Journal of Lie Theory Volume 4 (1994) 17–29 C ◯1994 Heldermann Verlag

Maximal Ol’shanskiĭ Semigroups

J. D. Lawson ^{∗}

Communicated by K. H. Hofmann

Abstract. In this article we show for simple complex groups G_{ℂ} that if a subsemigroup S contains a real form G such that G is not isolated in S, then S contains a minimal Ol’shanskiĭ semigroup and is contained in a maximal Ol’shanskiĭ semigroup. It follows in particular that maximal Ol’shanskiĭ semigroups are maximal semigroups.

Let G be a real form for a complex Lie group G_{ℂ}. A complex Ol’shanskiĭ semigroup in G_{ℂ} is one of the form Γ(C) = exp(iC)G, where C is an Ad G-invariant convex cone in the Lie algebra 𝔤 of G. These semigroups are noncommutative analogues of tube domains and play an important role in the representation theory of G ([13], [12]) and also arise in a variety of contexts of harmonic analysis on G ([5]). Their structure is presented in more detail in Section 1.

When one tries to establish in a concrete situation that a certain semigroup is an Ol’shanskiĭ semigroup, it is generally much easier to show that the appropriate Ol’shanskiĭ semigroup is contained in the given semigroup than to show the reverse containment. This paper provides an important theoretical tool for showing the reverse containment in the case of simple groups: if a subsemigroup S contains a real form G of G_{ℂ} such that G is not isolated in S, then S contains a minimal Ol’shanskiĭ semigroup and is contained in a maximal Ol’shanskiĭ semigroup. The preliminary background is recalled in the first two sections and the proof is given in Sections 3 and 4. Section 5 discusses some important consequences and applications of this result.

It is a corollary of what is done here that in the context of simple groups maximal complex Ol’shanskiĭ semigroups are maximal semigroups, a fact first established by Hilgert and Neeb [7, Chapter 8]. The proof given here relies heavily on their earlier work, but there is also a significant simplication. In particular, one no longer needs the theory of highest weight representations nor San Martin’s theory of invariant control sets [17]. This not only makes the proof shorter and more accessible, but also enables the generalization given in this paper. For related results in the real analytic case, see [8].

ISSN 0940–2268 / $2.50 ^{_{C} ◯} Heldermann Verlag

∗ The author gratefully acknowledges the support of NSF Grant DMS-9104582

1. Complex Ol’shanskiĭ Semigroups

By a wedge or cone in a finite dimensional real or complex vector space, we mean a non-empty subset that (i) is closed under addition, (ii) is closed with respect to scalar multiplication by non-negative scalars, and (iii) is topologically closed. The terms “wedge” and “cone” are thus interchangeable; we shall, however, use the term “wedge” when we wish to emphasize the likelihood that the edge W ∩ −W of W is non-trivial. We shall tend to think of cones C as pointed, i.e., as satisfying C ∩ −C = {0}. We shall generally be interested in cones or wedges C that are generating, i.e., that satisfy C − C is the whole vector space. Note that C is generating if and only if it has non-empty interior, which we denote by C^{∘}. In particular, C always has interior in C − C; we call this interior the algebraic interior and denote it by algint(C).

In the Lie theory of semigroups, one assigns to a closed subsemigroup S of G an appropriate wedge L(S) (consisting of subtangent vectors to S at e) in the Lie algebra L(G) of G, a process that extends the the usual assigning to an analytic subgroup of the group G its subalgebra in L(G).

We recall basic facts from the Lie theory of semigroups concerning this assignment (see, for example, [6]). Let S be a closed subsemigroup of a Lie group G with Lie algebra L(G). We define the subtangent set of S at e by

L(S) = {X ∈ L(G): exp(tX) ∈ S for all t ≥ 0}.

It turns out that W := L(S) is always a Lie wedge, i.e., W is a wedge which satisfies (i) the edge W ∩ −W (which is the largest vector subspace contained in W) is a subalgebra, and (ii) the wedge W is invariant under the adjoint action of any connected analytic subgroup with Lie subalgebra the edge (equivalently W is invariant under e^{ad X} for all X ∈ W ∩ −W). We say that S is infinitesimally generated if the subsemigroup generated by the exponential image of L(S) is dense in S and is strictly infinitesimally generated if the semigroup generated by the exponential image of L(S) is all of S.

Let 𝔤 be a Lie algebra, and let G be a corresponding Lie group. We denote by Aut(𝔤) the group of automorphisms of 𝔤. The group of inner automorphisms is the group Inn(𝔤) := ⟨e^{ad 𝔤}⟩; if G is connected, it agrees with the group Ad(G).

A cone C ⊆ 𝔤 is called an invariant cone if C is invariant under the adjoint action of G_{0} , the identity component of G. Note that this is equivalent to the invariance of the cone under all automorphisms of the form exp(ad X) for X ∈ 𝔤, since these automorphisms generate the adjoint group. Thus invariance may be viewed as strictly a Lie algebra phenomenon.

These invariant cones give rise to one of the most standard and basic constructions for Ol’shanskiĭ semigroups (see [13], Section V.4 of [6], Chapter VII of [7] and [10]).

Definition 1.1. Let C be an invariant generating pointed cone in a finite dimensional real Lie algebra 𝔤. Let 𝔤_{ℂ} = 𝔤+i𝔤 be the complexification of 𝔤 and let G_{ℂ} be a connected Lie group with Lie algebra 𝔤_{ℂ}. A closed subsemigroup S of G_{ℂ} is called the complex Ol’shanskiĭ semigroup associated to C and G_{ℂ} if the following conditions are satisfied:

(i) The semigroup S is closed in G_{ℂ} and satisfies S = (exp iC)G_{0} = G_{0}(exp iC), where G_{0} is the analytic subgroup generated in G_{ℂ} by exp 𝔤 and is itself a closed subgroup of G_{ℂ}.

(ii) (X, h) ↦ (exp iX)h: C × G_{0} → S is a homeomorphism (and a diffeomorphism if that is appropriately defined for this context). Thus each s ∈ S admits a unique (Ol’shanskiĭ) polar decomposition s = (exp iX)h, X ∈ C, h ∈ G_{0} . The left-right dual also holds.

(iii) The interior S^{∘} of S is a dense semigroup ideal, S^{∘} = (exp iC^{∘})G_{0} , where C^{∘} is the interior of C in 𝔤, S^{∘} is a complex manifold, and the multiplication on S^{∘} is holomorphic.

(iv) The mapping s ↦ s^{∗}: S → S defined by s^{∗} = g^{−1} exp iX if s = (exp iX)g is a continuous antiautomorphism of order 2 on S, called the adjoint involution. It is antiholomorphic on S^{∘}. For s ∈ S, s^{∗} = s^{−1} if and only if s ∈ G_{0} and s^{∗} = s if s ∈ exp iC.

(v) The polar decomposition is uniquely determined as s = exp(iX)h ∈ exp(iC)G_{0} , where X is the unique member of C such that exp(2iX) = ss^{∗} and h = exp(−iX)s.

(vi) The semigroup S is strictly infinitesimally generated and L(S) = 𝔤 ⊕ C. The complex Ol’shanskiĭ semigroup associated to C and G_{ℂ} is denoted by Γ(C, G_{ℂ}), or simply Γ(C) if G_{ℂ} is understood.

The next theorem is an existence theorem in the semisimple setting for complex Ol’shanskiĭ semigroups which is very close to the original one of Ol’shanskiĭ [13]. More general results can be found in [10] and [7].

Theorem 1.2. Let 𝔤 be finite dimensional semisimple Lie algebra which contains a pointed generating invariant cone C, let G_{ℂ} be a connected Lie group with Lie algebra 𝔤_{ℂ} = 𝔤 + i𝔤, and let G_{0} be the analytic subgroup of G_{ℂ} generated by the exponential image of 𝔤 in G_{ℂ}. Then the complex Ol’shanskiĭ semigroup Γ(C) always exists for the data C and G_{ℂ}. Furthermore, if complex conjugation on 𝔤_{ℂ} integrates to a conjugation involution on G_{ℂ}, then G_{0} is the identity component of the fixed point set, and the adjoint involution on Γ(C) is the restriction of the anti-isomorphism on G_{ℂ} given by g ↦ g^{∗} := \overline{g}^{−1} .

2. Invariant Cones in Semisimple and Hermitian Algebras

In the remainder of the paper let 𝔤 denote a finite dimensional semisimple Lie algebra which contains a compact Cartan algebra 𝔱 (i.e. e^{ad 𝔱} is a compact subgroup of Inn 𝔤). We summarize the basic theory of invariant cones in such algebras as it appears in [15], [16], and Chapter 7 of [7] (see also [19], [14], [18], Chapter III of [6], and [11]). Associated to the Cartan subalgebra 𝔱_{ℂ} in the complexification 𝔤_{ℂ} is a root decomposition as follows. For a linear functional λ ∈ 𝔱_{ℂ}^{∗} we set

𝔤_{ℂ}^{λ} := {X ∈ 𝔤_{ℂ} : (∀Y ∈ 𝔱_{ℂ})[Y, X] = λ(Y)X}

and

Δ := Δ(𝔤_{ℂ}, 𝔱_{ℂ}) := {λ ∈ 𝔱_{ℂ}^{∗} \ {0} : 𝔤_{ℂ}^{λ} ≠ {0}}.

Then

𝔤_{ℂ} = 𝔱_{ℂ} ⊕ ⨁_{λ∈Δ} 𝔤_{ℂ}^{λ},

λ(𝔱) ⊆ iℝ for all λ ∈ Δ and σ(𝔤_{ℂ}^{λ}) = 𝔤_{ℂ}^{−λ}, where σ denotes complex conjugation on 𝔤_{ℂ} with respect to 𝔤. Let 𝔨 ⊇ 𝔱 denote a maximal compactly embedded subalgebra; 𝔨 always exists and is uniquely determined by 𝔱. Then a root is said to be compact if 𝔤_{ℂ}^{λ} ⊆ 𝔨_{ℂ}.

We write Δ_{k} for the set of compact roots and Δ_{p} for the set of noncompact roots. A subset Δ^{+} ⊆ Δ is called a positive system of roots if there exists an element X ∈ i𝔱 such that Δ^{+} = {α ∈ Δ: α(X) ≥ 0} and Δ^{+} ∩ −Δ^{+} = Ø. The Weyl group associated to 𝔱 is the group

W_{𝔨} := N_{G}(𝔱)/Z_{G}(𝔱) ≅ N_{K}(𝔱)/Z_{K}(𝔱)

which coincides with the Weyl group of the compactly embedded Lie algbra 𝔨. A positive system Δ^{+} is said to be 𝔨-adapted if Δ_{p}^{+} is invariant under the Weyl group.

For a positive system Δ^{+} of roots we define the cone

𝔠_{max} := 𝔠_{max}(Δ_{p}^{+}) := {X ∈ 𝔱: (∀α ∈ Δ_{p}^{+})iα(X) ≥ 0}

and

𝔠_{min} := 𝔠_{min}(Δ_{p}^{+}) := cone\{i[\overline{X}_{α}, X_{α}]: α ∈ Δ_{p}^{+}, X_{α} ∈ 𝔤_{ℂ}^{α}\},

where for a subset A of a vector space cone(A) denotes the smallest closed convex cone containing A. For a cone C in a vector space V we write C^{⋆} := {ν ∈ V^{∗}: ν(C) ⊆ ℝ^{+}} for the dual cone.

The next theorem is the major one on the existence of invariant cones in the semisimple case.

Theorem 2.1. Let 𝔤 be a finite dimensional semisimple real Lie algebra, which contains a pointed generating invariant cone C. Then compact Cartan algebras 𝔱 exist. For any compact Cartan subalgebra 𝔱 and the (unique) maximal compact algebra 𝔨 containing it, there exists 0 ≠ Z ∈ 𝔷(𝔨) ⊆ 𝔱 and a 𝔨-adapted positive system of roots Δ^{+} such that such that (i) Z is in the interior of C, (ii) Z is in the interior of the elliptic (= compactly embedded) elements of 𝔤, (iii) iα(Z) > 0 for all α ∈ Δ_{p}^{+} , and (iv) for Δ_{p}^{+}

𝔠_{min} ⊆ C ∩ 𝔱 ⊆ 𝔠_{max}.

The invariant pointed generating cones containing 𝔠_{min} are in one-to-one correspondence via intersection with 𝔱 with those cones in 𝔱 containg 𝔠_{min} , contained in 𝔠_{max} , and invariant under the action of the Weyl group W_{𝔨}. In particular, there exist largest and smallest pointed generating invariant cones C_{max} and C_{min} containing 𝔠_{min} and these satisfy C_{max} ∩ 𝔱 = 𝔠_{max} and C_{min} ∩ 𝔱 = 𝔠_{min} .

The simple Lie algebras which contain pointed generating invariant cones are precisely the hermitian simple ones, those for which the maximal compactly embedded subalgebras have a non-trivial (one-dimensional) center. Let 𝔤 be a hermitian simple Lie algebra, let 𝔨 be a maximal compact subalgebra, and let B_{θ}(·, ·) be the positive-definite form on 𝔤 given by B_{θ}(X, Y) = −κ(X, θ(Y)), where κ is the Cartan-Killing form and θ is the Cartan involution which fixes 𝔨. Let Z ∈ 𝔨 span the center of 𝔨, and let 𝔱 be a compact Cartan subalgebra of 𝔤 contained in 𝔨. Then Z ∈ 𝔱, by maximal commutativity of 𝔱, and there exists a 𝔨-adapted positive system of roots Δ^{+} such that iα(Z) > 0 for all α ∈ Δ_{p}^{+} . The following conditions hold (see [15]):

(I) 𝔠_{min} contains Z in its algebraic interior, and 𝔠_{max} = (𝔠_{min})^{∗}, where the

dual is taken in 𝔱 with respect to the restriction of B_{θ}(·, ·). (II) Each pointed generating invariant cone in 𝔤 contains either Z or −Z in

its interior. (III) For each pointed generating invariant cone C, we have (C∩𝔱)^{∗} = C^{∗} ∩𝔱,

where the first dual cone is computed in 𝔱 and the second in 𝔤, both with respect to B_{θ}(·, ·).

3. Maximal Semigroups and Cartan Subalgebras

Let S be a subsemigroup of a Lie group G such that the interior S^{∘} of S in G is non-empty. We set

L_{∞}(S) := {X ∈ G: exp(tX) ∈ S^{∘} for some t > 0}.

We list some elementary properties of L_{∞}(S), which hold in an arbitrary Lie algebra.

Lemma 3.1. Let S ≠ G be a subsemigroup with non-empty interior in a connected Lie group G with Lie algebra 𝔤. Then the following hold:

(i) The set L_{∞}(S) is topologically open and is closed under multiplication by positive scalars.

(ii) If X ∈ L_{∞}(S), then there exists T > 0 such that t ≥ T implies

exp(tX) ∈ S. (iii) If exp(sX) ∈ \overline{S} for some s > 0 (in particular, if X ∈ L_{∞}(S)) and if

Y ∈ L_{∞}(S) and [X, Y] = 0, then X + Y ∈ L_{∞}(S). (iv) If X ∈ L_{∞}(S), then exp(−tX) ∉ \overline{S} for t > 0. In particular, 0 ∉

L_{∞}(S).

(v) If g, g^{−1} ∈ \overline{S} and X ∈ L_{∞}(S), then Ad g(X) ∈ L_{∞}(S).

Proof. (i) Consider the mapping Φ: (0, ∞) × 𝔤 → G defined by Φ(t, X) = exp(tX). Then Φ is continuous and L_{∞}(S) is projection into the second coordinate of the open set Φ^{−1}(S^{∘}), hence open. That L_{∞}(S) is closed under multiplication by positive scalars is immediate.

(ii) Suppose that X ∈ L_{∞}(S) and exp(tX) ∈ S^{∘} for t > 0. Then there exists ε > 0 such that exp(sX) ∈ S^{∘} for t − ε < s < t + ε. Note that for each n > 0, exp(n · (t − ε, t + ε)X) ⊆ S^{∘}, since the latter is a semigroup. As n is chosen larger, the intervals n·(t−ε, t+ε) expand in length. Hence if N is chosen large enough, then the intervals n·(t−ε, t+ε) overlap each other for consecutive integers greater than N, and thus their union over all n > N contains the open ray (Nt, ∞). (Indeed one can choose N so that 1/N < (t + ε)/(t − ε) − 1.) One now chooses T = Nt + 1.

(iii) Suppose that exp(sX) ∈ \overline{S} for s > 0. By (ii) there exists T > 0 such that exp(tY) ∈ S^{∘} for t ≥ T. Pick n ∈ ℕ such that ns > T. Then for t := ns,

exp(t(X + Y)) = exp(tX) exp(tY) ∈ \overline{S}S^{∘} ⊆ S^{∘},

so X + Y ∈ L_{∞}(S).

(iv) Suppose X ∈ L_{∞}(S) and exp(−tX) ∈ \overline{S} for some t > 0. By parts (i) and (iii), 0 ∈ L_{∞}(S). Thus S^{∘} contains some open neighborhood exp(B) of e, and hence is all of G, since G is connected, a contradiction.

(v) Suppose g, g^{−1} ∈ \overline{S}, and s := exp(tX) ∈ S^{∘} for t > 0. Pick an open symmetric neighborhood B of the identity e so small that BsB ⊆ S^{∘}. Then there exist b, d ∈ B such that gb, dg^{−1} ∈ S. Then

gsg^{−1} = gbb^{−1}sd^{−1}dg^{−1} ∈ SBsBS ⊆ SS^{∘}S ⊆ S^{∘}.

Thus exp t( Ad g(X)) = exp(Ad g(tX)) = g exp(tX)g^{−1} ∈ S^{∘}. It follows that Ad g(X) ∈ L_{∞}(S).

We record a lemma that will be useful for our later calculations.

Lemma 3.2. Let B be a neighborhood of 0 in sl(2, ℂ) and let 𝔞 be the onedimensional subspace of diagonal matrices in SL(2, ℝ). Then there exists m ∈ ℕ such that A^{m} = SL(2, ℂ), where A := exp 𝔞 ∪ exp B. Proof. Let U be an open set containing the identity e and contained in exp B. Then the maximal compact subgroup K := SU(2) is contained in U^{n} for some n, since these increasing open sets cover K. Let D = exp 𝔞 be the one-dimensional subgroup of diagonal matrices of determinant one with positive real entries on the diagonal. Then

SL(2, ℂ) = KDK ⊆ U^{n}DU^{n} ⊆ A^{2n+1},

where the first equality follows from the Cartan decomposition of a semisimple Lie group (see e.g. Theorem IX.1.1 of [4]), or can be deduced directly by writing any matrix in its polar decomposition as a product of a special unitary and a positive definite hermitian matrix, and then converting the latter to a diagonal matrix by an orthonormal change of basis.

In the next lemma we let 𝔤 be a semisimple Lie algebra containing a pointed generating invariant cone C.

We assume the notation and setting of Theorem 2.1. We thus have in the complexification 𝔤_{ℂ} of 𝔤 a positive system of roots Δ^{+} such that iα(Z) > 0 for all α ∈ Δ^{+} , where Z ∈ 𝔷(𝔨). For each non-compact positive root α ∈ Δ_{p}^{+} , we set

𝔤_{ℂ}(α) := 𝔤_{ℂ}^{α} + 𝔤_{ℂ}^{−α} + [𝔤_{ℂ}^{α}, 𝔤_{ℂ}^{−α}].

Note that 𝔤_{ℂ}(α) is isomorphic to sl(2, ℂ).

Let G_{ℂ} be a complex Lie group for 𝔤_{ℂ} for which complex conjugation integrates to G_{ℂ}, and let G be the connected component of the fixed point set. Then G is closed and has Lie subalgebra 𝔤 in 𝔤_{ℂ}. We henceforth consider only closed subsemigroups S of G_{ℂ} with non-empty interior S^{∘} and with the property that G ⊆ S ≠ G_{ℂ}.

Lemma 3.3. Suppose iZ ∈ L_{∞}(S). If for α ∈ Δ_{p}^{+} , there exists X ∈ 𝔱 such that iX ∈ L_{∞}(S) and iα(X) < 0, then 𝔤_{ℂ}(α) ∩ i𝔱 ⊆ \overline{L_{∞}(S) ∩ i𝔱}.

Proof. By hypothesis, we have that iZ ∈ L_{∞}(S)∩i𝔱, where Z is appropriately chosen as above in the center of 𝔨, and by choice of the positive system α(iZ) > 0. Thus there exists a positive number r such that iα(rZ + X) = 0. By Lemma 3.1(iv), Y := i(rZ + X) ≠ 0, and by Lemma 3.1(iii), Y ∈ L_{∞}(S). We can replace Y with a positive multiple such that exp(tY) ∈ S^{∘}; it remains true that α(tY) = 0. We again call this new element Y . Consider the subalgebra 𝔤_{ℂ}(α) = 𝔤_{ℂ}^{α} + 𝔤_{ℂ}^{−α} + [𝔤_{ℂ}^{α}, 𝔤_{ℂ}^{−α}]. Then 𝔤_{ℂ}(α) is isomorphic to sl(2, ℂ) and 𝔤^{[α]} := 𝔤_{ℂ}(α) ∩ 𝔤 is a real form isomorphic to sl(2, ℝ) (see e.g. Section 3.6 of [6] or Sections 7.1 and 7.2 of [7]). Since α(Y) = 0, we see directly that [𝔤_{ℂ}(α), Y] = 0. Thus exp Y commutes with exp U for all U ∈ 𝔤_{ℂ}(α). Pick an open set B containing 0 in 𝔤_{ℂ}(α) such that exp(B + Y) ⊆ S^{∘}. Set A := exp B ∪exp(𝔤^{[α]} . Since the subgroup generated by exp(𝔤_{ℂ}(α)) is a homomorphic image of the simply connected group SL(2, ℂ), it follows directly from Lemma 3.2 that H := ⟨exp 𝔤_{ℂ}(α)⟩ ⊆ A^{m} for some positive m. Let U ∈ 𝔤_{ℂ}(α). Then exp U ∈ A^{m}, so exp U = exp X_{1} ⋯ exp X_{m}, where for each i, X_{i} ∈ B or X_{i} ∈ 𝔤^{[α]} . Then

exp(mY + U) = exp(mY) exp U = exp Y exp X_{1} exp Y exp X_{2} ⋯ exp Y exp X_{m}
= exp(Y + X_{1}) ⋯ exp(Y + X_{m}).

If X_{i} ∈ B, then exp(Y + X_{i}) ∈ S^{∘} and if X_{i} ∈ 𝔤^{[α]} , then exp(Y + X_{i}) = exp Y · exp X_{i} ∈ S^{∘}S ⊆ S^{∘} since X_{i} ∈ 𝔤. Thus the product exp(mY + U) ∈ S^{∘}. Suppose further that U ∈ 𝔤_{ℂ}(α)∩i𝔱. From the preceding paragraph, mY +nU ∈ L_{∞}(S) ∩ i𝔱 for each n, and (1/n)(mY + nU) converges to U. Hence U is in the closure of L_{∞}(S) ∩ i𝔱.

We specialize now to the case that 𝔤 is hermitian simple. As noted above, these are precisely the simple Lie algebras which contain an invariant cone.

Lemma 3.4. Let 𝔱 be a compact Cartan subalgebra of 𝔤. If L_{∞}(S) ∩ i𝔱 ≠ Ø, then either ±iZ ∈ L_{∞}(S), where Z spans the center of the maximal compact subalgebra 𝔨 of 𝔤 containing 𝔱.

Proof. Pick X ∈ 𝔱 such that iX ∈ L_{∞}(S). Consider the element Y := ∑_{γ∈W_{𝔨}} γ(iX). Then Y ∈ 𝔱 and Y ∈ L_{∞}(S) by (iii) and (v) of Lemma 3.1 since K ⊆ S, where L(K) = 𝔨. It then follows from 3.1(iv) that Y ≠ 0. Since Y is fixed by the action of the Weyl group W_{𝔨}, it must be in the center of 𝔨. Since the center of 𝔨 is one-dimensional, the result follows from 3.1(i).

Remark 3.5. By interchanging Z and −Z and the positive and negative roots if necessary, we can in light of Lemma 3.4 assume that iZ ∈ L_{∞}(S). We henceforth assume this to be the case.

Theorem 3.6. Let 𝔤 be a hermitian simple Lie algebra, let G_{ℂ} be a complex Lie group for 𝔤_{ℂ} for which complex conjugation integrates to G_{ℂ}, and let G be the connected component of the fixed point set.

Furthermore assume that 𝔱 is a compact Cartan subalgebra of 𝔤, 𝔨 is the unique maximal compact subalgebra containg 𝔱, Z ∈ 𝔱 spans the one-dimensional center of 𝔨, and Δ^{+} is a 𝔨-adapted system of positive roots satisfying α(iZ) > 0 for each α ∈ Δ_{p}^{+} .

If S is a subsemigroup of G_{ℂ} containing G and having non-empty interior, then either L_{∞}(S) ∩ i𝔱 ⊆ 𝔠_{max} or L_{∞}(S) ∩ i𝔱 ⊆ −𝔠_{max} . Proof. We assume that L_{∞}(S) ∩ i𝔱 ≠ Ø, otherwise there is nothing to prove. By Lemma 3.4 and Remark 3.5, we may assume without loss of generality that iZ ∈ L_{∞}(S). In this case we show L_{∞}(S) ∩ i𝔱 ⊆ 𝔠_{max} . Indeed, if this is not the case, then there exists X ∈ 𝔱 such that iX ∈ L_{∞}(S) and iα(X) < 0 for some α ∈ Δ_{p}^{+} , by the definition of 𝔠_{max} .

Let 0 ≠ X_{α} ∈ 𝔤_{ℂ}^{α}. Then it is standard that \overline{X}_{α} ∈ 𝔤_{ℂ}^{−α} (see e.g., Theorem 7.4 of [7]), and thus −i[\overline{X}_{α}, X_{α}] ∈ 𝔤_{ℂ}(α) ∩ −𝔠_{min} , by the definitions of 𝔤_{ℂ}(α) and 𝔠_{min} . Since i𝔤_{ℂ}(α) = 𝔤_{ℂ}(α), We have

Y := [\overline{X}_{α}, X_{α}] = i(−i)[\overline{X}_{α}, X_{α}] ∈ 𝔤_{ℂ}(α) ∩ −i𝔠_{min} ⊆ 𝔤_{ℂ}(α) ∩ i𝔱.

It follows from Lemma 3.3 that Y ∈ \overline{L_{∞}(S) ∩ i𝔱}.

Consider U := ∑_{γ∈W_{𝔨}} γ(Y). Since Y ∈ −i𝔠_{min} , which is invariant under the action of the Weyl group W_{𝔨}, we conclude that 0 ≠ U ∈ −i𝔠_{min} . Since U is invariant under the action of W_{𝔨} and is in −i𝔠_{min} , we conclude that U is some negative multiple of iZ. If Y_{n} → Y , where Y_{n} ∈ i𝔱 ∩ L_{∞}(S), then U_{n} := ∑_{γ∈W_{𝔨}} γ(Y_{n}) is in L_{∞}(S) from parts (iii) and (v) of Lemma 3.1 and the sequence converges to U. Then −U_{n} converges to −U ∈ (ℝ^{+} \ {0})(iZ) ⊆ L_{∞}(S). It follows (Lemma 3.1(i)) that −U_{n} ∈ L_{∞}(S) for large n, and hence 0 = −U_{n} + U_{n} ∈ L_{∞}(S) by Lemma 3.1(iii), but a contradiction to Lemma 3.1(iv).

4. Maximal Subsemigroups

In this section we work in the following setting. Let 𝔤 be a hermitian simple Lie algebra, let 𝔤_{ℂ} be its complexification, let G_{ℂ} be a corresponding complex Lie group for which complex conjugation integrates to σ: G_{ℂ} → G_{ℂ}, and let G be the identity component of the fixed point set. Then G is a closed connected Lie subgroup with Lie algebra 𝔤. Let g^{∗} := σ(g)^{−1} denote the adjoint involution, the involutive antiautomorphism arising from σ.

We further assume that S is a subsemigroup of G_{ℂ} which contains G such that G is not isolated in S, i.e., (S∩GU)\G ≠ Ø for all open neighborhoods U of the identity. Our goal in this section and the main goal of the paper to establish the following:

Theorem 4.1. A subsemigroup S ≠ G_{ℂ} of G_{ℂ} containing G such that G is not isolated in S contains a minimal Ol’shanskiĭ semigroup and is contained in the corresponding maximal Ol’shanskiĭ semigroup. In particular, a maximal Ol’shanskiĭ semigroup is maximal in the class of all proper subsemigroups.

After the work of Section 3, the proof requires only minor modification of the proof of Hilgert and Neeb of the maximality of the maximal Ol’shanskiĭ semigroup. We recall the following proposition from their work which, although it is non-trivial, can be derived rather quickly using fairly standard Lie theory (see Proposition 8.48 of [7] and the preceding Proposition 8.32 and Lemma 8.47).

Proposition 4.2. Let 𝔤 be a simple hermitian Lie algebra, let 𝔱 ⊆ 𝔤 be a compact Cartan subalgebra contained in the maximal compact subalgebra 𝔨, let Δ^{+} be a 𝔨-adapted system of positive roots, and let 𝔠_{max} be the corresponding maximal cone in 𝔱. If S ⊆ G_{ℂ} is a closed subsemigroup which contains G and has dense interior and if

S ∩ exp(i𝔱) ⊆ exp(i𝔠_{max} ∪ −i𝔠_{max}),

then

S ⊆ \overline{GN_{G_{ℂ}}(𝔱)G},

where N_{G_{ℂ}}(𝔱) is the normalizer of 𝔱 in G_{ℂ}.

Proof of Theorem 4.1. Let S be a subsemigroup of G_{ℂ} containing G such that G is not isolated in S. For a countable base U_{n} at e, pick t_{n} = g_{n}u_{n} ∈ S \ G, where g_{n} ∈ G and u_{n} ∈ U_{n}. Then the sequence s_{n} := g_{n}^{−1}t_{n} is a sequence in S \ G converging to e.

Step 1: S contains a minimal Ol’shanskiĭ semigroup. For large n, we may write s_{n} = (exp X_{n})(exp iY_{n}), where X_{n}, Y_{n} ∈ 𝔤 converge to 0 (see, e.g., Lemma IV.7.10 of [6]). Then t_{n} := exp(−X_{n})s_{n} = exp(iY_{n}) is also a sequence in S converging to e, since G ⊆ S. So the tangent Lie wedge L(S) must meet i𝔤 non-trivially, since any limit point of iY_{n}/∥Y_{n}∥ will be in L(S). Since i𝔤 and L(S) are invariant under Ad G, their intersection is an invariant cone in i𝔤, and hence must contain a minimal invariant cone iC. Since S is closed, it is standard that S contains exp(L(S)), and hence contains the minimal Ol’shanskiĭ semigroup exp(iC)G.

Step 2: S is contained in the closure of GN_{G_{ℂ}}(𝔱)G. Since the minimal Ol’shanskiĭ semigroup has the identity e in the closure of its interior, so does S, and hence S has dense interior. We choose a compact Cartan subalgebra 𝔱 of 𝔤, let 𝔨 be the corresponding maximal compactly embedded subalgebra, and choose a 𝔨-adapted positive root system Δ^{+} so that iC ∩ i𝔱 = i𝔠_{min} ⊆ exp^{−1}(S) ∩ i𝔱. We deduce from Theorem 3.6 that the hypotheses of Proposition 4.2 are satisfied, and we thus conclude

S ⊆ \overline{GN_{G_{ℂ}}(𝔱)G}.

Before we proceed to Step 3, we need some additional notation. Let 𝔠_{max} be the corresponding maximal cone in 𝔱 for the positive root system Δ^{+} just introduced. Let W denote the big Weyl group of i𝔱 with respect to 𝔤_{ℂ} (i.e., the normalizer of i𝔱 in Ad(G_{ℂ}) modulo the centralizer of i𝔱 in Ad(G_{ℂ})), and let W_{𝔨} denote those members of W which have representatives of the form Ad(g), where g ∈ K = exp 𝔨. Note this agrees with our previous definition of W_{𝔨}, except now we consider the action on i𝔱 instead of 𝔱.

Step 3: If γ ∈ W and γ·i𝔠_{min} ⊆ i𝔠_{max} , then γ ∈ W_{𝔨}. Pick iX ∈ i𝔠_{min} such that the stabilizer of iX in W is trivial (this is possible since i𝔠_{min} has interior in i𝔱). Then γ·iX ∈ i𝔠_{max} , and from the definition of 𝔠_{max} , α(γ·iX) = iα(γ·X) ≥ 0 for all α ∈ Δ_{p}^{+} . We can rotate γ·iX to a Δ_{k}^{+} -positive chamber by an element of W_{𝔨}, i.e., there exists γ′ ∈ W_{𝔨} such that α(γ′γ·iX) ≥ 0 for each α ∈ Δ_{k}^{+} . Now since Δ^{+} is 𝔨-adapted, it is still the case that α(γ′γ·iX) ≥ 0 for each α ∈ Δ_{p}^{+} . Similarly since iX ∈ i𝔠_{min} ⊆ i𝔠_{max} , we can find γ′′ ∈ W_{𝔨} such that α(γ′′·iX) ≥ 0 for all α ∈ Δ^{+} . Since each closed Weyl chamber is a fundamental domain for W (e.g., [1, Ch. V, §3, no. 3.3, Thm. 2]), it follows that γ′γ·iX = γ′′·iX, and hence that γ′γ = γ′′ from the triviality of the stabilizer of iX. Thus γ = (γ′)^{−1}γ′′ ∈ W_{𝔨}.

Step 4: S^{∘}∩N_{G_{ℂ}}(𝔱) ⊆ S_{max} , where S_{max} is the unique maximal Ol’shanskiĭ semigroup containing the minimal one appearing in step 1. Let s ∈ S^{∘} ∩ N_{G_{ℂ}}(𝔱). Let γ := Ad(s)|_{i𝔱} be the member of the Weyl group corresponding to s. It follows from the finiteness of the Weyl group W that there exists n ∈ ℕ such that γ^{n} = Ad(s^{n})|_{i𝔱} = id_{i𝔱}. Then

exp(γ·i𝔠_{min})s^{n} = Ad(s)(exp(i𝔠_{min}))s^{n} = s exp(i𝔠_{min})s^{n−1} ⊆ S^{∘}SS^{∘} ⊆ S^{∘},

since exp(i𝔠_{min}) ⊆ S from step 1.

On the other hand, it is standard that s^{n} ∈ Z_{G_{ℂ}}(𝔱) = exp 𝔱_{ℂ}, so that there exists t ∈ T := exp 𝔱 and Y ∈ i𝔱 such that s^{n} = t exp(Y). Let A := exp(i𝔱). It follows that

exp(γ·i𝔠_{min})t exp(Y) = t exp(γ·i𝔠_{min} + Y) ⊆ S^{∘} ∩ (T A) = T (S^{∘} ∩ A) ⊆ T exp(i𝔠_{max}).

The last equality follows from T ⊆ G ⊆ S and the last containment follows from Theorem 3.6.

It now follows from the uniqueness of the polar decomposition G_{ℂ} = K′ exp(𝔭 + i𝔨), where K′ = exp(𝔨 + i𝔭), that γ·i𝔠_{min} + Y ⊆ i𝔠_{max} . For each X ∈ 𝔠_{min} , γ ·iX +(1/n)Y = \frac{1}{n}(nγ ·iX +Y) ∈ i𝔠_{max} . Thus γ ·iX ∈ i𝔠_{max} . Hence γ · i𝔠_{min} ⊆ i𝔠_{max} , and thus from step 3, γ ∈ W_{𝔨}. Consequently γ = Ad(k)|_{i𝔱} for some k ∈ N_{K}(𝔱). Now Ad(k^{−1}s) restricted to i𝔱 is the identity, i.e., k^{−1}s ∈ Z_{G_{ℂ}}(i𝔱) = exp 𝔱_{ℂ}. Thus s ∈ N_{K}(i𝔱)(exp 𝔱_{ℂ}) ∩ S. Since N_{K}(i𝔱) ⊆ K ⊆ G ⊆ S and exp 𝔱_{ℂ} = T A, we have

N_{K}(i𝔱)(T A) ∩ S = N_{K}(i𝔱)((T A) ∩ S) = N_{K}(i𝔱)T (S ∩ A)
⊆ N_{K}(i𝔱)T exp(i𝔠_{max}) ⊆ G(iC_{max}) = S_{max},

and thus s ∈ S_{max} .

Step 5: S ⊆ S_{max} , the appropriate maximal Ol’shanskiĭ semigroup. Since S^{∘} is dense in S and S_{max} is closed, it suffices to show S^{∘} ⊆ S_{max} . Let s ∈ S^{∘}, and let B be any open set containing s such that B ⊆ S^{∘}. We have already seen that S is contained in the closure of GN_{G_{ℂ}}(𝔱)G, so there exists x ∈ B ∩ GN_{G_{ℂ}}(𝔱)G. Pick g_{1}, g_{2} ∈ G and y ∈ N_{G_{ℂ}}(𝔱) such that x = g_{1}yg_{2} . Then y = g_{1}^{−1}xg_{2}^{−1} ∈ GS^{∘}G ⊆ S^{∘}. It follows from step 4 that y ∈ S_{max} . Since also G ⊆ S_{max} , we conclude that x = g_{1}yg_{2} ∈ S_{max} . Since B was arbitrary containing s, we conclude s ∈ \overline{S}_{max} = S_{max} . This concludes the proof.

The last assertion follows by beginning with S_{max} , supposing it is contained in some proper subsemigroup S, and applying the first part of the theorem to conclude that S ⊆ S_{max} .

Corollary 4.3. Let G_{ℂ} satisfy the hypotheses at the beginning of this section and let S ≠ G_{ℂ} be a closed subsemigroup which contains G and exp(i𝔠_{max}). Then S is the maximal Ol’shanskiĭ semigroup corresponding to 𝔠_{max} .

Proof. Since S contains exp(i𝔠_{max}), the hypotheses of Theorem 4.1 are satisfied. Hence S ⊆ S_{max} , the maximal Ol’shanskiĭ semigroup corresponding to 𝔠_{max} . On the other hand, L(S) ∩ i𝔤 is a G-invariant wedge in i𝔤 containing i𝔠_{max} , and hence must be the maximal invariant cone iC_{max} (see Section 2). Thus S_{max} = exp(iC_{max})G ⊆ (exp L(S))G ⊆ SS = S.

5. Applications and Reflections

We state first of all what is essentially a reformulation of the major result, Theorem 4.1.

Theorem 5.1. Let G be a complex Lie group with complex Lie algebra 𝔤, and let 𝔤_{0} ⊆ 𝔤 be a simple algebra which is a real form of 𝔤. If S ≠ G is a closed subsemigroup of G such that S contains the connected subgroup G_{0} with Lie algebra 𝔤_{0} and G_{0} is not isolated in S, then 𝔤_{0} is a hermitian Lie algebra, and there exist unique minimal and maximal Ol’shanskiĭ semigroups S_{min} and S_{max} each with group of units G_{0} such that S_{min} ⊆ S ⊆ S_{max} .

Proof. We first consider the G_{0} -invariant Lie wedge L(S). It follows directly that iC := L(S) ∩ i𝔤_{0} is invariant and proper in i𝔤 (since S ≠ G). Thus C is an invariant wedge in 𝔤_{0} , and hence is a pointed generating cone since 𝔤_{0} is simple. Thus 𝔤_{0} is hermitian. One is now in a position to apply Theorem 4.1.

Remark 5.2. The Theorems 4.1 and 5.1 give general conditions for concluding that a semigroup is a subsemigroup of an Ol’shanskiĭ semigroup. Once one obtains this conclusion, then one has the Ol’shanskiĭ polar factorization in S, and each of the factors will also belong to S since S contains G_{0} . The availabilty of this factorization can be a useful theoretical tool. For example, one easily derives the polar decompositions in contraction semigroups found in the work of Brunet and Kramer ([3] and [2]) by this means.

Remark 5.3. It is a consistent feature of Lie theory to try to reduce the study of various problems and objects to the study of Cartan subalgebras. We remark that this is the case for those semigroups arising in Theorems 4.1 and 5.1. Indeed if S is such a closed semigroup, then S = \overline{G(S ∩ i𝔱)G}, where 𝔱 is any compact Cartan subalgebra of the Lie algebra of the group of units of S. This follows directly from the facts that S has dense interior, that the interior of the maximal Ol’shanskiĭ semigroup is of the form G exp(iC^{∘})G, and that G ⊆ S ⊆ S_{max} . Hence determining S reduces to determining S ∩ i𝔱. The latter is generally a much more manageable problem. It might be an interesting problem to try to determine precisely which sets between i𝔠_{min} and i𝔠_{max} are traces of such semigroups.

Remark 5.4. Ol’shanskiĭ [13] has shown that that the compression semigroups of simple hermitian symmetric spaces are maximal Ol’shanskiĭ semigroups. The difficult direction in his proof is showing that the compression semigroup is contained in the maximal Ol’shanskiĭ semigroup, but this can be easily deduced using Corollary 4.3. Similar observations apply to the compression semigroup of complex linear transformations of determinant 1 preserving the set of positive (or negative) elements of a pseudohermitian form.

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J. D. Lawson Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803 USA

LAWSON@MARAIS.MATH.LSU.EDU

Received June 30, 1994