1 Introduction
Let G be a group. For k∈ℕ (a natural number) we denote by Pk the kth power map of G, defined by Pk(g)=gk for
all g∈G. Inspired by the question of surjectivity of exponential maps of Lie groups there has been interest in understanding
conditions for Pk to be surjective. The question was studied by Pralay Chatterjee for various classes groups, beginning with connected
solvable Lie groups, in [2], algebraic groups over algebraically closed fields (see [1]), groups of rational points of algebraic groups
defined over real and p-adic fields etc. (see [3] and references there for details).
Recently, in [4], the first named author extended the study of surjectivity of the exponential of a solvable Lie
group to describing conditions for certain subsets (specifically cosets of certain nilpotent normal Lie subgroups) to be contained in the image of the exponential map; the results were applied in particular to describe conditions under which the radical of an exponential Lie group is exponential, and to generalise a result of Moskowitz and Sacksteder [6] for complex Lie groups to a large class of Lie groups, on centers of exponential Lie groups.
In this paper we study the analogous question for power maps of a large class of solvable groups; see
below for the definition of the class and Theorem 1.1 for the statement of the main result. The groups considered include connected solvable Lie groups, and in this case we deduce from Theorem 1.1 some of the results of [2] concerning the question of surjectivity of the power maps.
Our results also yield the corresponding results
for the exponential map proved in [4], via McCrudden’s criterion [5] that an element in a Lie group is exponential if and only if it admits roots of all orders. Theorem 1.1 is also applied to deduce surjectivity of the power map of the radical R of a connected Lie group G, in analogy with the result in [4] mentioned above, when the corresponding power map of G is surjective, and G/R satisfies a condition, as in [4].
Let 𝔽 be a field.
By an 𝔽-nilpotent group, we mean a nilpotent group N such that if N=N0⊃N1⊃⋯⊃Nr={e}
is the central series of N (e being the identity element of G), then each Nj/Nj+1 is a finite-dimensional 𝔽-vector space.
Let N be an 𝔽-nilpotent group and {Nj} its central series. Let G be a group acting on N as a group of automorphisms of N.
The G-action on N is said to be 𝔽-linear if the induced action of G on Nj/Nj+1 is 𝔽-linear for all j.
We note that if G is a connected solvable Lie group with nilradical N and if the latter is simply connected, then N is an ℝ-nilpotent group and the conjugation action of G on N is ℝ-linear. Also, for any field 𝔽, if G is the group of 𝔽-points of a Zariski-connected solvable algebraic group and N is the unipotent radical of G, then N is 𝔽-unipotent and the conjugation action of G on N is 𝔽-linear. Starting with these examples, one can also construct examples of non-algebraic groups G with 𝔽-nilpotent normal subgroups N of G such that the conjugation action of G on N is 𝔽-linear; note that the condition holds in particular for any subgroup of G as above containing N, in place of G itself, and these include non-algebraic groups.
In our results 𝔽 is allowed to be of positive characteristic.
In the following, if 𝔽 is a field of characteristic p, we say that a natural number k is coprime to the characteristic of 𝔽 if either p=0 or (k,p)=1, namely k is not divisible by p. For any group G and k∈ℕ we shall denote, throughout, by Pk the power map of G defined by Pk(g)=gk for all g∈G, and by
Pk(G) the image of Pk.
The following is the main technical result of the paper.
Theorem 1.1.
Let G be a group and N a normal subgroup of G. Suppose that N is
F-nilpotent with respect to a field F, and that the conjugation action of
G on N is F-linear. Let N=N0⊃N1⊃⋯⊃Nr={e}
be the central series of N. Let A=G/N, x∈G and a=xN∈A.
Let k∈N be coprime to the characteristic of F. Let B={b∈A∣bk=a}
and let B* be the subset consisting of all b in B such that for any j=1,…,r any element of Nj-1/Nj which is fixed under the
action of a is also fixed under the action of b. Then we have the following:
For any b∈B*,
n∈N, there exists y∈G such that yN=b and yk=xn.
If A is abelian, and xn∈Pk(G) for all n∈N, then B* is nonempty.
For groups of 𝔽-rational points we deduce the following Corollary;
in the case when 𝔽 is an algebraically closed field of characteristic zero, the result is contained in [1].
Corollary 1.2.
Let G be the group of F-points of a solvable algebraic group G defined over F and let N be the group of F-points of the unipotent radical Ru(G) of G. Let k be coprime to the characteristic of F. Let x be a semisimple element in G. Then xn∈Pk(G) for all n∈N if and only if there exists y∈Z(ZG(x)) such that yk=x.
Theorem 1.1 yields in particular the following generalisations of certain results proved in [2], thereby putting them in a broader
perspective. For any Lie subgroup S of a Lie group G we denote by L(S) the corresponding Lie subalgebra. For any X∈L(G) and a Lie subgroup T we denote by ZT(X), the centraliser of X in T, namely {t∈T∣Ad(t)(X)=X}.
Corollary 1.3.
Let G be a connected solvable Lie group, N the nilradical of G and H a Cartan subgroup of G. Let h∈H and k∈N. Then hn∈Pk(G) for all n∈N if and only if there exists
g∈H such that gk=h and g∈ZH(X) for every X∈L(N) such that h∈ZH(X).
In particular, Pk:G→G is surjective if and only if Pk:ZH(X)→ZH(X) is
surjective for all X∈L(N).
Via McCrudden’s criterion [5] recalled above Theorem 1.1 yields
the following variation of [4, Theorem 2.2].
Corollary 1.4.
Let G be a connected solvable Lie group and N a connected nilpotent closed normal subgroup of G such that G/N is abelian. Let A=G/N and N=N0⊃N1⊃⋯⊃Nr={e}
be the central series of N. Let x∈G and a=xN/N. Then xn is exponential in G for all n∈N if and only if
there exists a one-parameter subgroup B of A containing a such that for any j=1,…,r any point of Nj-1/Nj which is fixed by the
action of a is also fixed by the action of B.
In analogy with the criterion from [4] for exponentiality of radicals we deduce the following criterion for surjectivity of the power maps of radicals.
Corollary 1.5.
Let G be a connected Lie group such that Pk:G→G is
surjective. Let R be the (solvable) radical of G and S=G/R. Suppose that S has a unipotent
one-parameter subgroup U such that ZS(U) does not contain any element whose order divides k.
Then Pk:R→R is surjective.
The paper is organised as follows. In Section 2 we prove a preliminary result which is applied in Section 3 to complete the proof of Theorem 1.1. In Section 4 we discuss the case of power maps of algebraic groups and prove
Corollary 1.2. Corollary 1.3 is proved in Section 5, where
we discuss some more
applications of Theorem 1.1 (see Corollaries 5.5 and 5.6) for the power maps of solvable Lie groups. Corollary 1.5 on power maps of radicals is proved in Section 6.
2 Results for linear actions on vector spaces
In this section we prove various results for groups which are semidirect products of cyclic groups with vector spaces, to be used later to deduce the main theorem.
Let 𝔽 be a field and V a finite-dimensional 𝔽-vector space. We denote by GL(V) the
group of nonsingular 𝔽-linear transformations of V. For τ∈GL(V) we denote by F(τ) the set of points fixed by τ, viz. F(τ)={v∈V∣τv=v}.
We note that if τ∈GL(V) and k∈ℕ are such that τ is not unipotent and τk is unipotent,
then F(τ) is a proper subspace of F(τk). Also, for any τ∈GL(V) and k∈ℕ there exists a unique minimal τ-invariant subspace V′ such that the (factor) action of τk on V/V′ is trivial while that of τ has no nonzero fixed point. These results can be proved by a straightforward application of
the decomposition into generalised eigenspaces, or equivalently the Jordan canonical form of matrices; we omit the details.
Proposition 2.1.
Let F and V be as above.
Let G be a group with (a copy of) V as a normal subgroup such that G/V is cyclic, and the conjugation action of G on V is F-linear. Let σ:G→GL(V) denote the induced action.
Let k∈N be coprime to the characteristic of F. Let g∈G and x=gk. Then the following statements hold:
If F(σ(x))=(0), then for every v∈V, there exists a unique w∈V such that xv=wxw-1.
If F(σ(x))=F(σ(g)), then xv∈Pk(G) for all v∈V.
Proof.
(i) Let I denote the identity transformation. The hypothesis implies that (σ(x)-1-I)∈GL(V).
Hence for any v∈V there exists an element w∈V such that
v=(σ(x)-1-I)(w)=x-1wxw-1,
in
the group structure of G; hence we have xv=wxw-1.
The uniqueness of w follows from the fact that if xv=w1xw1-1=w2xw2-1, then w2-1w1∈F(σ(x)) which is given to be trivial.
(ii) Let W be the largest σ(x) invariant subspace of V such that the restriction of σ(x) to W is unipotent. Then W is also
σ(g)-invariant, and the hypothesis together with the remark preceding the proposition (applied to W in place of V) implies that the restriction of σ(g) to W is unipotent. We now proceed by induction on the dimension of W. We note that if W is zero-dimensional then F(σ(x))=0 and in this case the desired assertion follows from part (i). Now consider the general case. Let U=F(σ(x))
which by the hypothesis is also F(σ(g)). Then U is a normal subgroup of G and G/U contains V/U as a normal subgroup with cyclic quotient and the action of its generator on V/U is given by the quotient of the σ(g)-action on V/U. As the restriction of σ(g) to W is unipotent, U is of positive dimension and W/U has dimension less than W. We note also that W/U is the largest σ(g)-invariant subspace of V/U on which the σ(x)-action is unipotent. Hence by the induction hypothesis for any v∈V there exists η∈G such that xvU=ηkU.
Thus there exists u′∈U such that xv=ηku′. Since k is coprime to the characteristic of 𝔽 there exists
u∈U such that u′=uk (we use the multiplicative notation since U is now being viewed as a subgroup of G). Also, as σ(g) fixes u, the latter is contained in the center of G. Hence
xv=ηku′=ηkuk=(ηu)k. Thus xv∈Pk(G).
∎
Now let V be a finite-dimensional 𝔽-vector space and let A be an abelian subgroup of GL(V). Let
𝔚 denote the collection of all minimal A-invariant subspaces W of V such that the factor action of any element of A on V/W is semisimple (diagonalisable over the algebraic closure of 𝔽) and can be decomposed into irreducible components which are isomorphic to each other as A-modules; the subspaces from 𝔚
may be arrived at by considering the largest semisimple quotient of the A-action on V and decomposing it into isotypical components (putting together irreducible submodules isomorphic to each other). We note that 𝔚 is a finite collection of proper subspaces of V. We shall denote by X(A,V) the subset ⋃W∈𝔚W. We note that X(A,V) is a proper subset of V.
Proposition 2.2.
Let V be a finite-dimensional F-vector space. Let G be a group with V as a normal subgroup such that G/V is abelian. Let σ:G→GL(V) be the induced action. Let k∈N, g∈G and x=gk.
Let A=G/V, a=xV, b=gV∈A. Suppose that F(a)≠F(b). Let v∈V be such that there exists y∈G satisfying yV=b and yk=xv. Then v∈X(σ(A),V).
Proof.
Using the Jordan canonical form, we see that there exists a unique minimal a-invariant subspace, say U, of V such that the action of a on V/U is trivial while the b-action on V/U has no nonzero fixed point. The condition in the hypothesis implies that U is a proper subspace of V.
As A is abelian it follows that U is A-invariant. Moreover, U is contained in X(σ(A),V) as we can find a
subspace W∈𝔚 containing it. We show that any v satisfying the condition in the hypothesis is contained in U. Let such a v be given and let y∈G be such that yV=b and yk=xv. Then there exists w∈V such that y=gw
and we have
xv=yk=(gw)k=(gw)⋯(gw)=gk(g-(k-1)wg(k-1))⋯(g-1wg)w=gkθw=xθw,
where θ=σ(g)-(k-1)+⋯+σ(g)-1+I, and hence v=θw. Let V′=V/U. Then any eigenvalue of the factor of σ(g) to V′ is contained in {λ∈ℂ∣λk=1,λ≠1}.
Also, (σ(g)-1-I)θ=σ(g)-k-I=σ(x)-1-I factors to the zero transformation on V′. Since σ(g) has no nonzero fixed point on V′, the restriction of (σ(g)-1-I) to V′ is invertible. Hence θ factors to the zero transformation on V′=V/U. Therefore v=θw∈U, as desired.
∎
3 Proof of Theorem 1.1
In this section we shall deduce Theorem 1.1 from the results of Section 2. In the sequel, given a group H,
for h∈H we denote by 〈h〉 the cyclic subgroup generated by h.
Proof of (i).
In proving this part, without loss of generality we may assume that G is the product of
〈g〉 with N, where g∈G is such that gN=b, with b a given element from the subset B*; while the assumption is not crucial it makes the proof more transparent.
Let N=N0⊃N1⊃⋯⊃Nr={e} be the central series of N. We proceed by induction on r. For r=1 the desired statement is immediate from Proposition 2.1 (ii). Now consider
the general case. Let n∈N be given. Now G/Nr-1 is a product of
〈gNr-1〉 and N/Nr-1 in G/Nr-1, with g∈G as above, and
the condition in the hypothesis of the theorem is satisfied for G/Nr-1 (with N/Nr-1, a1=g1kNr-1 and b1=g1Nr-1 in place of Na and b, respectively); we note that the
corresponding actions on Nj/Nj+1 coincide with those of a and b, respectively. Since
G/Nr-1 is an
𝔽-nilpotent group of length less than r, by the induction hypothesis there exists
g1∈G such that g1N=b and xnNr-1=g1kNr-1.
Hence there exists v∈Nr-1 such that xn=g1kv.
Now consider the product, say G1, of the subgroups 〈g1〉 and
Nr-1 in G; we note that Nr-1 is an 𝔽-vector space, and the condition as in Proposition 2.1 (ii) is satisfied. Hence we get that there exists y∈G1 such that
g1kv=yk;
thus we have y∈G such that yN=b and xn=yk, and so xn∈Pk(G), which proves (i).
∎
Proof of (ii).
For any subset S of B we denote by ES the set of n in
N for which there exists y∈G such that yN∈S and yk=xn. We shall show that
ES=N if and only if
S∩B* is nonempty; when the condition in (ii) holds, for S=B we have ES=N and this implies
assertion (ii) in the theorem. We proceed by induction on r, the length of the central series. For r=1
the assertion follows immediately from Propositions 2.1 and 2.2. Now consider the general case.
Let S be a subset of B. If S∩B* is nonempty, then assertion (i) of the theorem, proved
above, shows that ES=N. Now suppose that ES=N.
Let V=Nr-1, G′=G/V and let
B′ be the subset of B consisting of b such that for all indices j=1,…,r-1 any point of Nj-1/Nj
which is fixed by a is also fixed by b. Since ES=N, for all
n∈N there exists an element y∈G such that yN∈S and ykV=xnV, and hence the induction hypothesis
implies that S∩B′ is nonempty. Also, applying the induction hypothesis to the set S\B′,
we get that there exists n0∈N for which there does not exist any y∈G such that yN∈S\B′
and ykV=xn0V. Now consider xn0v, with v∈V. Since ES=N, there exists y∈G
(depending on v) such that yN∈S and
yk=xn0v, and the preceding conclusion shows that in fact yN∈S∩B′. To show that S∩B* is
nonempty, we have to show that there exists b∈S∩B′ such that every fixed point of a on V is fixed by b. Suppose that this is not true; thus F(a)≠F(b) for all b∈S∩B′, in the notation as before, with respect to the action on V as above. Since the action of a on V is the same as the (conjugation) action of xn0,
Proposition 2.2 implies that any v∈V for which there exists an element y∈G such that yN∈S∩B′ and
yk=xn0v is contained in X(σ(A),V), in the notation as before, σ being the conjugation action. This contradicts the conclusion as above,
since X(σ(A),V) is a proper subset of V. This shows that S∩B* is nonempty, and completes the proof of the theorem.
∎
4 Power maps of solvable algebraic groups
In this section we discuss power maps of solvable algebraic groups and prove Corollary 1.2.
Let G be the group of 𝔽-points of a solvable algebraic group 𝐆 defined over 𝔽. By Levi decomposition G=T⋅N (semidirect product), where T consists of the group of 𝔽-points of a maximal torus in 𝐆 defined over 𝔽, and N is the group of 𝔽-points of the unipotent radical Ru(𝐆) of 𝐆.
Remark 4.1.
Let N=N0⊃N1⊃⋯⊃Nr={e} be the central series of N, with e the identity element. For j=0,…,r-1 let pj:Nj→Nj/Nj+1 be the quotient homomorphism.
For j=0,1,…,r-1 let 𝔐𝔧 be the Ad(T)-invariant subspace of the Lie subalgebra of Nj
complementary to the Lie subalgebra of Nj+1, and let 𝔓𝔧 be the
collection of (algebraic) one-parameter subgroups ρ corresponding to one-dimensional subspaces of 𝔐𝔧. Let 𝔓=⋃j𝔓𝔧. By considering the decomposition of the Lie algebra
of N with respect to the action of T it can be seen that, for any t∈T, ZN(t) is generated by the collection of one-parameter subgroups ρ∈𝔓j, j=0,…,r-1, such that any v∈pj(ρ) is fixed under the action of t on Nj/Nj+1. Moreover, any one-dimensional subspace of Nj/Nj+1
which is pointwise fixed under the action of t is of the form pj(ρ) for some ρ∈𝔓j centralised by t.
Proof of Corollary 1.2.
Let T be the subgroup as above.
Then every semisimple element of G has a conjugate in T and hence it suffices to prove the assertion
in the corollary for x in T. Let x∈T be given and let a=xN∈A=G/N.
We note that for any t∈T, ZG(t)=TZN(t). Suppose there exists y∈Z(ZG(x)) such that yk=x.
Thus y has the form sn, s∈T,
n∈ZN(x), with sn, and hence n, commuting with all elements of T. Hence x=yk=sknk, which
implies that sk=x and nk=e. Since k is coprime to the characteristic of 𝔽, we get that n=e and hence y=s∈T. Since any ρ∈𝔓j, 0≤j≤r-1, which is centralised by x is also centralised by y, by Remark 4.1 any v∈Nj/Nj+1, 0≤j≤r-1, is fixed under the action of y. Thus for a=xN as above there exists b=yN for which the condition of Theorem 1.1 is satisfied.
Hence by the theorem xn∈Pk(G) for all n∈N.
Conversely, suppose that xn∈Pk(G) for all n∈N. Then by Theorem 1.1 there exists b∈G/N
such that bk=a and any v∈Nj-1/Nj, j=1,…,r, which is fixed under the action of a is
also fixed under the action of b. Let g∈G be such that gN=b. Let g=yn with y∈T and n∈N.
Then as bk=a we get yk=x. Also, the actions of b and y on any Nj/Nj+1, j=0,…,r-1
coincide. Thus any v∈Nj-1/Nj, j=1,…,r which is fixed under the action of x is
also fixed under the action of y. From Remark 4.1 we get that every one-parameter subgroup ρ∈𝔓j,
0≤j≤r-1, which is centralised by x is centralised by y, and in turn that ZN(x)=ZN(y). Since ZG(x)=TZN(x), we get that ZG(x)=ZG(y), which shows that y∈Z(ZG(x)).
∎
5 Power maps of solvable Lie groups
Let G be a connected solvable (real) Lie group and let N be a simply connected nilpotent Lie group of G such that G/N is abelian.
Let
N=N0⊃N1⊃⋯⊃Nr-1⊃Nr={e}
be the central series of N. Note that as N is simply
connected, Nj/Nj+1 are real vector space and therefore the G-action on N is ℝ-linear.
In this case Theorem 1.1 implies the following.
Corollary 5.1.
Let G be a connected Lie group and let N be a simply connected nilpotent closed normal subgroup of G such that G/N is abelian.
Let A=G/N, x∈G and xN=a∈A. Let N=N0⊃N1⊃⋯⊃Nr={e}
be the central series of N. Let k∈N. Then xn∈Pk(G) for all n∈N if and only if there exists b∈A with bk=a such that for any j=0,…,r-1 any fixed point of a in Nj/Nj+1 is also fixed by b.
Corollary 5.1 may be reformulated as follows, in a form comparable to the result for the exponential maps
proved in [4, Theorem 2.2].
If M and M′ are closed connected normal subgroups of G contained in N such that M⊃M′ and
the G-action on M/M′ is irreducible, then the pair (M,M′) is called an irreducible
subquotient of N (with respect to the G-action) (cf. [4]). Using Jordan canonical form, it can be seen that condition as in Corollary 5.1 is
equivalent to the condition that there exists b∈A such that for any irreducible subquotient (M,M′) for which
the action of a on M/M′ is trivial the action of b on M/M′ is also trivial. Hence we get the following:
Corollary 5.2.
Let the notation be as in Corollary 5.1. Then xn∈Pk(G) for all n∈N if and only if there exists b∈A such that
for any irreducible subquotient (M,M′), where M and M′ are closed connected normal subgroups of G contained in N, if the action of a on M/M′ is trivial, then the action of b on M/M′ is trivial.
We recall also
that an element g in a connected Lie group G is exponential if and only if it is contained in Pk(G)
for all k (see [5]). In view of this, Corollary 5.1 implies Corollary 1.4, which is a variation of the characterisation in [4].
We now describe some more applications of our results, after recalling some structural aspects of solvable Lie groups.
Let G be a connected solvable Lie group and N the nilradical of G.
We denote by N=N0⊃N1⊃⋯⊃Nr={e} the central series of N, with e
the identity element.
Let H be a Cartan subgroup of G. We note that H is a connected nilpotent subgroup and G can be written as G=HN.
We have a weight space decomposition of L(N) (the Lie algebra of N, as before) with respect to the adjoint action of H, as
L(N)=⊕s∈ΔL(N)s, where Δ is a set of weights, and
L(N)s, s∈Δ, are Ad(H)-invariant subspaces of L(N). For x∈H, the restriction of Ad(x) to L(N)s has either only one real eigenvalue or a pair of complex numbers as eigenvalues; in either case we shall denote the eigenvalue(s) by λ(x,s) and λ¯(x,s).
We note the following observations; the proofs are straightforward and will be omitted.
Remark 5.3.
Let x,y∈H be such that yk=x. Then for all j=0,…,r-1, every v∈Nj/Nj+1 which is fixed
under the action of x is also fixed under the action of y if and only if for every X∈L(N) such that Ad(x)X=X
we also have Ad(y)X=X.
Remark 5.4.
The nilradical N has a unique maximal compact subgroup C; moreover C is
a connected subgroup contained in the center of G and N/C is simply connected. Consequently,
for g∈G, for any k∈ℕ, g∈Pk(G) if and only if gC∈Pk(G/C).
Proof of Corollary 1.3.
In view of Remark 5.4 we may assume N to be simply connected. Thus N is a ℝ-nilpotent group and the G-action on N is ℝ-linear. Let h∈H be given.
Suppose that there exists an element g∈H such that gk=h and g∈ZH(X) for all X∈L(N) such that h∈ZH(X). Then Theorem 1.1 together with Remark 5.3 implies that hn∈Pk(G) for all n∈N.
Conversely, suppose that
hn∈Pk(G) for all n∈N. Let a=hN. By Theorem 1.1 there exists b∈A=G/N such that any
v∈Nj/Nj+1, 0≤j≤r-1, which is fixed by a is also fixed by b. Since G=HN and H is a connected nilpotent Lie group, there exists g∈H such that gN=b and gk=h. By Remark 5.3 we also have g∈ZH(X) for all X∈L(N) such that h∈ZH(X), since on each Nj/Nj+1 the action
of g is the same as that of b. This proves the first assertion in the corollary.
Now suppose that Pk:G→G is surjective. Let X∈L(N) be given. For any h∈ZH(X) we have
hn∈Pk(G) for all n∈N, and hence by the above there exists g∈ZH(X) such that gk=h,
which shows that Pk:ZH(X)→ZH(X) is surjective. Conversely, suppose that Pk:ZH(X)→ZH(X) is surjective for all X∈L(N). Since G=HN it suffices to show that hn∈Pk(G) for all h∈H and n∈N. Let h∈H be given, and let Δ′={s∈Δ∣λ(h,s)=1}.
For all s∈Δ′ let Xs∈L(N)s be such that
Xs≠0, and let X=∑s∈Δ′Xs. Then h∈ZH(X) and hence by the first part there exists g∈ZH(X) such that gk=h. Then g∈H and, since Ad(g)(X)=X, we have Ad(g)Xs=Xs for all Xs, for all s∈Δ′, and hence
λ(g,s)=1 for all s∈Δ′. This in turn implies that g∈ZH(Y) for all Y∈L(N) such that h∈ZH(Y). The first part of the corollary proved above now implies that hn∈Pk(G) for all n∈N, as sought to be shown.
∎
For g∈G we denote by Spec(g) the set of all (complex) eigenvalues of Ad(g).
An element g∈G is said to be Pk-regular if Spec(g)∩{λ∣λk=1,λ≠1}=∅.
Corollary 5.5.
Let G be a connected solvable Lie group and k∈N. For any x∈Pk(G) there exists a Pk-regular element y in G such that yk=x. If Pk is surjective, then Pk:Z(G)→Z(G) is surjective.
Proof.
We note that in view of Remark 5.4 we may assume that N simply connected.
First suppose x∈H, a Cartan subgroup.
Since xn∈Pk(G) for all n∈N
by Corollary 1.3 there exists y∈H such that yk=x and Ad(y)X=X for every X∈L(N) such that Ad(x)X=X. The latter condition implies in particular that y is Pk-regular.
Now consider any x∈G, say x=hu, h∈H and u∈N. Then there exists a Pk-regular element
g∈H such that gk=h. It is known that for a Pk-regular element g and u∈N there exists
a w∈N such that gku=(gw)k (see [2, Proposition 3.5]). Thus x=hu=gku=(gw)k, we get y=gw as a Pk-regular element such that yk=x.
Now suppose that Pk is surjective and let x∈Z(G) be given. By the above assertions there exists a Pk-regular element y such that yk=x. Since Ad(x) is trivial,
y being a Pk-regular element with yk=x implies that Ad(y) is trivial, namely y∈Z(G). Hence Pk:Z(G)→Z(G) is surjective.
∎
Corollary 5.6.
Let G be a simply connected solvable Lie group, x∈G and k≥2. Let N be the nilradical of G.
Then the following are equivalent:
There exists g∈G such that gk=x and
Spec(g)∩{λ∈ℂ∣|λ|=1,λ≠1}=∅.
Spec(g)∩{λ∈ℂ∣|λ|=1,λ≠1}=∅ for all g∈G such that gk=x.
Proof.
That statement (i) implies (ii) follows immediately from Corollary 1.3 and the fact that if g=hu, where h∈H, a Cartan subgroup, and u∈N, then Spec(g)=Spec(h); this part does not involve G being
simply connected.
To prove the other two assertions, we first note the following. Let A=G/N and a=xN. Since G is simply connected, it follows that A is a vector space. Hence a has a unique kth root b in A. Thus for any g∈G such that gk=x we have
gN=b. This firstly shows that (ii) implies (iii), since Spec(g) is determined by b.
Finally, suppose (iii) holds and let g∈G be such that gN=b, the unique kth root of a in A. If there exists v∈Nj/Nj+1, where N=N0⊃⋯⊃Nr={e} is the central series of N, which is fixed by the action of a but not by that of b, then Ad(g) would have an eigenvalue λ
such that λ≠1 and λk=1. Since this is ruled out,
by Theorem 1.1 we get that xN∈Pk(G), thus proving (i).
∎
6 Surjectivity of the power maps of the radicals
In this section, we consider power maps of radicals and prove Corollary 1.5, which is the analogue of [4, Theorem 1.2], in the present context. We note some preliminary results before going over to the proof of Corollary 1.5.
Lemma 6.1.
Let k∈N. Let U be a one-parameter subgroup of S such that ZS(U) does not contain any element whose order
divides k. Let u be a nontrivial element in U and v∈ZS(U) be such that vk=u. Then v∈U.
Proof.
As U is a one-parameter subgroup, there exists w∈U such that wk=u. Since v∈ZS(U),
it follows that
(vw-1)k=vkw-k=uu-1=e,
the identity element. Thus vw-1∈ZS(u) and its order divides k. Hence by hypothesis vw-1=e, and so v=w∈U.
∎
Proposition 6.2.
Let G be a connected Lie group, let R be the radical of G, and let S=G/R. Let k∈N be such that Pk:G→G is surjective. Let U be a unipotent one-parameter subgroup of S such that ZS(U) does not contain any nontrivial element whose order divides k.
Let H be the closed subgroup of G containing R such that H/R=U.
Let h∈H be an element not contained in R. Then h∈Pk(H).
Proof.
As Pk is surjective, there exists g∈G such that gk=h. Let v=gR∈S and u=hR∈U. Since v commutes with u and U is a unipotent one-parameter subgroup, it follows that v commutes with all elements of U, viz. v∈ZS(U). As h∉R, u is a nontrivial element of U. Also, since
gk=h, we have vk=u, and hence by Lemma 6.1 we get v∈U. Therefore g∈H and in turn
h∈Pk(H).
∎
Proof of Corollary 1.5.
Let x∈R be given. Let H be as above. We note that since H/R is isomorphic to ℝ, to prove that
x∈Pk(R) it suffices to prove that x∈Pk(H). Also, since N is the nilradical of G, it follows that G/N is
reductive, and hence H/N is abelian.
Now let {ut} be the one-parameter subgroup of G such that U={utR}. Let u=u1 and y=ux∈H. Let A=H/N and a=xN, a′=yN∈A.
We note that by Proposition 6.2, yn∈Pk(H) for all n∈N. Hence by Corollary 5.2 there exists
b′∈A such that b′k=a′ and if the a′-action on an irreducible subquotient (M,M′) of N, with respect to the H-action (see Section 4 for definition) is trivial, then the b′-action is also trivial. Let b=(u1kN)-1b′∈A; then we have bk=a.
We note that the action of U on the Lie algebra of N is unipotent and by the
irreducibility condition this
implies that for any t the action of utN on M/M′ is trivial. Hence the action of a on M/M′ coincides
with the action of a′, and similarly the action of b coincides with that of b′. Thus we see that
bk=a and if the action of a on an irreducible subquotient M/M′ is trivial then so is the action of b.
Hence by Corollary 5.2xn∈Pk(H) for all n. Thus x∈Pk(H) and as noted above it follows
that x∈Pk(R).
∎
It is noted in [4] that if S is a complex semisimple Lie group of the group of ℝ-points of a quasi-split semisimple algebraic group defined over ℝ, then it contains a unipotent one-parameter subgroup U such that ZS(U) does not contain any compact subgroup of positive dimension. In this case the set of primes dividing the orders of elements of ZS(U) is finite, say F, and for k∈ℕ which is not divisible by any p in F the one-parameter subgroup U satisfies the condition as in the hypothesis of Corollary 1.5.
und
Arunava Mandal
