Journal of Lie Theory Volume 8 (1998) 189–209 C ◯1998 Heldermann Verlag

Normalizers of compact subgroups, the existence of commuting automorphisms, and applications to operator semistable measures

W. Hazod, K. H. Hofmann, H.-P. Scheffler, M. Wüstner, and H. Zeuner

Communicated by D. Poguntke

Abstract. Let G be a Lie group with finitely many components and K a compact subgroup with identity component K_{0} . The normalizer, respectively, centralizer of K in G is denoted by N(K,G), respectively, Z(K,G). It is shown that N(K,G)/K_{0}Z(K,G) is finite. This is applied to a problem in theoretical probability theory, namely, to the characterisation of operator semistable measures. The article explains these concepts and puts the present application into perspective.

1. A Structure Theorem on Lie Groups and Some Consequences

Motivated by applications to probability theory, notably to semistable laws on vector spaces and groups which we shall discuss in detail in Section 3 below, we present some results on the structure of Lie groups and locally compact groups which are of independent interest.

If K is a compact subgroup of a locally compact group L and C is a closed subgroup isomorphic to either ℝ or ℤ such that L = KC, then the structure of L is known (see e.g. [14], pp.60–64; cf. also [11].)

Example 1.1. (The infinite dihedral group.) Let 𝕊^{0} be the multiplicative group of integers {1, −1}. On the product ℤ × 𝕊^{0} we consider the multiplication (m_{1}, ε_{1})(m_{2}, ε_{2}) = (m_{1} + ε_{1}·m_{2}, ε_{1}ε_{2}). Then K := {0} × 𝕊^{0} is one maximal compact and not normal subgroup and C := ℤ × {1} is an infinite cyclic normal subgroup. In fact, each of the subsets K =^{def} {(0, 1), (m, −1)} is a maximal compact subgroup. Note that no nonidentity element commutes with (0, −1).

We shall make frequent reference below to the special case that K is assumed to be normal; Example 1 shows how this may fail, and the structure theorems ([14], p.61) say that (at least in the case C ≅ ℝ) it is not far from the

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag worst that can happen. If K is normal and C ≅ ℝ, the subgroup C can be chosen to commute elementwise with K, i.e., L ≅ K × ℝ as we shall reprove in 1.9 below. In the case that K is open, i.e. C ≅ ℤ, the situation is more involved.

Let us use the following notation. If A is a subgroup of a group B, we denote by

N(A, B) = {b ∈ B : bA = Ab} the normalizer and by
Z(A, B) = {b ∈ B : ba = ab for all a ∈ A} the centralizer

of A in B. Note that AZ(A, B) = Z(A, B)A ⊆ N(A, B).

Example 1.2. Let K_{1} be a compact group and α an automorphism of the topological group K_{1} . Form L = K_{1} × ℤ and define a multiplication

(k_{1}, m_{1})(k_{2}, m_{2}) = (k_{1}α^{m_{1}}(k_{2}), m_{1} + m_{2}).

Then K := K_{1} × {0} is a compact normal subgroup of L and C := {1} × ℤ is infinite cyclic and discrete. An element (k, m) is in the centralizer Z(K, L) if and only if α^{m} = 1 and k is in the center Z(K_{1}, K_{1}). Thus

Z(K, L) = \{ \begin{matrix}Z(K_{1}, K_{1}) × nℤ & if α has finite order n, \\ Z(K_{1}, K_{1}) × \{0\} & otherwise. \end{matrix}

In the former case, KZ(K, L) has finite index n in L, and KZ(K, L) = K_{1} ×nℤ is a direct product.

Example 1.2.1. We write K_{1} ×_{α} ℤ for the semidirect product L introduced in 1.2. Let 𝕋 = ℝ/ℤ be the torus and set K_{1} = 𝕋^{2} . Then by duality we have Aut(𝕋^{2}) ≅ Aut ℤ^{2} ≅ GL(2, ℤ). Since GL(2, ℤ) contains elements of infinite

order such as ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) we have an α ∈ Aut 𝕋^{2} of infinite order. Then 𝕋^{2} ×_{α} ℤ

is a two dimensional metabelian Lie group with Z(K, L) = K = 𝕋^{2}×{0}. In this Lie group, the maximal compact normal subgroup is far from splitting directly.

In Exercise 1.11 below we shall observe that Example 1.2.1 is typical in the following sense: if the dimension of the center of K_{1} is < 2 the phenomenon illustrated by Example 1.2.1 cannot occur.

Examples of the type of 1.2 arise as follows: Let K be a compact subgroup of a topological group G and let a be an element of the normalizer N(K, G) of K in G such that C := ⟨a⟩ = {a^{n} : n ∈ ℤ} is an infinite discrete subgroup. Then L := KC ⊆ G is isomorphic to K ×_{α} ℤ with α ∈ Aut(K) given by α(k) = aka^{−1} . The order of α is n iff n is the smallest natural number such that a^{n} ∈ Z(K, G).

We shall show that whenever G is a Lie group with finitely many components, then for each a ∈ N(K, G) there is an element k in the identity component of K and a natural number e such that ka^{e} ∈ Z(K, G). In the case that ⟨a⟩ is infinite discrete, this entails that K⟨a^{e}⟩ = K⟨ka^{e}⟩ is isomorphic to the direct product K × ℤ.

This turns out to be a fairly quick consequence of a noteworthy structure theorem on Lie groups. If H is a topological group, then H_{0} denotes its identity component. Our main theorem contributes to the finiteness results surrounding the concept of a Weyl group in a general vein.

Theorem A. Let G be a Lie group with finitely many components and K a compact subgroup. Then N(K, G)/K_{0}Z(K, G) is finite.

The proof will be given in Section 2. Here we firstly point out sufficient conditions which secure the hypothesis that G has finitely many components and afterwards draw some consequences of Theorem A.

Lemma 1.3. Let A be a subgroup of a topological group G and let K be a subgroup of A. Then

N(K, A)/K_{0}Z(K, A) ≅ N(K, A)K_{0}Z(K, G)/K_{0}Z(K, G) ⊆ N(K, G)/K_{0}Z(K, G).

In particular, if N(K, G)/K_{0}Z(K, G) is finite, then N(K, A)/K_{0}Z(K, A) is finite as well.

Proof. We note that N(K, A) = N(K, G)∩A and Z(K, A) = Z(K, G)∩A and thus, by the modular law, K_{0}Z(K, A) = K_{0}Z(K, G)∩A = K_{0}Z(K, G)∩N(K, A). Thus

\begin{aligned}N(K, A)/K_{0}Z(K, A) & = N(K, A)/(N(K, A) ∩ K_{0}Z(K, G)) \\ ≅ N(K, A)K_{0}Z(K, G)/K_{0}Z(K, G) & ⊆ N(K, G)/K_{0}Z(K, G). \end{aligned}

The Lemma follows.

Corollary 1.4. If A is a closed subgroup of a real or complex affine algebraic group and K a compact subgroup of A, then N(K, A)/K_{0}Z(K, A) is finite.

Proof. Any real or complex affine algebraic group is contained up to isomorphism, as a closed subgroup, in some general linear group G = GL(n, ℝ) for a suitable n. Thus A is a real Lie group contained in some group GL(n, ℝ), n ≥ 1, which is a Lie group with two components. Thus Theorem A applies and shows the finiteness of N(K, G)/K_{0}Z(K, G).

In particular, the 2-dimensional metabelian Lie group of Example 1.2.1 cannot be (isomorphic to) a subgroup of a real or complex affine algebraic group; in particular, it is not isomorphic to any matrix group.

Corollary 1.5. Let H be a connected Lie group. If K is a compact subgroup of the automorphism group Aut H, then N(K, Aut H)/K_{0}Z(K, Aut H) is finite.

Proof. Every automorphism φ: H → H determines uniquely a unique lifting to an automorphism φ̃: H̃ → H̃ of the universal covering, and φ ↦ φ̃: Aut H → Aut H̃ is an isomorphism onto the closed subgroup of those automorphisms of H̃ which preserves the kernel of the covering map H̃ → H. On the other hand, Aut H̃ is a Lie group which is isomorphic to the linear Lie group Aut 𝔥 where 𝔥 is the Lie algebra of H and H̃. But Aut 𝔥 is an affine algebraic subgroup of GL(𝔥). Hence Corollary 1.4 proves the assertion.

Corollary 1.6. Let K be a compact subgroup of GL(V) for a finite dimensional real vector space V . Then N(K, GL(V))/K_{0}Z(K, GL(V)) is finite.

Proof. This is an immediate consequence of Corollary 1.5 which we apply with H = V .

Now we draw two conclusions from Theorem A; the first is Lie group theoretical, the second is arithmetical and group theoretical.

The Lie algebra of a Lie group G will be denoted by 𝔏(G); we abbreviate 𝔏(G) by 𝔤 and 𝔏(K) by 𝔨.

Corollary B. Let G be a Lie group with finitely many components and K a compact subgroup. Then 𝔏(N(K, G)) = 𝔨 + 𝔏(Z(K, G)).

Proof. By Theorem A, the Lie group N(K, G)/K_{0}Z(K, G) is finite, and thus K_{0}Z(K, G) is open in N(K, G). Consequently, 𝔏(N(K, G)) = 𝔏(K_{0}Z(K, G)). But 𝔏(K_{0}Z(K, G)) = 𝔏(K_{0}) + 𝔏(Z(K, G)). Since 𝔏(K_{0}) = 𝔏(K) = 𝔨, the assertion follows.

We remark that in the case that K and G are connected, the Lie algebra 𝔏(N(K, G)) is the normalizer of 𝔨 in 𝔤, and 𝔏(Z(K, G)) is the centralizer of 𝔨 in 𝔤.

Corollary 1.7. Let G be a Lie group with finitely many components and K a compact subgroup. Then the following conditions are equivalent.

(1) K is open in N(K, G).

(2) 𝔨 = 𝔏(N(K, G)).
(3) 𝔏(Z(K, G)) ⊆ 𝔨.
(4) Z(K, G)_{0} ⊆ K.

Proof. (1) and (2) are equivalent since for any closed normal subgroup A in a Lie group B the relation 𝔏(A) = 𝔏(B) is equivalent to A_{0} = B_{0} , and that is equivalent to A being open in the Lie group B. By Corollary B we have 𝔏(N(K, G)) = 𝔏(K) + 𝔏(Z(K, G)). Hence assertions (2) and (3) are equivalent. But (3) and (4) are both equivalent to Z(K, G)_{0} ⊆ K_{0} .

Now we draw a second conclusion. For this purpose recall that the exponent of a group F is

e(F) = \{ \begin{matrix}min\{n ∈ ℕ : (∀g ∈ G) g^{n} = 1\} & if \{|⟨g⟩| : g ∈ F \} is bounded, \\ ∞ & otherwise. \end{matrix}

If F is a finite group, then the exponent e(F) is finite.

Lemma 1.8. If K is a subgroup of a group G and if the exponent e = e(N(K, G)/K_{0}Z(K, G)) is finite, then for all a ∈ N(K, G) there is a k ∈ K_{0} such that (ka^{e})x = x(ka^{e}) for all x ∈ K. In particular, K⟨a^{e}⟩ = K⟨ka^{e}⟩ and the factors in the last product commute elementwise. Accordingly, (x, n) ↦ x(ka^{e})^{n}: K × Z → K⟨a^{e}⟩ is an isomorphism of groups.

Proof. By hypothesis and the definition of e we have a^{e} ∈ K_{0}Z(K, G), i.e., there is a k ∈ K_{0} such that a^{e} = k^{−1}z for some z ∈ Z(K, G). The claim follows.

Now Lemma 1.8 and Theorem A imply Corollary C. Let G be a Lie group with finitely many components and K a compact subgroup. Let e be the exponent of N(K, G)/K_{0}Z(K, G). Then for all a ∈ N(K, G) there is a k ∈ K_{0} such that (ka^{e})x = x(ka^{e}) for all x ∈ K.

Moreover, (x, n) ↦ x(ka^{e})^{n}: K×ℤ → K⟨a^{e}⟩ is a morphism of topological groups which is an isomorphism if and only if K ∩ ⟨a^{e}⟩ = {1}.

Corollary D. Let G be a linear Lie group and K a compact normal subgroup such that G/K is either isomorphic to ℝ or to ℤ. Then in the first case

(i) G ≅ K × ℝ. In the second case,

(ii) there is a natural number e such that for each a ∈ G there is an element k ∈ K_{0} such that ka^{e} ∈ Z(K, G). Moreover, if a ∉ K, then the open normal subgroup K⟨a^{e}⟩ of G is isomorphic to K × ℤ. If ⟨Ka⟩ = G/K, then K⟨a^{e}⟩ has index e in G.

Proof. Since K is normal in G, we have N(K, G) = G.

Case 1. Here K is not open in G. Hence by Corollary 1.7, Z(K, G)_{0} ⊈ K, and since the only connected subgroups of ℝ are {0} and ℝ, we have G = KZ(K, G)_{0} . Since Z(K, G)_{0}/(Z(K, G)_{0} ∩K) ≅ ℝ, there is a one parameter subgroup E ≅ ℝ contained in Z(K, G)_{0} . Then K ∩ E = {1} and G = KE. Therefore (k, x) ↦ kx: K × E → G is an isomorphism.

Case 2. Since G is a linear Lie group, Corollary 1.4 (or Corollary 1.6) shows that G/K_{0}Z(K, G) is finite. Now Corollary C applies and shows that (x, n) ↦ x(ka^{e})n: K × ℤ → K⟨a^{e}⟩ is an isomorphism of topological groups. Clearly G = K⟨a⟩ implies |G/K⟨a^{e}⟩| = |ℤ/eℤ| = e.

The structure of an extension of a compact normal subgroup by a group isomorphic to ℝ has long been settled. (See e.g [4]; or [14], Proposition 9.4; or [11], Lemma 1.25.) We can use the present results to present an independent proof.

Remark 1.9. Let K be a compact normal subgroup of a locally compact group G such that G/K ≅ ℝ. Then G ≅ K × ℝ. Proof. Claim 1. G/G_{0} is compact. Again, since the only connected subgroups of ℝ are {0} and ℝ, we know that KG_{0}/K = G/K and thus G = KG_{0} . Hence G/G_{0} is compact as asserted.

Claim 2. K contains a compact normal subgroup N of G such that G/N is a Lie group. There are arbitrarily small compact normal subgroups N of G such that G/N is a Lie group. (See e.g. [23], p.175, Theorem 4.6.) As K is the unique maximal compact normal subgroup, N ⊆ K follows.

Now G/N is a Lie group with finitely many components by Claims 1 and 2, while K/N is a compact normal subgroup and (G/N)/(K/N) ≅ G/N ≅ ℝ. Then by Corollary D we know G/N ≅ K/N × ℝ. Then G ≅ K × ℝ follows at once.

A straightforward generalisation of the Corollary D, G/K ≅ ℤ along the lines of Remark 1.9 is not possible. Let K_{∗} = (ℤ/2ℤ)^{ℤ} and let α: ℤ → Aut(K_{∗}) be defined by

α(m)((x_{n})_{n∈ℤ}) = (x_{(n−m)})_{n∈ℤ}.

Set G = K_{∗} ×_{α} ℤ, K = K_{∗} × {0}. Then G/K ≅ ℤ and the unique smallest nondegenerate compact normal subgroup is K. Also K_{0}Z(K, G) = K. The essential conclusions of Corollary D fail. What remains is the following exercise.

Exercise 1.10. Let G be a locally compact group and K a compact normal subgroup such that G/K ≅ ℤ and that G has arbitrarily small compact normal subgroups N such that G/N is a linear Lie group. Then there is a natural number e such that for each a ∈ G \ K the open normal subgroup K⟨a^{e}⟩ of G is isomorphic to K × ℤ. If ⟨Ka⟩ = G, then K⟨a^{e}⟩ has index e in G.

The following exercise rounds off the discussion illustrated by Example 1.2.1 which is shown to depend on the dimension of the center of K_{1} .

Exercise 1.11. Let K_{1} be a compact connected Lie group, let Z be the identity component of the center, and let S be the (semisimple) commutator subgroup of K_{1} . The automorphism group of S is a finite extension of the inner automorphism group of S and is therefore compact. Now α ↦ (α|Z, α|S) : Aut(K_{1}) → Aut(Z) × Aut(S) is an embedding and thus Aut(K_{1}) is compact if Aut(Z) is compact, and this is the case if and only if dim Z < 2. If Aut(K_{1}) is compact, then the natural semidirect product P =^{def} K_{1} × Aut(K_{1}) is a compact Lie group G with the normal subgroup K =^{def} K_{1} × {1}. Trivially the number of components of G is finite. Hence, by Corollary C there is a natural number e such that for any α ∈ Aut(K_{1}) there is an element k ∈ K_{1} such that (k_{α}, α^{e}) =^{def} (k, α)^{e} commutes elementwise with K.

Now let α ∈ Aut(K_{1}) and recall the notation of 1.2.1. Then σ: K_{1} ×_{α} ℤ → P, σ(x, m) = (x, α^{m}) is injective. Hence (k_{α}, e) commutes elementwise with K_{1} × {0}, and K_{1} × eℤ is a direct product.

2. The proof of Theorem A.

The proof proceeds through several steps. Case 1. The case that K is connected and abelian. Let K be a connected compact abelian subgroup of a Lie group G. Since K is compact, each ad X, X ∈ 𝔨 is semisimple because \overline{e^{ℝ}· ad X} = \overline{Ad(exp ℝ·X)} is a compact subgroup of GL(𝔤). Hence there is a Cartan subalgebra 𝔥 of 𝔤 containing 𝔨. (See [1], Chap. 7, §2, n^{o} 3, Proposition 10). Since the group exp 𝔨 = K is compact and contained in the nilpotent Lie subgroup H =^{def} exp 𝔥, it is central in H. (Recall that any connected nilpotent Lie group has a unique maximal compact subgroup which is central.) Hence [𝔨, 𝔥] = {0} = [𝔨, 𝔨]. Now Theorem 4.9 on p. 131 of [13] applies and shows that the Weyl group W(𝔨, 𝔤) is finite. By Lemma 4.3 on p. 128 of [13] this means that N(K, G_{0})/Z(K, G_{0}) is finite. From Lemma 1.3, recalling K = K_{0} ⊆ Z(K, G), we note

N(K, G_{0})/Z(K, G_{0}) ≅ N(K, G_{0})Z(K, G)/Z(K, G),

so this last group is finite. Further N(K, G)/(N(K, G) ∩ G_{0}) ≅ N(K, G)G_{0}/G_{0} . Now assume that G/G_{0} is finite. Then the group N(K, G)/N(K, G_{0}), being isomorphic to a subgroup of G/G_{0} , is finite. Then N(K, G)/N(K, G_{0})Z(K, G), as a quotient of N(K, G)/N(K, G_{0}), is finite, too. Hence N(K, G)/Z(K, G) is finite as asserted. Since K = K_{0} ⊆ Z(K, G), Theorem A is proved in this case.

Case 2. The case that K is connected. Any compact group is the product of the identity component Z_{0}(K) of the center and the commutator group K′ which is a semisimple Lie group; both of these subgroups are (fully) characteristic. We construct morphisms

\begin{aligned}φ & : N(K, G)/KZ(K, G) → N(Z_{0}(K), G)/Z(Z_{0}(K), G), \\ ψ & : N(K, G)/KZ(K, G) → Aut(K′)/Int(K′). \end{aligned}

The range of φ is finite in view of the case that K is connected abelian, the range of ψ is finite by a result of Iwasawa’s [15]. If we can show that ker φ∩ker ψ = {1}, the assertion of Theorem A follows in this case.

Now N(K, G) ⊆ N(Z_{0}(K), G) and Z(K, G) ⊆ Z(Z_{0}(K), G). Moroever, K′ ⊆ Z(Z_{0}(K), G) and Z_{0}(K) ⊆ Z(Z_{0}(K), G) whence

KZ(K, G) ⊆ Z_{0}(K)K′Z(Z_{0}(K), G) ⊆ Z(Z_{0}(K), G).

Hence φ is well defined by φ(nKZ(K, G)) = nZ(Z_{0}(K), G) and has the kernel Z(Z_{0}(K), G)/KZ(K, G).

Recall that K′ is characteristic, hence normal in N(K, G). Each inner automorphism I_{n}, I_{n}(k) = nkn^{−1} , n ∈ N(K, G), k ∈ K′, gives us a morphism n ↦ I_{n}: N(K, G) → Aut(K′). This morphism maps KZ(K, G) = K′Z_{0}(K)Z(K, G) = K′Z(K, G) to the subgroup of inner automorphisms Int(K′) and therefore defines ψ via ψ(nZ(K, G)) = I_{n}·Int(K′). The kernel of this morphism is KZ(K′, G)/KZ(K, G). Hence

ker φ ∩ ker ψ = (Z(Z_{0}(K), G) ∩ KZ(K′, G))/KZ(K, G).

Now

K′ ⊆ Z(Z_{0}(K), G) and KZ(K′, G) = K′Z(K′, G).

Hence the numerator equals K′(Z(Z_{0}(K), G) ∩ Z(K′, G)) by the modular law, and Z(Z_{0}(K), G) ∩ Z(K′, G) = Z(K, G) because K = Z_{0}(K)K′. It follows that ker φ ∩ ker ψ is indeed singleton. Thus Theorem A is proved for connected K = K_{0} .

Case 3. The general case. Since K_{0} is characteristic in K we have N(K, G) ⊆ N(K_{0}, G). By the preceding N(K_{0}, G)/(K_{0}Z(K_{0}, G)) is finite, and then

(N(K, G)Z(K_{0}, G))/(K_{0}Z(K_{0}, G)) ≅ N(K, G)/(N(K, G) ∩ K_{0}Z(K_{0}, G))

is finite. We claim that

(N(K, G) ∩ K_{0}Z(K_{0}, G))/K_{0}Z(K, G)

is finite. This will imply that N(K, G)/K_{0}Z(K, G) is finite, as asserted in Theorem A.

We consider the morphism α: N(K, G) → Aut(K), α(n) = I_{n}|K. Then ker α = Z(K, G). Let Int_{K_{0}}(K) = α(K_{0}). By a result of Iwasawa’s [15], this group is the identity component Aut_{0}(K) of Aut(K). Thus α induces an injective morphism α_{0}: N(K, G)/K_{0}Z(K, G) → Aut(K)/ Aut_{0}(K). Let Aut_{K_{0}}(K) be the group of automorphisms of K fixing K_{0} elementwise. Then α_{0}(N(K, G) ∩ K_{0}Z(K_{0}, G)/K_{0}Z(K, G)) is contained in

Aut_{K_{0}}(K) Aut_{0}(K)/ Aut_{0}(K) ≅ Aut_{K_{0}}(K)/( Aut_{K_{0}}(K) ∩ Aut_{0}(K))
= Aut_{K_{0}}(K)/( Aut_{K_{0}}(K) ∩ Int_{K_{0}}(K))
= Aut_{K_{0}}(K)/ Int_{Z(K_{0})}(K).

It thus suffices for our purposes to verify that Aut_{K_{0}}(K)/ Int_{Z(K_{0})}(K) is finite. Now let Aut_{K_{0},K/K_{0}}(K) be the subgroup of Aut_{K_{0}}(K) of all automorphisms σ ∈ Aut(K) with σ(k) ∈ kK_{0} . Then Int_{K_{0}}(K) ⊆ Aut_{K_{0},K/K_{0}}(K) ⊆ Aut_{K_{0}}(K). The group Aut_{K_{0},K/K_{0}}(K)/ Int_{Z(K_{0})}(K) is finite according to Iwasawa [15]. It therefore remains to show that

Aut_{K_{0}}(K)/ Aut_{K_{0},K/K_{0}}(K)

is finite. Indeed, we notice that the homomorphism β: Aut_{K_{0}}(K) → Aut(K/K_{0}), β(σ)(kK_{0}) = σ(k)K_{0} has the precise kernel Aut_{K_{0},K/K_{0}}(K) and that the group Aut(K/K_{0}) is finite because K is a compact Lie group, whence K/K_{0} is finite. Thus Aut_{K_{0}}(K)/ ker β is finite. This completes the proof of the theorem.

3. Applications to probability theory

Theorem A, as we pointed out in the introduction, was motivated by an attempt to describe the structure of certain closed subgroups of GL(ℝ^{d}) which arise in the theory of infinitely divisible probability measures on ℝ^{d}. We describe the situation. Let ℝ_{+}^{×} denote the multiplicative group of positive real numbers. For a locally compact group G we let denote Aut(G) its group of automorphisms endowed with the standard refined compact open topology. If G is a connected Lie group, then Aut(G) is isomorphic to a subgroup of Aut(𝔤) ⊆ GL(𝔤) (with the Lie algebra 𝔤 of G) and is therefore a linear Lie group. In particular, every subgroup of Aut(G) has a countable basis for its topology. For any group G we let Z(G) denote the center of G. If G is a Lie group, 𝔤 is its Lie algebra, and 𝔷(𝔤) is the Lie algebra of Z(G).

Let μ_{•} := (μ_{t})_{t≥0} be a continuous convolution semigroup. Then μ_{0} is an idempotent, hence μ_{0} = ω_{L} a Haar measure on a compact subgroup L. To any probability measure μ define μ̃ by μ̃(A) =^{def} μ(A^{−1}) for Borel sets A.

Definition 3.1. For a locally compact group G and a continuous convolution semigroup μ_{•} := (μ_{t})_{t≥0} we set

Dec(μ_{•}) = {A ∈ Aut(G) : (∃a ∈ ℝ_{+}^{×}, X ∈ 𝔷(𝔤))(∀t ≥ 0) Aμ_{t} = μ_{ta} ∗ ε_{exp t·X}}
Sym(μ_{•}) = {A ∈ Aut(G) : (∃X ∈ 𝔷(𝔤))(∀t ≥ 0) Aμ_{t} = μ_{t} ∗ ε_{exp t·X}}.

The group Dec(μ_{•}) is called the decomposability group, Sym(μ_{•}) the group of symmetries. Indeed, it is easily shown that Dec(μ_{•}) and Sym(μ_{•}) are subgroups of Aut(G).

The projection f_{μ_{•}}: Dec(μ_{•}) → (ℝ_{+}^{×}, ·), f_{μ_{•}}(A) = a, is a continuous homomorphism with kernel Sym(μ_{•}) so that we have the following exact sequence before our eyes:

(†)  0 → Sym(μ_{•})⟶^{incl} Dec(μ_{•})⟶^{f_{μ•}}ℝ_{+}^{×} ⟶^{log}_{≅} ℝ.

Definition 3.2. A convolution semigroup μ_{•} on a locally compact group G is called

(i) semistable if and only if Dec(μ_{•}) ≠ Sym(μ_{•}), (ii) stable if and only if im(f_{μ_{•}}) = ℝ_{+}^{×}, (iii) properly semistable if and only if im(f_{μ_{•}}) = (a^{n} : n ∈ ℤ) for some

a ∈ ]0, 1[, (iv) (A, a)-semistable if A ∈ Dec(μ_{•}) and f_{μ_{•}}(A) = a, and

(v) strictly (A, a)-semistable if (∀t ≥ 0) A(μ_{t}) = μ_{at}.

Assume for a moment that for a semistable law μ_{•} on a connected Lie group G, we have the additional information that Dec(μ_{•}) is closed in Aut(G) and that Sym(μ_{•}) is compact. Then from (†) we know that the group Dec(μ_{•}) is the extension of the maximal compact normal subgroup Sym(μ_{•}) by a group isomorphic to ℝ (if μ_{•} is stable) or ℤ (if μ_{•} is properly semistable). In these circumstances, Corollary D applies. This is the probability theoretical interest of the discourse in Section 1.

Now it is important for us to explain sufficient conditions for Dec(μ_{•}) to be closed in Aut(G) and for Sym(μ_{•}) to be compact. As usual ε_{g} denotes the probability measure concentrated on {g}. We shall say that μ_{•} is degenerate if for each t there is a compact subgroup L_{t} of G and an element x_{t} ∈ G such that μ = ω_{L_{t}} ∗ ε_{x_{t}}, equivalently, if for each t the measure μ_{t} ∗ μ̃_{t} is the normalized Haar measure of a compact subgroup of G.

Proposition 3.3. Assume that G is a Lie group and that μ_{•} is nondegenerate. Then Dec(μ_{•}) is a closed subgroup of Aut(G). Moreover, if G is connected, then Dec(μ_{•}) is a linear Lie group with a countable basis for its topology.

Proof. The idempotent μ_{0} is a Haar measure ω_{L} on a compact subgroup L. Assume A_{n} ∈ Dec(μ_{•}), A = lim_{n} A_{n} in Aut(G) and A_{n}(μ_{t}) = μ_{ta_{n}} ∗ ε_{exp(tX_{n})} . We must show A ∈ Dec(μ_{•}). Now A_{n}(μ_{t} ∗ μ̃_{t}) = μ_{ta_{n}} ∗ μ̃_{ta_{n}} → A(μ_{t} ∗ μ̃_{t}). Suppose firstly that a_{n} → 0. Then A(μ_{t} ∗ μ̃_{t}) = μ_{0} ∗ μ̃_{0} = ω_{L}, contradicting the hypothesis that μ_{•} is nondegenerate. Suppose secondly a_{n} → ∞. As we shall show in an appendix, μ_{t} ∗ μ̃_{t} converges vaguely for t → ∞. Then A(μ_{t} ∗ μ̃_{t}) = ρ = lim_{s→∞} μ_{s} ∗ μ̃_{s} for all t > 0, again a contradiction. Hence the sequence a_{n} is bounded away from 0 and ∞. Assume a_{n} → a ∈ ]0, ∞[. As above we obtain now A(μ_{t}) = lim_{n→∞} μ_{a_{n}t} ∗ ε_{exp(tX_{n})} . Since μ_{a_{n}t} converges to μ_{at}, we obtain the boundedness of {exp(tX_{n})} for all t ≥ 0. Hence

(∗)  (∀t ≥ 0)(∃z(t) ∈ Z(G)_{0}) A(μ_{t}) = μ_{at} ∗ ε_{z(t)}.

Lemma Z. Assume that μ_{•} is a convolution semigroup of a Lie group G, that A ∈ Aut(G), and that 0 < a ∈ ℝ. If Condition (∗) is satisfied, then there is an X ∈ 𝔷(𝔤) such that

(∗∗)  (∀t ≥ 0) μ_{at} ∗ ε_{z(t)} = μ_{at} ∗ ε_{exp(t·X)}.

In particular, A ∈ Dec(μ_{•}). Once Lemma Z is proved it will be established that Dec(μ_{•}) is closed in Aut(G). Proof of Lemma Z. This requires that we find an X ∈ 𝔷(𝔤) such that μ_{at} ∗ ε_{z(t)} = μ_{at} ∗ ε_{exp(t·X)} . Define I_{r} = {x ∈ Z(G)_{0} : μ_{r} ∗ ε_{x} = μ_{r}}. Then I_{r} is a compact subgroup for each r. Since the function r ↦ I_{r} is increasing and continuous from the right, G is a Lie group, and the set of closed compact subgroups of a Lie group satisfies the descending chain condition, there is a δ > 0 and a compact subgroup I such that I_{r} = I for all 0 ≤ r < δ. We compute μ_{a(s+t)} ∗ ε_{z(s+t)} = A(μ_{s+t}) = A(μ_{s}) ∗ A(μ_{t}) = μ_{as} ∗ ε_{z(s)} ∗ μ_{at} ∗ ε_{z(t)} = μ_{as} ∗ μ_{at} ∗ ε_{z(s)} ∗ ε_{z(t)} = μ_{a(s+t)} ∗ ε_{z(s)z(t)} since z(s) is central. Hence z(s)z(t) ∈ z(s + t)I_{a(s+t)} , and thus z(s)z(t) ∈ z(s + t)I for 0 ≤ s + t < \frac{δ}{a}. The local oneparameter subsemigroup r ↦ z(r)I : [0, \frac{δ}{a}[ → Z_{0}(G)I/I ≅ Z_{0}(G)/(Z_{0}(G) ∩ T) gives rise to an element X ∈ 𝔷(𝔤) such that z(t)I = exp(tX)I for 0 ≤ t < \frac{δ}{a}. Then A(μ_{t}) = μ_{at} ∗ ε_{z(t)} = μ_{at} ∗ ε_{exp(t·X)} for t ∈ [0, \frac{δ}{a}[. Since t ↦ A(μ_{t}), μ_{at} ∗ ε_{exp(t·X)}: [0, ∞[ → M^{1}(G) are continuous one-parameter semigroups which agree on a neighborhood of 0, they agree. This proves statement (∗∗) of Lemma Z. From (∗) and (∗∗) and from the definition of Dec(μ_{•}) in 3.1 it follows that A ∈ Dec(μ_{•}). Thus Lemma Z is proved.

If G is connected, then Aut(G) is a Lie subgroup of Aut(𝔤) ⊆ GL(𝔤). In these circumstances, as a closed subgroup of Aut(G), the group Dec(μ_{•}) is a linear Lie group, and we conclude that it has a countable basis for its topology.

In order to prove the compactness of Sym(μ_{•}) and closedness of im f_{μ_{•}} one needs a “Convergence of Types Theorem.” We explain what this means. Namely, we say that the Convergence of Types Theorem holds if there is a subgroup B ⊆ Aut(G) and an open subset F ⊆ M^{1}(G) such that for μ, ν ∈ F,

(μ_{n}, A_{n}, x_{n}) ∈ M^{1}(G) × B × G, n ∈ ℕ,

such that

(μ, ν) = lim_{n→∞}(μ_{n}, A_{n}(μ_{n}) ∗ ε_{x_{n}}),

it follows that {A_{n} : n ∈ ℕ} and {x_{n} : n ∈ ℕ} are relatively compact in B and G, respectively. We shall also say that a Convergence of Types Theorem holds for a convolution semigroup μ_{•} if a Convergence of Types Theorem holds in such a fashion that Dec(μ_{•}) ⊆ B and μ_{t} ∈ F for t > 0.

Proposition 3.4. Let G be a connected Lie group and μ_{•} a convolution semigroup. If a Convergence of Types Theorem holds for μ_{•}, then im f_{μ_{•}} is closed in ℝ_{+}^{×}, and Sym(μ_{•}) is compact. Proof. (Cf. [7], [9], [24].) Assume that (a_{n}) is a sequence in im f_{μ_{•}} converging to a ∈ ]0, ∞[. We must show that a ∈ im f_{μ_{•}}. Pick A_{n} ∈ Dec(μ_{•}) with f_{μ_{•}}(A_{n}) = a_{n} and find X_{n} ∈ 𝔷(𝔤) such that

(1)  μ_{a_{n}t} = A_{n}(μ_{t}) ∗ ε_{exp(−tX_{n})}.

Then μ_{a_{n}t} → μ_{at}, t ≥ 0. By the Convergence of Types Theorem for μ_{•} we obtain relative compactness of {A_{n} : n ≥ 1} and of {exp −tX_{n} : n ≥ 1} for each t > 0. Hence there exist accumulation points A ∈ Aut(G) and z(t) ∈ Z(G)_{0} , such that

(∗)  (∀t ≥ 0)(∃z(t) ∈ Z(G)_{0}) A(μ_{t}) = μ_{at} ∗ ε_{z(t)}.

By Lemma Z in the proof of 3.3 we find an element X ∈ 𝔷(𝔤) such that

(∗∗)  (∀t ≥ 0) A(μ_{t}) = μ_{at} ∗ ε_{exp(t·X)}

and that A ∈ Dec(μ_{•}). This shows that a ∈ im f_{μ_{•}} and thus complete the proof that im f_{μ_{•}} is closed. In order to see the compactness of Sym(μ_{•}) we take {A_{n} : n ∈ ℕ} in Sym(μ_{•}), and we have to show that this sequence has a cluster point. We find X_{n} ∈ 𝔷(𝔤) such that

(2)  μ_{t} = A_{n}(μ_{t}) ∗ ε_{exp(−t·X_{n})}.

This relation follows from (1) upon setting a_{n} = 1 for all n. Thus the argument in the first part of the proof shows that there is a cluster point A of {A_{n} : n ∈ ℕ} in Dec(μ_{•}) and an X ∈ 𝔷(𝔤) such that A(μ_{t}) = μ_{t} ∗ ε_{exp(t·X)} . Hence A ∈ Sym(μ_{•}), and the compactness of Sym(μ_{•}) is established.

We shall discuss below situations in which a Convergence of Types Theorem is available for a given semigroup μ_{•}.

Corollary 3.5. Assume the hypotheses of Proposition 3.4 and, in addition, that μ_{•} is stable. Then Dec(μ_{•}) ≅ Sym(μ_{•}) × ℝ.

Proof. The surjectivity of f_{μ_{•}} implies that the following sequence is exact:

0 → Sym(μ_{•})⟶^{incl} Dec(μ_{•})⟶^{log ∘f_{μ•}}ℝ → 0.

Set Γ =^{def} Dec(μ_{•})/ Sym(μ_{•}). Then log ∘f_{μ_{•}} induces canonically a bijective morphism f: Γ → ℝ. Since Sym(μ_{•}) is compact and im f_{μ_{•}} is closed in ℝ_{+}^{×} by 3.4, by the Open Mapping Theorem for morphisms between locally compact groups, f_{μ_{•}} is open onto its image, and thus is also a closed map because its kernel Sym(μ_{•}) is compact. Hence f is also closed and thus is an isomorphism. Now Remark 1.9 proves the corollary.

The conclusion of the previous Corollary may be obtained with slightly weaker hypotheses as follows. We continue to set Γ =^{def} Dec(μ_{•})/ Sym(μ_{•}) and let f: Γ → ℝ be the morphism induced by log ∘f_{μ_{•}}.

Remark 3.6. Let G be a locally compact group and μ_{•} a stable convolution semigroup such that Dec(μ_{•}) is locally compact and Sym(μ_{•}) is compact. If Dec(μ_{•}) has a countable basis for its topology, which is the case if G is a connected Lie group and Dec(μ_{•}) is closed in Aut(G), then Dec(μ_{•}) ≅ Sym(μ_{•})×ℝ.

Proof. Since μ_{•} is stable, f: Γ → ℝ is a bijective morphism of topological groups. The connected subgroup f(Γ_{0}) of ℝ is either {0} or ℝ since ℝ does not contain any proper connected nonsingleton subgroups. In the latter case, the bijectivity of f implies Γ_{0} = Γ. Then, as a locally compact and connected group, Γ is σ -compact. Hence f is an isomorphism. If this fails, then Γ_{0} = {1}, i.e., Γ is a totally disconnected locally compact abelian group. Hence it has an open compact subgroup C. Then f(C) = {0} since ℝ has no nondegenerate compact subgroups. Hence C = {1} and thus Γ is discrete. Then Dec(μ_{•}) has uncountably many open components and thus cannot be contained in the group Aut(G) which has a countable basis for its topology. So this is impossible. The splitting is then concluded form 1.9 as in the proof of 3.5.

A particular consequence of Corollary 3.5 concerns information on convolution semigroups. Indeed, from 3.5 we obtain a one-parameter subgroup t ↦ τ_{t}: ℝ_{+}^{×} → Z( Sym(μ_{•}), Aut(G)) such that Dec(μ_{•}) = Sym(μ_{•})τ_{ℝ_{+}^{×}} and that this product is direct.

Corollary 3.7. Assume that G is a locally compact group and μ_{•} a convolution semigroup such that Dec(μ_{•}) ≅ Sym(μ_{•}) × ℝ; by 3.6, this hypothesis is satisfied if G is a connected Lie group, if a Convergence of Types Theorem holds for μ_{•}, and if μ_{•} is stable. Then there exists a continuous one-parameter group t ↦ τ_{t}: ℝ_{+}^{×} → Dec(μ_{•}), τ_{st} = τ_{s}τ_{t}, such that μ_{•} is (τ_{t}, t)-semistable for all t > 0. In other words, for all s ≥ 0 and t > 0 we have

τ_{t}(μ_{s}) = μ_{ts} ∗ ε_{exp(tsX)}

for a suitable X = X(t) ∈ 𝔷(𝔤).

Next we deal with the properly semistable variant of Proposition 3.4. For this purpose we introduce the following definition.

Definition 3.8. An element A ∈ Dec(μ_{•}) is called a commuting normalization if AS = SA for all S ∈ Sym(μ_{•}).

We note the following. Assume that A is a commuting normalisation. Then we define a ∈ ℝ_{+}^{×} \ {1} by f_{μ_{•}}(A) = a. Now the semidirect product

Dec_{a}(μ_{•}) =^{def} {A′ ∈ Dec(μ_{•}) : f_{μ_{•}}(A′) = a^{n}, n ∈ ℤ} = f_{μ_{•}}^{−1}{a^{n} : n ∈ ℤ}

splits as a direct product over the compact subgroup Sym(μ_{•}) in the form

Dec_{a}(μ_{•}) = Sym(μ_{•}) × {A^{n} : n ∈ ℤ} ≅ Sym(μ_{•}) × ℤ.

Theorem 3.9. Let G be a connected Lie group and μ_{•} a properly semistable convolution semigroup of probability measures on G. If a Convergence of Types Theorem holds for μ_{•}, then the following conclusions hold,

(I) Dec(μ_{•}) ≅ Sym(μ_{•})× ℤ.

(II) There exists a natural number e ∈ ℕ such that for any C ∈ Dec(μ_{•}) there is an S ∈ Sym(μ_{•})_{0} such that A =^{def} C^{e}S is a commuting normalization.

(III) Let c = f_{μ_{•}}(C). Then

f_{μ_{•}}^{−1}({c^{en} : n ∈ ℤ}) = Sym(μ_{•}){A^{n} : n ∈ ℕ},

and this product is direct and isomorphic to Sym(μ_{•}) × ℤ. (IV) Let C ∈ Dec(μ_{•}) be such that f_{μ_{•}}(C) is a generator of im(f_{μ_{•}}). Then

there is an element S ∈ Sym(μ_{•})_{0} such that the subgroup Sym(μ_{•})⟨C^{e}S⟩ of Dec(μ_{•}) is a direct product and Dec(μ_{•})/ Sym(μ_{•})⟨C^{e}S⟩ is a cyclic group of order e.

Proof. (I) It follows from 3.4 that im(f_{μ_{•}}) is a closed subgroup of ℝ_{+}^{×}; it is proper since μ_{•} is properly semistable and nondegenerate since μ_{•} is semistable. Therefore it is infinite cyclic discrete. Thus im(f_{μ_{•}}) is free in the category of topological groups and (I) follows.

(II) By (I) there is an element T ∈ Dec(μ_{•}) such that Dec(μ_{•}) = Sym(μ_{•})⟨T ⟩ with ⟨T ⟩ = {T^{n} : n ∈ ℤ}, and T normalizes Sym(μ_{•}). By 3.4 we may consider Dec(μ_{•}) as a subgroup of Aut(𝔤) and thus as a closed subgroup of GL(𝔤). Therefore the assumptions of Corollary D are fulfilled, and the assertion follows from Corollary D(ii).

(III) Moreover, since f_{μ_{•}}(⟨T ⟩) = im f_{μ_{•}} is infinite cyclic in ℝ_{+}^{×}, it follows that Dec(μ_{•})/ Sym(μ_{•}) is infinite cyclic discrete. Thus the assertion follows from Corollary D.

(IV) If we take C = T with T as in the proof of (II), then (II) applies and yields the assertions straightforwardly.

This theorem was an original motivation for the theory presented in Sections 1 and 2. In the remainder of the paper we illustrate situations in which we have a Convergence of Types Theorem so that 3.4 and its corollaries, notably 3.9 become applicable. Firstly, all simply connected nilpotent Lie groups admit a Convergence of Types Theorem with B = Aut(G), generalizing the well-known Convergence of Types Theorem for vector groups ℝ^{d}. A precise formulation will be given below. However, there are other types of Lie groups admitting a Convergence of Types Theorem, e.g., certain connected Lie groups without central tori of positive dimension or certain algebraic groups where the automorphisms are supposed to be algebraic [3]. Yet we shall restrict our considerations to simply connected nilpotent Lie groups. This is justified since the existence of a semistable continuous convolution semigroup on a connected Lie group causes these Lie groups to come into focus as was shown in [10]; we will overview this fact now.

Reduction to simply connected nilpotent Lie groups

We begin on a purely topological group theory level by considering a topological group G and an automorphism A ∈ Aut(G) of topological groups. Set

C(A) =^{def} {x ∈ G : lim_{n→∞} A^{n}(x) = 1}.

Then C(A) is an A-invariant subgroup of G. Let L be a compact subgroup of G with A(L) ⊆ L. We take x ∈ C(A) and g ∈ L. The space S =^{def} {Ax, A^{2}x, …} ∪ {1} is compact. For each n ∈ ℕ we write g_{n} = A^{n}gA^{−n} and get A^{n}(gxg^{−1}) = g_{n}(A^{n}x)g_{n}^{−1} . Assume that n(j) is a subnet of the sequence of natural numbers such that h = lim_{j} A^{n(j)}gxg^{−1} in the compact space LSL^{−1} . Since L is compact, there is a subnet n(j_{k}), such that c = lim_{k} g_{n(j_{k})} exists. Thus we obtain h = c1c^{−1} = 1. Hence lim_{n→∞} A^{n}(gxg^{−1}) = 1 and gxg^{−1} ∈ C(A). Therefore every A-invariant compact subgroup L of G normalizes C(A).

For a finite dimensional real vector space 𝔤 and an endomorphism α we write 𝔤_{0}(α) = {X ∈ 𝔤 : lim_{n→∞} α^{n}(X) = 0}. Now assume that G is a Lie group with Lie algebra 𝔤 = 𝔏(G) and that A ∈ Aut(G) is an automorphism. Denote by 𝔏(A) ∈ Aut(𝔤) the automorphism induced on the Lie algebra. As is shown in [10], C(A) is an analytic subgroup with Lie algebra 𝔏(C(A)) = 𝔤_{0}(𝔏(A)).

For a compact subgroup L of G we shall also consider the homogeneous space G/L = {xL : x ∈ G} and define the subgroup

C_{L}(A) =^{def} {x : lim_{k→∞} A^{k}(x)L = L in G/L},

the L-contractible part of G. Clearly C(A) ⊆ C_{L}(A). If A(L) = L, then L normalizes C(A) and A induces an autodiffeomorphism A_{L} of the homogeneous space G/L in such a way that x ∈ C_{L}(A) iff lim_{k→∞} A_{L}^{k}(xL) = L in G/L. In particular, L ⊆ C_{L}(A) and thus C(A)L ⊆ C_{L}(A). In Theorem 2.4 of [10] equality is shown. The normalizer of C(A) ∩ L contains C(A)L = C_{L}(A), and thus D =^{def} \overline{C(A) ∩ L} is a compact normal subgroup of \overline{C_{L}(A)}.

The case G = \overline{C_{L}(A)}. This suggests that the hypothesis G = \overline{C_{L}(A)} which we assume now is of particular interest. Then D is a compact normal subgroup of G and we can form the Lie groups

G =^{def} G/D,
L =^{def} L/D, and the autmorphism
A ∈ Aut(G) induced by A ∈ Aut(G).

The analytic subgroup C(A)D/D of G becomes a Lie group when it is endowed with its intrinsic Lie group structure; this Lie group we shall call N, and the inclusion morphism yields an injective morphism of Lie groups j_{N}: N → G with bijective image C(A)D/D. The automorphism A ∈ Aut(G) induces on N an automorphism A_{N} such that C(A_{N}) = N. It is shown in [10] that N is a simply connected nilpotent Lie group. The inner automorphisms implemented by elements of L on C(A)D/D implement automorphisms of N and thus provide us with a morphism of Lie groups β: L → Aut(N). This allows us to form the Lie group Γ =^{def} N ×_{β} L. Since L as a compact Lie group has finitely many components, the Lie group Γ has finitely many components. We let C denote the subgroup C_{L}(A)/D of G. The definition of D implies C(A)D/D ∩L/D = {D}. By Theorem 2.4 of [10] we have C_{L}(A) = C(A)L. Hence the morphism of topological groups p: Γ → C ⊆ G, p(n, x) = j_{N} (n)x is bijective. The subgroup p(Γ_{0}) ⊆ C of G is analytic and has finite index in C. The group C is σ - compact (i.e. is a countable union of compact subspaces) since it is a continuous image of the σ -compact space Γ; hence C is Borel measurable, and the inverse map p^{−1}: C → Γ is Borel measurable. In passing we remark from the fact that D and C_{L}(A)/D have finitely many arc components, that C_{L}(A) has finitely many arc components. The arc component C_{L}(A)_{0} of the identity is an analytic subgroup of G which is dense in G_{0} . In particular, C_{L}(A) is Borel measurable. Thus G/G_{0} is finite and (G_{0})′, the commutator subgroup of G_{0} , is contained in C_{L}(A)_{0} .

The desintegration of convolution semigroups. Now assume that μ_{•} is a convolution semigroup of probability measures on G. Then μ_{0} is an idempotent probability measure and therefore is a Haar measure ω_{L} of a compact subgroup L of G. In addition we assume that μ_{•} is semistable with A ∈ Dec(μ_{•}), 0 < a = f_{μ_{•}}(A) < 1. Then L =^{def} supp μ_{0} is an A-invariant compact subgroup. It is no loss of generality if we restrict our attention to the smallest closed Ainvariant subgroup supporting μ_{•}. The smallest closed subgroup supporting μ_{•}

is H =^{def} \overline{⟨⋃_{t>0} supp(μ_{t})⟩}. If

(+)  H = \overline{⟨⋃_{t>0} supp(μ_{t} ∗ μ̃_{t})⟩},

then H is A-invariant and thus agrees with G by our assumption. With the methods of [10] it follows that for any t > 0 μ_{t} ∗ μ̃_{t} is concentrated on the Borel measurable subgroup C_{L}(A). Thus C_{L}(A) is dense in G if (+) holds. In [10] it is shown that, if μ_{•} is strictly (A, a)-semistable (see 3.2(v)), then, more strongly,

(++)  μ_{t}(C_{L}(A)) = 1 for all t ≥ 0.

Thus again C_{L}(A) is dense in G in this case. We observe that the condition

1 ∈ supp(μ_{t}) for all t > 0

implies both (+) and (++); it therefore implies in particular, that C_{L}(A) is a dense subgroup of G.

The bimeasurable homomorphism p defines an injective weakly continuous homomorphism M^{1}(Γ) → M^{1}(G) of convolution semigroups. Indeed, p establishes a bijection between the set of measures ν × ω_{L}, where ν is an L-invariant measure on the subgroup N of Γ, and the set of measures λ = ω_{L} ∗ λ ∗ ω_{L} in M^{1}(G) with the property λ(C) = 1. This applies, in particular, to semistable continuous convolution semigroups on Γ and on G, respectively. Let π: G → G denote the quotient morphism. For a measure ν on G we write π(ν) for its image on G.

The significance of these structural results for the semigroup μ_{•} is summarized in the following proposition ([10]). We denote the automorphism induced by A on N by A_{N} ∈ Aut(N).

Let μ_{•} be an (A, a)-semistable convolution semigroup on a connected Lie group G in which we consider a compact subgroup L. For the sake of the formulation of the following proposition let us say that μ_{•} satisfies hypothesis (S) if (i) μ_{0} = ω_{L} (ii) μ_{•} is strictly semistable, or 1 ∈ supp(μ_{t}) for all t ≥ 0.

The property (S) guarantees that C_{A}(L) is dense and the results we discussed apply. We have seen that for a μ_{•} satisfying (S), μ_{•} and π(μ_{•}) are concentrated on C_{L}(A) and on C_{L}(A), respectively. The notation introduced in the preceding discussion is maintained.

Proposition 3.10. Let G be a connected Lie group G. Then the following conclusions hold.

(i) For each convolution semigroup μ_{•} on G satisfying hypothesis (S), there is a continuous, L-invariant, and (A_{N} , a)-semistable convolution semigroup λ_{•} = (λ_{t})_{t≥0} on N such that λ_{0} = ε_{1} and

(#)  π(μ_{•}) = p(λ_{•} × ω_{L}) = j_{N} (λ_{•}) ∗ ω_{L} = j_{N} (λ_{•}) ∗ π(ω_{L})

(ii) Conversely, given an L-invariant (A_{N} , a)-semistable convolution semigroup λ_{•} = (λ_{t})_{t≥0} on N, then t ↦ λ_{t} × ω_{L} lifts uniquely to an (A, a)- semistable convolution semigroup μ_{•} on G in the sense that (#) holds.

Proof. After the preceding comments, only the last assertion needs proof. Note that D ⊆ L and thus π sets up a bijection between L-biinvariant measures on G and L-biinvariant measures on G.

The following consequence of Proposition 3.10 collects relevant conclusions.

Corollary 3.11. For a given connected Lie group G, an A ∈ Aut(G), the compact subgroup L of G, and the corresponding simply connected nilpotent Lie group N defined by A and L, there is a bijective correspondence between strictly (A, a)-semistable convolution semigroups μ_{•} on G with μ_{0} = ω_{L} on the one hand, and strictly (A_{N} , a)-semistable, and hence L-invariant convolution semigroups λ_{•} with λ_{0} = ε_{1} on the simply connected nilpotent Lie group N on the other.

Therefore, in order to investigate (strictly) semistable convolution semigroups on Lie groups G it is largely sufficient to investigate semistable convolution semigroups on simply connected nilpotent Lie groups N. We aim to apply Theorem 3.9 to this situation.

Convergence of Types Theorem on simply connected nilpotent Lie groups

Let G be a simply connected nilpotent Lie group. The subsequent remarks follow [9]. We define π be the homomorphism resulting from the composition

G⟶^{quot}G/[G, G]⟶^{≅}𝔤/[𝔤, 𝔤].

Recall that in the classical situation a probability measure on a vector space ℝ^{d} is called full if supp(μ) is not concentrated on a coset of a proper linear subspace. A probability convolution semigroup μ_{•} on G is said to be full if π(μ_{•}) is a full probability convolution semigroup on the vector space 𝔤/[𝔤, 𝔤]. Let F denote the set of full measures and B =^{def} Aut(G). In this case an analog of the Convergence of Types Theorem for Vector Spaces ([9], Theorem 2.3, [7]) holds. In particular, for full convolution semigroups μ_{•} on G, the morphism f_{μ_{•}} is a proper map, and the group Sym(μ_{•}) is compact. (The above description implies directly that μ_{t}, t > 0 is full iff μ_{1} is full.)

An application of Theorem 3.9 to this situation yields

Theorem 3.12. Let G be a simply connected nilpotent Lie group and μ_{•} a full convolution semigroup of probability measures on G. Assume that μ_{•} is properly semistable. Then there exists a natural number e with the following property. For each C ∈ Dec(μ) there is an S ∈ Sym(μ)_{0} such that A =^{def} C^{e}S is a commuting normalization. Accordingly, the subgroup Sym(μ_{•})⟨C^{e}S⟩ of Dec(μ_{•}) is a direct product and Dec(μ_{•})/ Sym(μ_{•})⟨C^{e}S⟩ is a cyclic group of order e.

We set c = f_{μ_{•}}(C) and note that μ_{•} is (A, c^{e})-semistable in these circumstances.

The case G = ℝ^{d}

We assume now that μ is a full probability on ℝ^{d}, a special case of what we discussed in Theorem 3.12. Nevertheless, from the point of view of applications we prefer to reformulate Theorem 3.11 in the context of semistable laws on finite dimensional vector spaces. In this context, (semi-)stable probabilities μ are traditionally called operator (semi-)stable (see [16], [17], [18], [25]).

On ℝ^{d} infinitely divisible probabilities μ are uniquely embeddable into continuous convolution semigroups μ_{•} such that μ = μ_{1} . Therefore, semistability is a property of a probability μ = μ_{1} , and the notations f_{μ} =^{def} f_{μ_{•}}, Sym(μ) =^{def} Sym(μ_{•}), and Dec(μ) =^{def} Dec(μ_{•}) are justified.

For simply connected nilpotent Lie groups only a weaker result is available: Let μ_{•} and μ_{•}′ be convolution semigroups on G which are strictly semistable and assume μ_{1} = μ′_{1} . Then μ_{t} = μ′_{t} for all t ≥ 0 (see [8]). Hence strict semistability is a property of a probability μ = μ_{1} , and then again the notations f_{μ} =^{def} f_{μ_{•}}, Sym(μ) =^{def} Sym(μ_{•}), and Dec(μ) =^{def} Dec(μ_{•}) are justified.

Several remarks are in order. Firstly,

Remark 3.13. Semistable, respectively, stable laws are frequently defined as limit laws of operator-normalized sums of i.i.d. random variables. In the case of full measures these definitions coincide.

Secondly, for a probability μ on ℝ^{d} the decomposability semigroup is defined as

D(μ) := {A ∈ End(ℝ^{d}) : (∃ν ∈ M^{1}(ℝ^{d})) μ = A(μ) ∗ ν}.

If A ∈ Dec(μ) and f_{μ}(A) < 1, then μ_{t} = A(μ_{t}) ∗ μ_{t−at} for any t > 0 . Thus we have Remark 3.14. The subset Dec(μ)_{+} := {A ∈ Dec(μ) : a = f_{μ_{•}}(A) ∈ ]0, 1]} is a subsemigroup of the decomposability semigroup D(μ).

The semigroup D(μ) is an important tool for investigations of selfdecomposable (or Lévy’s) probabilities. See e.g., [17] or [12].

On ℝ^{d} one-parameter groups τ: ℝ_{+}^{×} → Aut(ℝ^{d}) = GL(ℝ^{d}) with τ_{st} =

τ_{s}τ_{t} are all of the form τ_{t} = t^{E} =^{def} exp ((log t)·E), E ∈ M(ℝ, d). Now Corollary 3.7 applies to the present situation and yields instantly

Remark 3.15. For full operator stable laws there exist commuting exponents, i.e., there exists a one-parameter group (t^{E_{c}})_{t>0} with exponent E_{c} ∈ GL(ℝ^{d}) such that t^{E_{c}} ∈ Dec(μ) with f_{μ}(t^{E_{c}}) = t for all t > 0 and such that t^{E_{c}} commutes with all S ∈ Sym(μ).

For a direct proof see e.g. [17]. For the corresponding result for groups see e.g. [5] or [6]. Recall that according to Definition 3.8 an automorphism A ∈ Dec(μ) is called commuting normalization if AS = SA for all S ∈ Sym(μ). In particular, for full stable laws the existence of commuting exponents provides commuting normalizations t^{E_{c}} for all t > 0.

In contrast with the situation of stable laws, for properly semistable laws on ℝ^{d}, up to now, only partial results were known, namely, under the hypothesis that the symmetry group Sym(μ) is finite [19] or that d ≤ 3 [20], Theorem 7. As a corollary of 3.12 we obtain now:

Corollary 3.16. Let μ be a full semistable law on ℝ^{d}. Then there is a natural number e depending only on Sym(μ) such that for each C ∈ Dec(μ) there exists an S ∈ Sym(μ)_{0} for which A =^{def} C^{e}S is a commuting normalization.

It can easily be shown that, in contrast with the stable case, the exponent e in Theorem 3.12, will in general not be 1. Thus in general Dec(μ) itself does not split as a direct product. We saw in Example 1.2.1 that a semidirect extension of a compact group K by ℤ is in general not a direct product even if K is a Lie group; however, not every compact subgroup of GL(ℝ^{d}) is representable as Sym(μ). For a discussion see [21]. Thus we shall now exhibit examples K ⊆ GL(ℝ^{d}) representable as the symmetry group K = Sym(μ) of a full semistable law μ with a semidirect nonsplitting extension:

Example 3.17. Consider ℝ^{2} and let K =^{def} \{±I, ±D\}, where D = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ).

Then A := ( \begin{matrix}0 & α \\ β & 0 \end{matrix} ) normalizes K, but the centralizer Z(K, GL(ℝ^{d})) of K is

the group of diagonal matrices. Therefore AC ∉ Z(K, GL(ℝ^{d})) for all C ∈ K, but of course A^{2} = αβI is central. Hence we have e = 2.

Remark 3.18. If 0 < α < β and αβ < a < 1 then there exist (A, a)- semistable laws μ on ℝ^{d} with Sym(μ) = K.

Proof. We see this easily by constructing a suitable K -invariant Lévy measure η on the union of orbits ⋃_{X∈K}{XA^{k}e_{1} : k ∈ ℤ}, e_{i} denoting the i-th unit vector, such that A(η) = aη. Note that Spec(A) = {±(αβ)^{1/2}}. Hence such Lévy measures η exist, cf. [16].

Theorem 3.12 illuminates the fine structure of the decomposability group Dec(μ) and has various applications. As an example we briefly sketch one particular result which we now derive easily from Theorem 3.12. (For corresponding results in the situation of stable laws cf. [22], Theorem 2.1 and Corollary 2.1.)

Proposition 3.19. Let μ be a full semistable law. Assume A ∈ Dec(μ) and f_{μ}(A) = a. Then the absolute values of the eigenvalues of A, thus the spectral radii ρ(A) and ρ(A^{−1}), and therefore the rate of decay of ∥A^{n}∥ depend only on a = f_{μ}(A) and not on the particular choice of A.

Proof. (1) For T ∈ GL(ℝ^{d}) let T = S_{T} U_{T} be the (multiplicative) Jordan decomposition, S_{T} and U_{T} denoting the semisimple and unipotent part of T, respectively. Assume P, Q ∈ Dec(μ) such that Q is a commuting normalization and that f_{μ}(P) = f_{μ}(Q). Then, since P Q^{−1} ∈ Sym(μ) = ker f_{μ}, we obtain that Q and P Q^{−1} commute, hence P Q = QP and thus S_{P} , S_{Q}, U_{P} , and U_{Q} commute. Therefore P Q^{−1} = (S_{P} S_{Q}^{−1})(U_{P} U_{Q}^{−1}) is the Jordan decomposition. From the fact that Sym(μ) is compact we conclude U_{P} U_{Q}^{−1} = I, hence U_{P} = U_{Q}. Moreover, S_{P} S_{Q}^{−1} = S_{Q}^{−1}S_{P} ∈ Sym(μ) is semisimple and therefore Spec(P) = Spec(Q) up to multiplication with unimodular constants.

(2) Let now A ∈ Dec(μ). According to Theorem 3.12, let e ∈ ℕ and C ∈ Sym(μ) such that Q =^{def} A^{e}C ∈ Z(Sym(μ), GL(ℝ^{d})). We apply Part (1) to P =^{def} A^{e} and find that the absolute values of the eigenvalues of A^{e} and of Q are equal, r_{1}, …, r_{d}, say. Therefore, by the Spectral Mapping Theorem, the absolute values of the eigenvalues of A are r_{i}^{1/e}, i = 1, …, d. The assertion follows.

Appendix: Vague Convergence.

Let μ_{•} a continuous convolution semigroup on a second countable locally compact group G. Then the vague limit lim_{t→∞} μ_{t} ∗ μ̃_{t} exists.

Proof. (1) According to a theorem by Csiszàr [2] for any sequence ν_{n} in M^{1}(G), there exists a sequence x_{n} in G such that for λ_{n} =^{def} ν_{1}∗⋯∗ν_{n} either λ_{n}∗ ε_{x_{n}} converges weakly or the concentration function sup_{x∈G} λ_{n}(Kx) converges to 0 for all compact K ⊆ G. In either case we obtain weak, respectively, vague convergence of λ_{n} ∗ λ̃_{n}.

(2) We apply this to continuous convolution semigroups μ_{•}: For any increasing

sequence t_{n} ↗ ∞ put ν_{n} =^{def} μ_{t_{n}−t_{n−1}} ; hence λ_{n} = ν_{1} ∗ ⋯ ∗ ν_{n} = μ_{t_{n}}, and we

obtain that the vague limit lim_{n→∞} μ_{t_{n}} ∗ μ̃_{t_{n}} =^{def} ρ exists. Assume ρ and σ to be the limits for sequences s_{n} and t_{n} ↗ ∞. Without losing generality we may assume that t_{n} ≤ s_{n} < t_{n+1} . Combining these sequences we obtain a new sequence r_{n}, and again the limit exists. We conclude that ρ = σ. This proves our assertion.

References

[1] Bourbaki, N., “Groupes et algèbres de Lie, Chapitres 7 et 8,” Hermann, Paris, 1975.

[2] Csiszàr, I., On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups, Z. Wahrscheinlichkeitstheorie verw. Geb. 5 (1966), 279–295.

[3] Dani, S. G., Invariance groups and convergence of types of measures on Lie groups, Math. Proc. Cambridge Phil. Soc. 112 (1992), 91-108.

[4] Freudenthal, H., La structure des groupes à deux bouts et des groupes triplements transitifs, Indag. Math. 13 (1951), 288–294.

[5] Hazod, W., On the decomposability group of a convolution semigroups, In: Probabilities on Vector Spaces, Lecture Notes in Math. (Springer) 1391 (1988), 99–111.

[6] —, Stable probability measures on groups and on vector spaces. A survey, in: Probability Measures on Groups VIII, Lecture Notes in Math. (Springer) 1375 (1989), 304–352.

[7] —, Remarks on the convergence of types theorem for finite dimensional vector spaces, Publ. Math. Debrecen 50 (1997), 197–219.

[8] —, On some convolution semi- and hemigroups appearing as limit distributions of normalized products of group valued random variables, Preprint Univ. Dortmund, 1997.

[9] Hazod, W., and S. Nobel, Convergence of types theorem for simply connected nilpotent groups, in: Probability Measures on Groups IX, Lecture Notes in Math. (Springer) 1375 (1989), 99–106.

[10] Hazod, W., and E. Siebert, Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups, J. Theor. Probability 1 (1988), 211–226.

[11] Hofmann, K. H., Locally compact semigroups in which a subgroup with compact complement is dense, Trans. Amer. Math. Soc. 106 (1991) 19– 63.

[12] Hofmann, K. H., and Z. J. Jurek, Some analytical semigroups occuring in probability theory, J. of Theor. Probability 9 (1996), 745–763.

[13] Hofmann, K. H., J. D. Lawson, and W. A. F. Ruppert, Weyl groups are finite—and other finiteness properties Math. Nachrichten 179 (1996), 119-143.

[14] Hofmann, K. H., and P. S. Mostert, “Splitting in Topological Groups,” Memoir Amer. Math. Soc. 43, 1963, 82pp.

[15] Iwasawa, K., On some types of topological groups, Annals of Math. 50(3) (1949), 507–558.

[16]  Jajte, R., Semistable measures on ℝ^{N}, Studia Math. 61 (1977), 29–39.

[17] Jurek, Z. J., and J. D. Mason, “Operator-Limit Distributions in Probability Theory,” J. Wiley, New York etc., 1993.

[18] Luczak, A., Elliptical symmetry and characterization of operator stable and operator semistable measures, Ann. Prob. 12 (1984), 1217–1223.

[19] —, On some symmetry properties of infinite divisible and operator semistable measures, in: Stochastic Methods in Experimental Design, World Scientific 1990, 304–311.

[20] —, Remarks on operator semistable measures on finite dimensional vector spaces, to be published.

[21] Meerschaert, M., and J. A. Veeh, Symmetry-groups in d-space, Statistics and Probability Letters 22 (1995), 1–6.

[22] —, The structure of the exponents and symmetries of an operator stable measure, J. Theor. Probability 6 (1993), 713–726.

[23] Montgomery, D., and L. Zippin, “Topological Transformation Groups,” Interscience Publishers, New York, 1955.

[24] Nobel, S., Limit theorems for probability measures on simply connected nilpotent Lie groups, J. Theor. Probability 4 (1991), 261–284.

[25] Siebert, E., Supplements to operator stable and operator semistable laws on Euclidian vector spaces, J. Mult. Analysis 19 (1986), 329–341.

W.Hazod and H.-P.Scheffler Institut für Mathematik Universität Dortmund Campus Nord, Mathematikgebäude D-44221 Dortmund, Germany hazod@math.uni-dortmund.de K.H.Hofmann and M.Wüstner Fachbereich Mathematik Technische Universität Darmstadt Schlossgartenstr. 7 D-64289 Darmstadt, Germany hofmann@mathematik.tu-darmstadt.de

H.Zeuner

Medizinische Universität zu Lübeck

Technisch-Naturwissenschaftliche Fakultät Wallstr. 40 D-23560 Lübeck, Germany zeuner@math.mu-luebeck.de

Received August 5, 1996 and in final form March 11, 1998