exponential.

Proof. Let g ∈ G. Since G is splittable, there is a semisimple g_{s} ∈ G and a unipotent g_{u} ∈ G with g = g_{s}g_{u} = g_{u}g_{s}. By Lemma 1.3, we find a nilpotent x ∈ 𝔤 with g_{u} = exp x. Since g_{s} is semisimple it is Ad-semisimple. We observe that Z_{G}(x) is connected. By Proposition 2.6, it is even Ad-semisimple in Z_{G}(x) and contained in a Cartan subgroup of Z_{G}(x). By Theorem 2.1, the Cartan subgroups of Z_{G}(x) are connected. Thus there is a y_{s} ∈ 𝔷_{𝔤}(x) with g_{s} = exp y_{s}. This implies g = exp y_{s} exp x = exp(y_{s} + x). □

We summarize our results in the following theorem:

Theorem 3.5. Let G ⊆ Gl(V) be a splittable analytic group. Then the following statements are equivalent:

(i) G is exponential. (ii) For each nilpotent x ∈ 𝔤 and each semisimple g_{s} ∈ Z_{G}(x) there is a semisimple x_{s} ∈

𝔷_{𝔤}(x) with g_{s} = exp x_{s}. (iii) For each nilpotent x ∈ 𝔤 the centralizer Z_{G}(x) is weakly exponential.

Proof. (i)⇒(ii) is Lemma 3.1 and (ii)⇒(iii) Lemma 3.3. The implication (iii)⇒(i) is Lemma 3.4. □

4. Example

To illustrate our result we consider the analytic subgroup G of Gl(ℝ, 4) defined as follows:

G = \{( \begin{matrix}e^{t} cos t & e^{t} sin t & a & b \\ −e^{t} sin t & e^{t} cos t & c & d \\ 0 & 0 & e^{t} cos t & e^{t} sin t \\ 0 & 0 & −e^{t} sin t & e^{t} cos t \end{matrix} ) | t, a, b, c, d ∈ ℝ\} .

A standard computation shows that G is splittable but not algebraic. Thus G is a splittable analytic group. Its Lie algebra is given by

𝔤 = \{( \begin{matrix}t & t & a & b \\ −t & t & c & d \\ 0 & 0 & t & t \\ 0 & 0 & −t & t \end{matrix} ) | t, a, b, c, d ∈ ℝ\} .

The gl(4)-nilpotent elements of 𝔤 are exactly those with t = 0. Now we consider the centralizer of an element x where only a is not equal to 0. We get

Ad ( \begin{matrix}e^{t} cos t & e^{t} sin t & a & b \\ −e^{t} sin t & e^{t} cos t & c & d \\ 0 & 0 & e^{t} cos t & e^{t} sin t \\ 0 & 0 & −e^{t} sin t & e^{t} cos t \end{matrix} ) ( \begin{matrix}0 & 0 & a & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) =

( \begin{matrix}0 & 0 & a cos^{2} t & −a cos t sin t \\ 0 & 0 & −a cos t sin t & a sin^{2} t \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) .

So, for each element of Z_{G}(x) we have cos^{2} t = 1 and sin t = 0. We get

Z_{G}(x) = \{( \begin{matrix}e^{t} cos t & e^{t} sin t & a & b \\ −e^{t} sin t & e^{t} cos t & c & d \\ 0 & 0 & e^{t} cos t & e^{t} sin t \\ 0 & 0 & −e^{t} sin t & e^{t} cos t \end{matrix} ) | t ∈ πℤ\} .

Hence, Z_{G}(x) is not connected, in particular not weakly exponential. Thus G is not exponential.

5. Exponential covering groups of splittable analytic groups

Now we generalize Theorem 3.5. Assume that G is a covering group of a splittable analytic subgroup of a Gl(V). Let π denote the covering map. Note that G has a splittable Lie algebra. If G is exponential, then π(G) ⊆ Gl(V) is exponential and if G is weakly exponential then π(G) is weakly exponential.

Theorem 5.1. Let G be a covering group of a splittable analytic group of a Gl(V). Then the following statements are equivalent:

(i) G is exponential. (ii) For all nilpotent x ∈ 𝔤 and each g_{s} ∈ Z_{G}(x) where π(g_{s}) is semisimple, there is a

semisimple x_{s} ∈ 𝔷_{𝔤}(x) such that g_{s} = exp x_{s}. (iii) For all nilpotent x ∈ 𝔤 the centralizer Z_{G}(x) is weakly exponential.

Proof. First, assume that G is exponential. Let x ∈ 𝔤 be nilpotent and g_{s} ∈ Z_{G}(x) such that π(g_{s}) is semisimple. We consider the element g exp x. Since G is exponential there is a y ∈ 𝔤 with g_{s} exp x = exp y. Since 𝔤 is splittable there is a semisimple y_{s} and a nilpotent y_{n}, both in 𝔤, with [y_{s}, y_{n}] = 0 and g_{s} exp x = exp(y_{s} + y_{n}) = exp y_{s} exp y_{n}. Thus, we have π(g_{s})π(exp x) = π(exp y_{s})π(exp y_{n}). By the uniqueness of the Jordan decomposition we get π(g_{s}) = π(exp y_{s}) and π(exp x) = π(exp y_{n}). Lemma 1.3 implies x = y_{n}. Then we get exp x = exp y_{n} and hence g_{s} = exp y_{s}. This implies g_{s} ∈ exp 𝔷_{𝔤}(x).

Now suppose that (ii) is true. We prove that Z_{G}(x) is weakly exponential for all nilpotent x ∈ 𝔤. By assumption and Theorem 3.5, the centralizer Z_{π(G)}(x) is weakly exponential, and hence π(G) is exponential. In particular, the Cartan subgroups of π(G) are connected. Now let g ∈ Z_{G}(x). Since by Proposition 3.2, Z_{π(G)}(x) is splittable, there is a semisimple g_{s} and a unipotent g_{u}, both in Z_{π(G)}(x), with π(g) = g_{s}g_{u} = g_{u}g_{s}. By Lemma 1.3, we find a nilpotent x_{n} ∈ 𝔤 with g_{u} = π(exp x_{n}). Furthermore, we know that g_{s} is in a (connected) Cartan subgroup of Z_{π(G)}(x), hence there is a semisimple x_{s} ∈ 𝔷_{𝔤}(x), with π(g_{s}) = π(exp x_{s}). Thus g = z exp x_{s} exp x_{n} with some z ∈ ker π, and π(z exp x_{s}) = π(exp x_{s}) is semisimple. By (ii) there is a y ∈ 𝔷_{𝔤}(x) with z exp x_{s} = exp y. Thus g ∈ (Z_{G}(x))_{0}, hence Z_{G}(x) is connected.

Now suppose that, in addition, g is regular in Z_{G}(x). Then z exp x_{s} is regular in Z_{G}(x). Thus, y is exp-regular in 𝔷_{𝔤}(x). Because of e^{ad y}x_{n} = x_{n} and the exp-regularity of y, by Proposition 1.1 we conclude [y, x_{n}] = 0 and hence g = exp(y + x_{n}). Furthermore, we have [y + x_{n}, x] = 0. This implies the claim.

Next, we prove (iii)⇒(i). Let g ∈ G. We consider the Jordan decomposition π(g)_{s}π(g)_{u} = π(g)_{u}π(g)_{s} = π(g). Since π(g)_{u} is unipotent, there is a nilpotent x ∈ 𝔤 with π(g)_{u} = π(exp x). Moreover, we have π(g)_{s} ∈ Z_{π(G)}(x). Now we consider the element g exp(−x). Since Ad_{G} = Ad_{π(G)} ∘π, we get Ad_{G}(g exp(−x)) = Ad_{π(G)}(π(g)π(exp(−x))) = Ad_{π(G)}(π(g)_{s}). So g exp(−x) is Ad-semisimple. Moreover, it is contained in Z_{G}(x). Since Z_{G}(x) is weakly exponential it is connected. So by Proposition 2.6, g exp(−x) is contained in a Cartan subgroup of Z_{G}(x). Theorem 2.1 together with the weak exponentiality of Z_{G}(x) implies that there is a w ∈ 𝔷_{𝔤}(x) with g exp(−x) = exp w. We conclude g = exp w exp x = exp(w + x). This implies statement (i). □

This theorem can be applied to real semisimple Lie groups. Recall that in this case an element x ∈ 𝔤 ⊆ gl(V) is nilpotent if and only if it is ad-nilpotent. We note that Ad(G) is an analytic semisimple subgroup of Gl(𝔤) and according to [4, Proposition 4], it is closed. Because of the splittability of linear real semisimple Lie groups Theorem 5.1 still holds for the linear semisimple case. Thus we have:

Corollary 5.2. Let G be a real semisimple Lie group and 𝔤 ⊆ gl(V). Then G is exponential if and only if for each nilpotent x ∈ 𝔤 the centralizer Z_{G}(x) is weakly exponential.

Very recently, D. Ž. Djoković told me that he found the same characterization for real semisimple exponential Lie groups. Note that in particular a real semisimple Lie group cannot be exponential if there is any nilpotent x ∈ 𝔤 ⊆ gl(V) such that the centralizer Z_{G}(x) is not connected.

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Received January 27, 1996