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In this context, we obtain results for density of the image of Pk on real points G(R) of a complex algebraic group G, which is defined over R and is not necessarily Zariski connected (see Theorem 1.1). Indeed, Theorem 1.1 provides an extension of [Ch1, Theorem 5.5], which deals with the surjectivity of the power map on the set of semisimple elements of G(R). As a consequence of Theorem 1.1, we get a characterization of the surjectivity of the power map for (possibly disconnected) group G(R) (see Corollary 1.2). Let G be a complex algebraic group defined over R. Let G(R) be the set of R-points of G. Note that G(R) is equipped with the real topology and has a real manifold structure. We denote the Zariski connected component of identity of G by G0. Let G(R)∗ be the connected component of identity of G(R) in Hausdorff or real topology. We denote (a, b) = 1 if two integers a and b are co-prime. Also, the order of a finite group F is denoted by ◦(F ). For a subset A of an algebraic group G, we denote the set of semisimple elements of A by S(A). The following is the main result of this article. It characterizes the density of the images of the power maps for a disconnected algebraic group which is not necessarily reductive, and provides an extension of [Ch1, Theorem 5.5]. In the context of surjectivity of the power map, an analogous result (corresponding to (1) ⇔ (2) in Theorem 1.1) was proved by P. Chatterjee (see [Ch1, Theorem 1.8]). Also, he showed (3) ⇔ (4) in Theorem 1.1 (see [Ch1, Theorem 5.5]).

Theorem 1.1. Let G be a complex algebraic group, not necessarily Zariski-connected, defined over R. Let A be a subgroup of G with G(R)∗ ⊂ A ⊂ G(R) and k ∈ N. Then the following are equivalent.

1. Pk(A) is dense in A.

∗ ∗ 2. (k, ◦(A/G(R) ))=1 and Pk(G(R) ) is dense.

∗ ∗ ∗ 3. (k, ◦(A/G(R) ))=1 and Pk : S(G(R) ) → S(G(R) ) is surjective.

4. Pk : S(A) → S(A) is surjective.

In particular, if G is a Zariski connected algebraic group defined over R then for

2 ∗ an odd k ∈ N, Pk(G(R)) is dense in G(R) if and only if Pk(G(R) ) is dense in G(R)∗. The connection of the power maps with weak exponentiality is discussed in §4 and §5. The next result (see §5) characterizes the surjectivity of the power map for a disconnected group G(R). Corollary 1.2. Let G be an algebraic group defined over R, which is not necessarily Zariski connected. Let k ∈ N. Then Pk : G(R) → G(R) is surjective if and only if Pk(ZG(R)(u)) is dense in ZG(R)(u) for any unipotent element u ∈ G(R). An application of Corollary 1.2 is given in Corollary 5.1, which characterizes the exponentiality of G(R). Corollary 1.2 can also be thought as an extension of [Ch1, Theorem 1.7].

2 Preliminaries

Let G be a complex algebraic group. Let G0 denote the Zariski-connected compo- nent of the identity of G. The maximal, Zariski connected, Zariski closed, normal unipotent subgroup of G is called the unipotent radical of G and is denoted by 0 Ru(G). Note that Ru(G)= Ru(G ). We now include the following results which will be used subsequently: Theorem 2.1. ([HP, Theorem 8.9(c)]) Let G be a complex algebraic group, which is not necessarily connected. Suppose the connected component G◦ is reductive. Then for each a ∈ G the set of semisimple elements S(G) ∩ G◦a is a non-empty Zariski open subset of the coset G◦a. In particular S(G) is Zariski dense in G. Proposition 2.2. Let G be a complex algebraic group, not necessarily Zariski connected, defined over R such that G0 is reductive. Then S(G(R)) contains an open dense subset of G(R) in the Hausdorff topology of G(R). Proof. Since G(R)/G0(R) embeds in G/G0, it follows that G(R)/G0(R) is a finite group. Let g1,...,gp ∈ G(R) such that 0 0 G(R)= g1G (R) ∪···∪ gpG (R).

0 It is enough to prove that for all i =1,...,p, the set S(giG (R)) contains an open 0 dense subset of giG (R) in the Hausdorff topology of G(R). By Theorem 2.1, there 0 is a Zariski W of giG consisting of semisimple elements of G. Let Xsm be the set of smooth points of an irreducible affine variety X defined over R, and M := Xsm(R) =6 ∅. Let W =6 ∅ be a Zariski open set of X. Then W ∩ M is an open dense subset of M in the Hausdorff topology of M. 0 R 0 R Now observe that giG sm( ) = giG ( ). Hence by the above fact, it follows 0 0 that W ∩ giG (R) is open dense in giG (R).

3 We recall that for a connected G, an element g ∈ G is said to be regular if the nilspace N(Adg −I) is of minimal dimension (see [Bou]). An element g is Pk-regular if (dPk)g is non-singular. Remark 2.3. Let G be a complex algebraic group defined over R. Then any element g ∈ G(R) can be written as g = gsgu where gs,gu ∈ G(R) and gs is semisimple, gu is unipotent, and g,gs,gu commute with each other. Then we have

Adg = Adgs Adgu . Since all the eigenvalues of Adgu are 1, the generalized eigenspace

for eigenvalue 1 of Adg is the same as that of Adgs . Note that N(Adgs − I) =

ker(Adgs − 1) as Adgs is semisimple. The dimension of ker(Adgs − 1) is equal to the dimension of the centralizer ZG(R)(gs). So g is regular in G(R) if and only if ZG(R)(gs) has minimal dimension. In particular, g is regular in G(R) if and only if gs is regular in G(R). Remark 2.4. The definition of a regular element, mentioned above, coincide with the definition of a regular element in §12.2 of [B] for the algebraic case. In liter- ature, there is another notion of regular element of an algebraic group (see [S]). Let G be an algebraic group over an algebraically closed field. An element g ∈ G is called regular, if dimZG(g) ≤ dimZG(x) for all x ∈ G. Note that according to this definition, a unipotent element may be a regular element; in fact, if G is connected and semisimple, then it admits a regular unipotent element (see [S]). However, our notion of regular element is different from that. For a semisimple group G as above, it is easy to see that unipotent elements are not regular in our sense. Indeed, let u be a unipotent element of a semisimple algebraic group G, then Adu is unipotent, and hence the nilspace N(Adu − I) is the whole space, which is not of minimal dimension.

To prove Theorem 1.1, we need the following lemma.

Lemma 2.5. Let G be a complex algebraic group, not necessarily Zariski-connected, defined over R. Let k ∈ N. Then the following are equivalent.

∗ ∗ 1. Pk(S(Reg(G(R) ))) ⊃ S(Reg(G(R) )).

∗ ∗ 2. Pk : S(G(R) ) → S(G(R) ) is surjective.

∗ ∗ 3. The image of the map Pk : G(R) → G(R) is dense. Proof. First, we will show that each of (1) and (2) implies (3). In view of [BM, ∗ ∗ Theorem 1.1], it is enough to show that Reg(G(R) ) ⊂ Pk(G(R) ). Let g ∈ ∗ Reg(G(R) ). Let g = gsgu where gs and gu are respectively the semisimple part and unipotent part of the Jordan decomposition of g. By Remark 2.3, g is regular if and only if gs is regular. Hence, either (1) or (2) implies that there exists R ∗ R ∗ k hs ∈ S(Reg(G( ) )) (hs ∈ S(G( ) )) such that hs = gs. Since gu is unipotent,

4 k there exists a unique unipotent element hu such that hu = gu. As the Zariski closure of the cyclic subgroups generated by gu and hu are the same, gs commutes with hu. ∗ Since hu is unipotent, there exists a unique nilpotent element X ∈ Lie(G(R) ) −1 such that hu = expX. As gs commutes with hu, we have gsexpXgs = expX, which gives

exp(Adgs (X)) = expX. R ∗ Now by the uniqueness of the element X ∈ Lie(G( ) ), we get Adgs (X) = X. k Therefore, Adhs (X)= X. By [BM, Lemma 2.1], hs is regular and Pk-regular. This implies Adhs (X)= X, which in turn shows that hs commutes with hu. Hence, we have k k k g = gsgu = hs hu =(hshu) . Now suppose that (3) holds. Let g ∈ S(G(R)∗) or S(Reg(G(R)∗)). Then there exists a Cartan subgroup C ⊂ G(R)∗ such that g ∈ C. By [BM, Theorem 1.1], we k have Pk(C)= C. Thus there exists h ∈ C such that h = g. Since g is semisimple (regular), h is also semisimple (regular), which proves the lemma.

Proposition 2.6. Let G be a complex reductive algebraic group, not necessarily Zariski-connected, defined over R. Let k ∈ N. Then the following are equivalent.

1. The image of Pk : G(R) → G(R) is dense.

∗ ∗ ∗ 2. Pk : G(R)/G(R) → G(R)/G(R) is surjective and the image of Pk : G(R) → G(R)∗ is dense.

0 0 0 3. Pk : G(R)/G (R) → G(R)/G (R) is surjective and the image of Pk : G (R) → G0(R) is dense.

Proof. This follows by applying [Ch1, Theorem 5.5] and Lemma 2.5 so we omit the details.

3 Proof of Theorem 1.1

We use the following additional lemma to prove Theorem 1.1.

Lemma 3.1. Let G be a complex reductive algebraic group, not necessarily Zariski- ∗ connected, defined over R. Let k ∈ N. Let G(R) ⊂ A ⊂ G(R). Suppose that Pk(A) is dense in A. Then for any s ∈ S(A) there exists a closed abelian subgroup Bs (containing s) of A with finitely many connected components such that Pk : Bs → Bs is surjective.

5 Proof. In view of Propositions 2.2 and 2.6, we note that Pk(A) is dense in A if and only if Pk : S(A) → S(A) is surjective. By [Ch1, Theorem 5.5], Pk : S(A) → S(A) is surjective if and only if k is co-prime to the order of A/G(R)∗ and ∗ ∗ Pk : S(G(R) ) → S(G(R) ) is surjective. We conclude our assertion by following the arguments of the proof of [Ch1, Theorem 5.5(1)]. Let s ∈ S(A). If the order of A/G(R)∗ is m then sm ∈ G(R)∗. On page- 230 of [Ch1, Theorem 5.5], it is shown that there exists a maximal R-torus T of ′ m −1 G = ZG(s ) such that sTs = T and Pk : ZT (R)∗ (s) → ZT (R)∗ (s) is surjective. Moreover, it is shown in Case 1 and Case 2 of [Ch1, Theorem 5.5] that s =(tcsd)k n for some c,d ∈ Z, where t ∈ ZT (R)∗ (s). Also, s ∈ ZT (R)∗ (s) for some positive integer n.

Now we consider the subgroup H of A generated by ZT (R)∗ (s) and the element s. ∗ Then H is a closed abelian subgroup of A and H/H is finite. Further, Pk : H → H c d k is surjective as Pk : ZT (R)∗ (s) → ZT (R)∗ (s) is surjective and s =(t s ) ∈ H. Now set Bs := H. Next we state a result from [DM] (see also [Da] and [Da1]) which we use in the proof of Theorem 1.1. Let G be a Lie group and N be a simply connected nilpotent of G. Let A = G/N be abelian (not necessarily connected). Let Nj, j =0, 1,...,r, be closed connected normal subgroups of G contained in N such that

N = N0 ⊃ N1 ⊃···⊃ Nr = {e} with [N, Nj] ⊂ Nj+1. For j ∈{0, 1,...,r−1}, let Vj = Nj/Nj+1. Then Vj is a finite dimensional vector space over R for each j and the G-action on N by conjugation induces an action on Vj for all j. Moreover, the restriction of the action to N is trivial, and hence it induces an action of G/N on each Vj, j ∈ {0, 1,...,r − 1}. The existence of such a series can be ensured by taking the central series of N. We refine the sequence Nj by inserting more terms in between if necessary, and assume that the G-action (and hence the G/N-action) on Nj/Nj+1 is irreducible for all j.

Theorem 3.2. ([DM]) Let G, N, A be as above. Let N = N0 ⊃ N1 ⊃ ··· ⊃ Nr = {e} be a series of closed connected normal subgroups Nj of N as above such that the action of A = G/N on the vector space Vj = Nj/Nj+1 (induced by conjugation) is irreducible for all j ∈{0, 1,...,r −1}. Let x ∈ G and a = xN ∈ A. k Suppose there exists b ∈ A with b = a such that for any Vj, if the action of a on Vj is trivial, then the action of b on Vj is trivial. Then for n ∈ N there exists y ∈ G such that yN = b and yk = xn.

The proof is immediate from Corollary 5.2 and Theorem 1.1(i) of [DM].

6 We now give a proof of Theorem 1.1.

Proof of Theorem 1.1: (1) ⇒ (2) : This is obvious. (2) ⇒ (1) : We first give a brief outline of the proof and later explain the steps. In step 1, we assume G(R)= L(R) ⋉ Ru(G)(R) for some Levi subgroup L (of G) defined over R and consider a dense set D = S(A∩L(R))Ru(G)(R) of A. By using the hypothesis, we see that Pk : S(A ∩ L(R)) → S(A ∩ L(R)) is surjective, i.e., Pk(A ∩ L(R)) is dense. In step 2, to each element x0 ∈ S(A ∩ L(R)) we associate a subgroup Bx0 of A such that Bx0 is abelian with finitely many components and

Pk : Bx0 → Bx0 is surjective. Then we construct a solvable subgroup Gx0 of A ⋉ R given by Gx0 = Bx0 Ru(G)( ). We show there exists a dense open set Wx0 of

Gx0 such that each element of Wx0 has k-th root in Gx0 (and hence in A). In step 3, we give an existence of a dense set W (of A) whose elements have k-th roots in A as required. Step 1: Since S(L(R)) is dense in L(R) by Proposition 2.2, and A ∩ L(R) is open in L(R), S(A ∩ L(R)) is dense in A ∩ L(R). Since G(R)∗ ⊂ A ⊂ G(R), we have L(R)∗ ⊂ A ∩ L(R) ⊂ L(R). Note that if (k, ◦(A/G(R)∗)) = 1, then ∗ ∗ ∗ (k, ◦((A ∩ L(R))/L(R) )) = 1. Also Pk(G(R) ) is dense in G(R) , which implies ∗ ∗ that Pk(L(R) ) is dense in L(R) . Now by Lemma 2.5 applied to the reductive ∗ ∗ ∗ group L(R) , we have Pk : S(L(R) ) → S(L(R) ) is surjective. So Pk : S(A ∩ L(R)) → S(A ∩ L(R)) is surjective by [Ch1, Theorem 5.5]. Hence Pk(A ∩ L(R)) is dense in A ∩ L(R).

Step 2: For a given x0 ∈ S(A ∩ L(R)), by Lemma 3.1, there exists a closed R abelian subgroup Bx0 of A∩L( ) containing x0 and Pk(Bx0 )= Bx0 . Also, Bx0 has finitely many connected components in real topology. Now consider the subgroup ⋉ R R Gx0 = Bx0 Ru(G)( ) of A. Note that the unipotent radical N = Ru(G)( ) is a simply connected nilpotent Lie group. Let

N = N0 ⊃ N1 ⊃···⊃ Nr = {e} be the series as in Theorem 3.2. Let Vj := Nj/Nj+1 for j =0, 1,...,r − 1. As N is simply connected, Vj’s are all finite dimensional real vector spaces, and Bx0 -action on Vj’s are irreducible.

Now for j = 0, 1,...,r − 1, let Fj be the set of elements of Bx0 which act trivially on Vj under conjugation action. Note that Fj is a closed subgroup of Bx0 for all j. We shall consider the following two cases separately.

i) dim(Fj) < dim(Bx0 ) for all j,

ii) for some j, dim(Fj) = dim(Bx0 ). ′ Suppose (i) holds. Then U := Bx0 −∪j Fj is a dense open set in Bx0 . Therefore ′ all elements in U act non trivially on all Vj’s, and hence by Theorem 3.2, gN ⊂

7 ′ ′ Pk(Gx0 ) for all g ∈ U . If we take Wx0 = U × N, then Wx0 is a dense open subset of Gx0 such that Wx0 ⊂ Pk(Gx0 ). Now suppose (ii) holds. Let I ⊂{0, 1,...,r − 1} be the set of indices such

that dim(Fj) = dim(Bx0 ) for all j ∈ I. Then ∪i/∈IFi is a proper closed analytic

subset of Bx0 of smaller dimension. Note that Bx0 \ ∪i/∈I Fi is a dense open set in

Bx0 .

Claim: the coset xN ⊂ Pk(Gx0 ) for all x ∈ Bx0 \ ∪i/∈I Fi.

If dim(Fj) = dim(Bx0 ), then Fj is the union of some connected components of

Bx0 (containing the identity component of Bx0 ). Fix j ∈ I. We will show that for

any x ∈ Fj \ ∪i/∈IFi the coset xN ⊂ Pk(Gx0 ). It might happen that x belongs to Fs (for s ∈ I) other than Fj. Let J = {m ∈ I|x ∈ Fm}. Clearly, j ∈ J and x ∈ ∩i∈J Fi ⊂ Fj. To prove the coset xN has a k-th root, we will apply Theorem 3.2, and for that we will show the existance of a k-th root of x in ∩i∈J Fi. ∗ ∗ Since Pk(Bx0 ) = Bx0 , we get Pk : Bx0 /Bx0 → Bx0 /Bx0 is surjective, and ∗ hence k is co-prime to the order of the component group Γ = Bx0 /Bx0 . Let the cardinality of Γ be n. Then there exist integers a and b such that ak + bn = 1. Without loss of generality, we can choose a to be a positive integer. Note that ak+bn a k n b n ∗ n b ∗ x = x =(x ) (x ) where x ∈ Bx0 and so does (x ) . As Bx0 is a connected ′ ∗ ′k n b abelian group, it is divisible. So there exists x ∈ Bx0 such that x =(x ) . This gives x =(xa)kx′k =(xax′)k as both xa and x′ commute with each other. Clearly a ′ x x ∈ ∩i∈J Fi ⊂ Fj. This implies that for l ∈ {0, 1,...,r − 1}, whenever the a ′ action of x on Vl is trivial there exists a k-th root of x (namely x x ) which acts trivially on Vl. Therefore by Theorem 3.2 we conclude that xn ∈ Pk(Gx0 ) for all n ∈ N. Finally, since j ∈ I is arbitrary, this proves the claim. Hence, there exists a ′ ′ ′ ′ dense open set W in Bx0 such that Wx0 ⊂ Pk(Gx0 ) where Wx0 = W × N. Thus for both cases we have a dense open set Wx0 in Gx0 such that each element of Wx0 has a k-th root in Gx0 , and hence in A. R Step 3: We note that S(A ∩ L( ))N = ∪x0∈S(A∩L(R))Gx0 . Now set

W = ∪x0∈S(A∩L(R))Wx0 .

Then W is a dense set in A such that each element of W has a k-th root in A, which proves (2) implies (1). (2) ⇔ (3) Follows from Lemma 2.5. (3) ⇔ (4) Follows from [Ch1, Theorem 5.5]. A. Borel and Tits showed that if G is a Zariski connected algebraic group defined over R, then either G(R) = G(R)∗ or G(R)/G(R)∗ is a direct product of

8 cyclic groups of order two (see [BT, Theorem 14.4]). Hence, the result follows immediately from the above.

Remark 3.3. Let G be an algebraic group over C, which is not necessarily Zariski- connected. Let G(C) denote the complex point of G. Let k ∈ N. Then it is immediate that Pk(G(C)) is dense in G(C) if and only if k is co-prime to the order of G/G0.

Theorem 1.1 asserts the following corollary.

Corollary 3.4. Let G be as in Theorem 1.1. Let G(R) = L(R) ⋉ Ru(G)(R) and k ∈ N. Then Pk(G(R)) is dense if and only if Pk(L(R)) is dense. Proof. The proof follows using the same procedure as in the proof of Theorem 1.1 applied to the group A = G(R).

4 Application to weak exponentiality

For a Lie group G with Lie algebra Lie(G), let exp : Lie(G) → G be the exponential map of G. We recall that G is weakly exponential if exp(Lie(G)) is dense in G. The group G is said to exponential if exp(Lie(G)) = G.

Remark 4.1. In view of (1) ⇒ (2) in Theorem 1.1, we observe that Pk(A) is ∗ ∗ dense implies k is co-prime to the order of A/G(R) and Pk(G(R) ) is dense. By [BM, Corollary 1.3], it follows that Pk(A) is dense for all k if and only if A is weakly exponential.

Remark 4.1 can be thought of as the analogous result of Hofmann and Lawson [HL, Proposition 1], which states the following: For a closed subgroup H (possi- bly disconnected) of a connected Lie group, H is exponential if and only if H is divisible, i.e., Pk(H)= H for all k ∈ N. Remark 4.2. Let G be a Zariski-connected complex algebraic group defined over R. Then the following are equivalent: (i) the map P2 : S(G(R)) → S(G(R)) is surjective, (ii) the image of the map P2 : G(R) → G(R) is dense, (iii) G(R) is weakly exponential. This can be seen from [Ch1, Theorem 1.6] and the fact that G(R)/G(R)∗ is a group of order 2m for some m. It can also be deduced from Theorem 1.1. Indeed, by [Ch1, Theorem 5.5] and Lemma 2.5, statement (i) implies ∗ ∗ 2 is co-prime to the order of the group G(R)/G(R) and P2(G(R) ) is dense. Thus by Theorem 1.1, P2(G(R)) is dense. Now (ii) ⇔ (iii) follows from Remark 4.1.

Note that if we replace G(R) by G(R)∗ in statements (i)-(iii) of Remark 4.2, then the equivalence follows from [Ch1, Theorem 1.6]. Also, Remark 4.2 generalizes

9 the result for a connected linear group (see [BM, Corollary 1.5]). Note that [BM, Corollary 1.5] (see Proposition 4.3 below) can be proved using results different from theorems in [BM]. For example, we now provide a proof which is due to P. Chatterjee.

Proposition 4.3. Let G be a connected linear Lie group. Then G is weakly expo- nential if and only if P2(G) is dense.

Proof. We assume that P2(G) is dense in G. Since G is linear so is G/Rad(G) (see p. 26. [OV, Proposition 5.2]). Now as G/Rad(G) is a connected linear semisimple Lie group, it is isomorphic to A(R)∗ for some Zarsiki connected (semisimple) alge- ∗ braic group A defined over R. Since P2(G) is dense in G, it follows that P2(A(R) ) ∗ ∗ ∗ is dense in A(R) . Using Theorem 1.1, we see that P2 : S(A(R) ) → S(A(R) ) is surjective. Now apply [Ch1, Theorem1.6] to see that A(R)∗ is weakly exponential. Thus G/Rad(G) is weakly exponential. As Rad(G) is connected solvable, it is weakly exponential and hence by [HM, Lemma 3.5] G is weakly exponential. The other part is obvious.

The following corollary establishes a special behaviour for a dense image of the power map on G(R), analogous to a result of [HM, Corollary 2.1A] for the exponential map. Moreover, it generalizes the result [HM, Corollary 2.1A] and [BM, Proposition 3.3] restricted to the group G(R).

Corollary 4.4. Let G be a Zariski-connected complex algebraic group defined over R. Let N be a Zariski-connected, Zariski-closed algebraic normal subgroup of G defined over R. If both Pk(G(R)/N(R)) and Pk(N(R)) are dense, then Pk(G(R)) is dense.

Proof. As G is Zariski connected for any odd k ∈ N, Pk(G(R)/N(R)), Pk(N(R)) and Pk(G(R)) are dense by Theorem 1.1. Suppose both the images of P2 : G(R)/N(R) → G(R)/N(R) and P2 : N(R) → N(R) are dense. Then by Remark 4.2, G(R)/N(R) and N(R) are weakly exponen- tial and hence connected. Recall that for any closed subgroup L of a H, H is connected if H/L and L are connected. This implies G(R)= G(R)∗. Since both the groups G(R)∗/N(R)∗ and N(R)∗ are weakly exponential, by [HM, ∗ Corollary 2.1A], we have G(R) is weakly exponential. Therefore P2(G(R)) is dense by Remark 4.2.

Remark 4.5. Suppose G is a Zariski connected complex algebraic group defined ∗ ∗ over R and k ∈ N. Then Pk is surjective on G(R)/G(R) and Pk(Reg(G(R) )) ⊃ ∗ Reg(G(R) ) together imply Pk(Reg(G(R))) ⊃ Reg(G(R)).

From Remark 4.5 we have the following.

10 Remark 4.6. For a Zariski connected complex algebraic group G defined over R, the density of image Pk(G(R)) is equivalent to the fact that the complement of the image is of measure zero. This follows since the density of image Pk(G(R)) implies Reg(G) ∩ G(R) ⊂ Pk(G(R)) (Reg(G) is non-empty Zariski-open in G).

5 Surjectivity of the power maps

Proof of Corollary 1.2: By [Ch1, Lemma 5.6], Pk : G(R) → G(R) is sur- ∗ jective if and only if for every unipotent element u ∈ G(R) , the map Pk : S(ZG(R)(u)) → S(ZG(R)(u)) is surjective. Note that ZG(R)(u) = ZG(u)(R). For ∗ each unipotent element u ∈ G(R) , let Hu = ZG(u). Since Pk is surjective on ∗ ∗ S(Hu(R)), by [Ch1, Theorem 5.5], Pk : Hu(R)/Hu(R) → Hu(R)/Hu(R) and ∗ ∗ ∗ Pk : S(Hu(R) ) → S(Hu(R) ) are surjective. By Lemma 2.5, Pk(Hu(R) ) is dense ∗ in Hu(R) , which implies Pk(Hu(R)) is dense in Hu(R) by Theorem 1.1. This completes the proof.

The following result is well known (see [DT, Theorem 2.2]) and was later de- duced by P. Chatterjee (see [Ch1, Corollary 5.7]). Here we prove it as an applica- tion of Corollary 1.2.

Corollary 5.1. Let G be a connected real algebraic group. Then G(R)∗ is expo- nential if and only if ZG(R)∗ (u) is weakly exponential for all unipotent elements u ∈ G(R)∗.

∗ ∗ Proof. By McCrudden’s criterion, G(R) is exponential if and only if Pk : G(R) → ∗ ∗ ∗ G(R) is surjective for all k ∈ N. Further, by Corollary 1.2, Pk : G(R) → G(R) is surjective if and only if Pk(ZG(R)∗ (u)) is dense in ZG(R)∗ (u) for all unipotent ∗ elements u ∈ G(R) . Now by Remark 4.1, we conclude that ZG(R)∗ (u) is weakly exponential.

Acknowledgement. I would like to thank P. Chatterjee for raising the ques- tion about dense image of the power map for a disconnected group, especially the statement of Theorem 1.1. He has also provided various comments and sugges- tions for other parts of the paper, and communicated the proof of Proposition 2.2. I thank U. Hartl for pointing out the reference [HP]. I would also like to thank S. G. Dani and Riddhi Shah for many comments on the manuscript. I sincerely appreciate the anonymous referee for useful comments and suggestions. I would like to thank Indian Statistical Institute Delhi, India for a post doctoral fellowship while most of this work was done.

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Arunava Mandal Theoretical Statistics and Unit, Indian Statistical Institute, Bangalore Centre, Bengaluru 560059, India. E-mail: [email protected]

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