Seminar Sophus Lie 1 (1991) 143–151

Convexity theorems in Harmonic Analysis

Karl-Hermann Neeb

I. Kostant’s Convexity Theorem

Let G be a connected semisimple Lie group, 𝔤 = 𝔏(G) its Lie algebra, and B(X, Y) = tr(ad X ad Y) its Cartan Killing form. Then a Cartan involution θ is an involutive automorphism of 𝔤 such that the bilinear form B_{θ}(X, Y) := −B(X, θY) is positive definite. It is well know that Cartan involutions always exist ([6, p.185]). Fix such an involution θ. Then we set

𝔨 := {X ∈ 𝔤 : θ(X) = X} and 𝔭 := {X ∈ 𝔤 : θ(X) = −X}.

Then 𝔨 is a compact subalgebra of 𝔤 and all operators ad X, X ∈ 𝔭 are semisimple with real spectrum because they are selfadjoint with respect to B_{θ} ([6, p.184]). We choose a maximal abelian subspace 𝔞 ⊆ 𝔭. Since all Cartan involutions on 𝔤 are conjugate under inner automorphisms ([6, p.183]) and all maximal abelian subspaces of 𝔭 are conjugate under e^{ad 𝔨} ([6, p.247]), we don’t loose any generality by making this choice. For a linear functional α ∈ 𝔞^{∗} we set 𝔤^{α} := {Y ∈ 𝔤 : (∀X ∈ 𝔞)[X, Y] = α(X)Y }. Since the operators ad X, X ∈ 𝔞 commute and are semisimple, we have a direct vector space decomposition

𝔤 = ⨁_{α∈Δ} 𝔤^{α},

where Δ := {α ∈ 𝔞^{∗} : 𝔤^{α} ≠ {0}} is called the system of restricted roots with respect to 𝔞. Let X_{0} ∈ 𝔞 be a regular element, i.e. an element of 𝔞 such that α(X_{0}) ≠ 0 for all non-zero elements of Δ (such an element exists because we only have to avoid a finite union of hyperplanes). A positive system Δ^{+} ⊆ Δ is a subset which is given by

Δ^{+} = {α ∈ Δ : α(X_{0}) > 0},

where X_{0} is a regular element. We fix a positive system Δ^{+} ⊆ Δ and define

𝔫 := ⨁_{α∈Δ^{+}} 𝔤^{α}.

This is a nilpotent subalgebra of 𝔤 and we have the direct vector space decomposition

𝔤 = 𝔨 + 𝔞 + 𝔫

called the Iwasawa decomposition of 𝔤.

On the group level we have an involution, which we also denote with θ, whose differential agrees with the corresponding involution on the Lie algebra and which has the following properties:

1) K := ⟨exp 𝔨⟩ agrees with the fixed point set of θ.

2) The mapping K × exp 𝔭 → G, (k, X) ↦ k exp(X) is a diffeomorphism ([6, p.253]). Now we set A := exp 𝔞 and N := exp 𝔫. These are closed simply

connected subgroups of G and the mapping

K × A × N → G, (k, a, n) ↦ kan

is a diffeomorphism ([6, p.270]). This is called an Iwasawa decomposition of the group. These decompositions are very useful in harmonic analysis on the Riemannian symmetric space G/K. They provide well behaved parametrizations of the group which can be used to compute integrals or at least to obtain estimations for certain integrals (see Faraut’s article in this volume). Clearly G = KAN is not a direct product decomposition of a group and therefore one has to analyse the way the pieces are built together. One theorem which gives some information in this direction is Kostant’s Convexity Theorem.

In the following we consider 𝔞 as an euclidean vector space, where the scalar product comes from the restriction of the positive definite form B_{θ} to 𝔞 × 𝔞. Since 𝔞 ⊆ 𝔭, this is the same as the restriction of the Cartan Killing form to 𝔞. For a root α ∈ Δ we write s_{α} for the orthogonal reflection on the hyperplane ker α in 𝔞. Then the Weyl group W = W(Δ) is the group generated by the reflections s_{α}, α ∈ Δ. It may be identified with the quotient

N_{K}(A)/Z_{K}(A),

where

N_{K}(A) = {k ∈ K : kAk^{−1} ⊆ A} and Z_{K}(A) = {k ∈ K : (∀a ∈ A)kak^{−1} = a}

([6, p.289]). Let us write L : KAN → 𝔞 for the analytic mapping determined by

g ∈ K exp L(g)N.

Now we have all definitions to state Kostant’s theorem:

Theorem 1. (Kostant’s Convexity Theorem) Let a ∈ A. Then the set L(aK) equals the convex hull of the Weyl group orbit W log a.

In the following we describe how this theorem can be generalized from Riemannian symmetric spaces to symmetric spaces of regular type, and which methods are used to prove this theorem.

II. Symmetric spaces of regular type

Let G be a connected semisimple Lie group, and τ an involutive endomorphism of G. We write H for the 1-component of the group G^{τ} of fixed points of τ in G. Note that this implies that M := G/H is a symmetric space. Let us use the same letter for the corresponding involution of the Lie algebra 𝔤 = 𝔏(G). We choose a Cartan involution θ such that τ and θ commute ([9, p. 153]). We set

𝔥 := {X ∈ 𝔤 : τ(X) = X} and 𝔮 := {X ∈ 𝔤 : τ(X) = −X}.

Then the compatibility of τ and θ show that we have a direct vector space decomposition

𝔤 = 𝔥_{𝔨} + 𝔥_{𝔭} + 𝔮_{𝔨} + 𝔮_{𝔭},

where 𝔥_{𝔨} := 𝔥 ∩ 𝔨, 𝔥_{𝔭} := 𝔥 ∩ 𝔭 etc., and all summands are invariant under τ and θ. It is convenient to assume that 𝔥 contains no non-trivial ideals of 𝔤, i.e. that Z_{𝔥}(𝔮) = {0}, and that G/H is an irreducible symmetric space. For all our main results formulated below an easy reduction shows that they remain true without this additional hypothesis ([10, I]).

Definition 2. Write 𝔥^{a} := 𝔥_{𝔨} + 𝔮_{𝔭} for the subalgebra of τθ-fixed points in 𝔤. Then it is shown in [8, Chapert 3] that

\begin{aligned}Z(𝔥^{a}) ∩ 𝔮 & = 𝔠 := \{X ∈ 𝔮_{𝔭} : [X, 𝔮_{𝔭}] = \{0\}\} \\ & = \{X ∈ 𝔮_{𝔭} : [X, 𝔥_{𝔨}] = \{0\}\}. \end{aligned}

Let us write Z_{𝔮}(𝔠) for the centralizer of 𝔠 in 𝔮. Then we say that the symmetric space G/H or equivalently the symmetric Lie algebra (𝔤, τ) is of regular type if the condition

Z_{𝔮}(𝔠) = 𝔮_{𝔭}

is satisfied.

It is explained in [8, Chapter 4] that this condition is related to the existence of certain H -invariant cones in 𝔮. As it stands it is not very instructive. But let us remark that it is satisfied by two essential classes of symmetric spaces.

Remark 3. If G/H is a Riemannian symmetric space, i.e. if τ is a Cartan involution, then 𝔥 = 𝔨 and 𝔮 = 𝔭. Therefore 𝔠 = {0} because [X, 𝔭] = {0} implies that

B(X, X) ∈ B(X, 𝔭) = B(X, [𝔭, 𝔨]) = B([X, 𝔭], 𝔨) = {0}.

So Z_{𝔮}(𝔠) = 𝔭 = 𝔮_{𝔭}.

Remark 4. Let 𝔤 be a simple hermitean Lie algebra and 𝔤_{ℂ} its complexification. We define τ(X + iY) := X − iY . Then (𝔤_{ℂ}, τ) is of regular type. To see this, let 𝔤 = 𝔨 + 𝔭 be a Cartan decomposition of 𝔤. Then 𝔤_{ℂ} = (𝔨 + i𝔭) + (i𝔨 + 𝔭) is a Cartan decomposition of 𝔤_{ℂ} and therefore 𝔮_{𝔭} = i𝔨. So 𝔠 = iZ(𝔨) and the claim follows from the fact that 𝔨 is the centralizer of its center ([6, p.382]).

Lemma 5. Let 𝔞 ⊆ 𝔮_{𝔭} be a maximal abelian subspace. Then 𝔞 is maximal abelian in 𝔮 and in 𝔭. Moreover Z_{𝔤}(𝔠) = 𝔥_{𝔨} + 𝔮_{𝔭}.

Proof. Since 𝔞 contains Z(𝔮_{𝔭}), the regularity condition implies that 𝔞 is maximal abelian in 𝔮.

Write 𝔤^{c} := 𝔥 + i𝔮 for the dual Lie algebra. Then 𝔨^{c} := 𝔥_{𝔨} + i𝔮_{𝔭} contains i𝔠 in its center and we have to show that Z_{𝔤^{c}}(i𝔠) = 𝔨^{c}. This is proved in [11, pp.130,131]. Note that the condition that 𝔥 contains no non-trivial ideal of 𝔤 is used in the proof.

Definition 6. As explained in Section I we get a root decomposition of 𝔤 with respect to 𝔞. We write

Δ_{p} := {α ∈ Δ : α(𝔠) ≠ {0}} and Δ_{k} := {α ∈ Δ : α(𝔠) = {0}}.

The elements of Δ_{k} are called the compact roots and the elements of Δ_{p} the non-compact roots. Note that a root α is compact if and only if 𝔤^{α} ⊆ Z_{𝔤}(𝔠) = 𝔥^{a} = 𝔥_{𝔨} + 𝔮_{𝔭} (Lemma 5). So Δ_{k} is the system of restricted roots with respect to the subalgebra 𝔥^{a}.

We choose a positive system corresponding to a regular element X_{0} with α(X_{0}) < β(X_{0}) for all β ∈ Δ_{p}^{+} and α ∈ Δ_{k}^{+} (One has to choose X_{0} near to a non-zero element of 𝔠). We write W := W(Δ_{k}^{+}) for the Weyl group generated by the reflections s_{α} with α ∈ Δ_{k}^{+} and set

C_{max} := {X ∈ 𝔞 : (∀α ∈ Δ_{p}^{+})α(X) ≥ 0}, C_{min} = C_{max}^{⋆},

and

C_{k} := {X ∈ 𝔞 : (∀α ∈ Δ_{k}^{+})α(X) ≥ 0}.

Then C_{k} ∩ C_{max} is the positive Weyl chamber of the root system and C_{k} is the Weyl chamber with respect to Δ_{k}^{+} . The subalgebra 𝔫, and the groups A and N are defined as in Section I.

Theorem 7. (Generalized Iwasawa decomposition)

1) 𝔤 = 𝔥 + 𝔞 + 𝔫 is a direct vetor space sum.

2) The mapping

G^{τ} × A × N → G, (h, a, n) ↦ han

is a diffeomorphism onto the open subset G^{τ} AN of G.

Proof. 1) Since 𝔞 is maximal abelian in 𝔭 and 𝔮, it follows that 𝔤^{0} ⊆ 𝔞 + 𝔥_{𝔨}. If X ∈ 𝔤^{−α}, where α ∈ Δ^{+} , then X = (X + τ(X)) − τ(X) ∈ 𝔥 + 𝔫. So the root decomposition shows that 𝔤 = 𝔥 + 𝔞 + 𝔫. To see that the decomposition is direct, let X = Y + Z ∈ 𝔥 ∩ (𝔞 + 𝔫) with Y ∈ 𝔞 and Z ∈ 𝔫. Then τ(X) = X = Y + Z = −Y + τ(Z), i.e. 2Y + Z − τ(Z) = 0. Now the directness of the root decomposition implies that Z = 0 and Y = 0, because τ(Z) ∈ ⨁_{α∈Δ^{+}} 𝔤^{−α}.

2) In view of 1) and the fact that G^{τ} and AN are subgroups of G, one only has to show that G^{τ} ∩ AN = {1}. But this follows as in 1) from τ(N) ∩ AN = {1} (Bruhat decomposition) ([6, p.406]).

The main deficiency of the generalized Iwasawa decomposition is that G^{τ} AN is in general different from the whole group (see Hilgert’s article for an example). Since G^{τ} is in general not connected, the same holds for the set G^{τ} AN and so we restrict our considerations to the connected component HAN. As in Section I we consider the analytic mapping:

L : HAN → 𝔞, g ↦ L(g)

which is determined by g ∈ H exp L(g)N.

The set L(aH) is only well defined if aH ⊆ HAN. The following Lemma tells us when this is true.

Lemma 8. The condition aH ⊆ HAN is equivalent to a ∈ exp(C_{max}).

Proof. The proof can be found in [10]. It bases on the fact that

\overline{H exp(C_{max})H} = \{g ∈ G : gHAN ⊆ HAN\}

which, more or less in this formulation, is proved in [8]. See also the discussion in Section 2 of [3]

The preceding lemma describes those candidates in A for which the set L(aH) is well defined. This explains the restriction in the assumptions of the following theorem.

Theorem 9. (The Convexity Theorem) Let a ∈ A with 0 ≠ log a ∈ C_{max} . Then

L(aH) = co(log a) + C_{min},

where co(log a) denotes the closed convex hull of the W -orbit of log a.

Remark 10. If τ = θ, then Δ_{k} = Δ and H = K. Therefore C_{min} = {0} and Theorem 9 reduces to Theorem 1.

In the remainder we explain the methods which are used in the proof of the convexity theorem. The details can be found in [10]

One starts with the invariance properties of the set L(aH).

Proposition 11. co(Y) ⊆ L(aH) for every Y ∈ L(aH).

Proof. (Sketch) The essential idea is to use rank-1-reduction. In view of Lemma 10.4 in [7] one only has to show that the line segments {s_{α}(Y), Y } are contained in L(aH) for every element Y ∈ L(aH) and every reflection s_{α}, α ∈ Δ_{k}^{+} . If this holds for the reflections associated to the elements α of a basis of the root system, then we conclude that L(aH) is invariant under the Weyl group, and the general assertion follows from the fact that every root is conjugate to a root in the basis. If α is a root in the basis, then N decomposes as a semidirect product N ≅ N^{α}N′, where N^{α} = exp 𝔤^{α} is contained in the subgroup H^{a} := ⟨exp 𝔥^{a}⟩. This paves the way to rank-1-reduction. For the details we refer to [10] or [7, pp.476,477].

The next step is a description of the sets co(Y) + C_{min} in a more convenient form.

Proposition 12. Let E be a euclidean vector space, H a finite system of hyperplanes in E, W the group generated by the reflections on the elements of H, and suppose that W is finite and leaves H invariant. If C is a chamber of H, Y ∈ \overline{C}, and C′ a W -invariant closed convex cone in E, then

co(Y) + C′ = co(Y + C′) = ⋂_{s∈W} s(X + C′ − C^{⋆}).

Proof. ([10, I.12]) The proof is rather elementary, one only has to deal with basic properties of finite groups generated by reflections ([1, Ch. V]) and the finite dimensional version of the Hahn-Banach Separation Theorem. The only difficulties come from the non-compactness of the set C′, but they can be overcome with the crucial observation that C′ − C^{⋆} is always closed in this situation.

Let us keep the notation from Proposition 12 for a moment.

Corollary 13. If X_{0} ∈ \overline{C} and C′ := ∑_{s∈W} ℝ^{+}s(X_{0}), then

co(Y) + C′ = co(Y + ℝ^{+}X_{0}) = ⋂_{s∈W} s(X + ℝ^{+}X_{0} − C^{⋆}).

Proof. One only has to show that the cones C′ − C^{⋆} and ℝ^{+}X_{0} − C^{⋆} are equal. But this follows from the fact that X_{0} − s_{α}(X_{0}) ∈ C^{⋆} ([10, I.3]).

Now we can already proof one inclusion of the convexity theorem. Let X := log a and assume that X ≠ 0 (otherwise there is nothing to prove). For a compact root α we choose an element X_{α} orthogonal to ker α such that α(X_{α}) = 1. If Y_{α} is a non-zero vector in 𝔤^{α}, then span{Y_{α}, θ(Y_{α}), X_{α}} is a subalgebra isomorphic to 𝔰𝔩(2, ℝ). After conjugation with W we may assume that X_{α} ∈ \overline{C_{k}} which implies that α(X) > 0. An explicit calculation in Sl(2, ℝ) (see Hilgert’s article) now shows that L(aH) contains the ray X + ℝ^{+}X_{α}. Therefore

co(X + ℝ^{+}X_{α}) = co(X) + C_{min} ⊆ L(aH)

because C_{min} = ∑_{s∈W} ℝ^{+}s(X_{α}) ([10]).

The other inclusion of the theorem is harder to prove. It bases on Harish Chandra’s construction of the holomorphic discrete series for a linear hermitean Lie group. The first observation is that, in view of Propositions 11 and 12, it suffices to show that

L(aH) ⊆ log a + C_{min} − C_{k}^{⋆}.

But −C_{min} + C_{k}^{⋆} = (−C_{max} ∩ C_{k})^{⋆} because it is closed, as we have already remarked above. So it would be sufficient to prove that

λ(L(ah) − log a) ≤ 0

for all functionals in a set P ⊆ (−C_{max}) ∩ C_{k} which has the property that ℝ^{+}P is dense. It is shown in [10, II.1] that the following set satisfies this condition. Let E ⊆ 𝔞 denote the abelian group generated by the vectors corresponding to the roots under the identification 𝔞 ≅ 𝔞^{∗}. As P we take the subset of all those elements λ in E which correspond to dominant integral weights of the subalgebra 𝔥_{𝔨} + 𝔮_{𝔭} and which satisfy the additional condition that

⟨λ + ρ, β⟩ < 0 ∀β ∈ Δ_{p}^{+},

where ρ = \frac{1}{2} ∑_{α∈Δ^{+}} α.

We collect the essential facts in the following theorem:

Theorem 14. Let P be as defined above, suppose that G is contained in a complex Lie group G_{ℂ} with 𝔏(G_{ℂ}) = 𝔤_{ℂ}, and set G^{c} := ⟨exp(𝔥 + i𝔮)⟩. Then the following assertions hold:

1) For every λ ∈ P there exists an irreducible unitary representation π_{λ} : G^{c} → U(H) on a Hilbert space H which has the following properties:

a) Let K^{c} := ⟨exp(𝔥_{𝔨} + i𝔮_{𝔭})⟩. Then the subspace H^{K} of K^{c} - finite vectors consists of analytic vectors and therefore carries the structure of a 𝔤_{ℂ} -module. As 𝔤_{ℂ} -module it is a module with highest weight λ.

b) For α ∈ 𝔞^{∗} we set H^{α} := {v ∈ H : (∀X ∈ 𝔞)dπ_{λ}(X)v = α(X)v} and call α a weight if H^{α} ≠ {0}. Then all weights are contained in the set

W(λ − C_{min}).

2) Let S_{min} denote the subsemigroup of G generated by H and exp(C_{min}). Then S_{min} = \overline{H exp(C_{min})H} and the restriction π_{λ}|_{H} permits a continuation to a representation

π̃_{λ} : S_{min} → C(H),

where C(H) denotes the semigroup of contractions on H. This representation is continuous with respect to the weak operator topology on C(H) and analytic on the interior of S_{min} .

3) The restriction π̃_{λ}|_{exp C_{min}} permits a continuation to a representation π̂_{λ} of exp(C_{max}) by bounded operators on H such that

||π̂_{λ}(exp X)|| = e^{λ(X)} ∀X ∈ C_{max}.

4) Let v ∈ H^{K} be a unit vector of highest weight λ, h ∈ H, and a ∈ exp C_{max} such that ah ∈ h_{1} exp L(ah)N. Then

π_{λ}(h_{1})^{−1}π̂_{λ}(a)π_{λ}(h) = e^{λ(L(ah))}v.

Proof. For a more precise formulation of the results we refer to [10]. 1) is essentially due to Harish Chandra ([4], [5]), and 2) is a result of Ol’shanskiĭ who uses a theorem on analytic continuation of Lüscher and Mack ([12]). The rest is proved in [10]. Assertion 3) bases on the fact, which follows from 1)b), that the spectrum of the essentially selfadjoint operators dπ_{λ}(X), X ∈ C_{max} is bounded from above. To prove 4) one has to make use of the analyticity of the representation π̃_{λ} on int(S_{min}) for analytic continuation arguments because neither π̃_{λ} nor π̂_{λ} is defined on the whole semigroup exp(C_{min})N ⊆ G.

With this theorem the proof of the convexity theorem is almost complete. So let λ ∈ P, a ∈ exp(C_{max} ∩ C_{k}), h ∈ H, ah ∈ h_{1} exp L(ah)N, and v a unit vector of highest weight λ. Then

\begin{aligned}e^{λ(L(ah))} & = ||e^{λ(L(ah))}v|| = ||π_{λ}(h_{1})^{−1}π̂_{λ}(a)π_{λ}(h)v|| \\ & ≤ ||π_{λ}(h_{1})^{−1}||||π̂_{λ}(a)||||π_{λ}(h)v|| = e^{λ(log a)}. \end{aligned}

This proves that λ(L(ah)) ≤ λ(log a) and the theorem follows.

References

[1] Bourbaki, N., Groupes et algèbres de Lie, Chapitres 4,5 et 6, Masson, Paris, 1981.

[2] Faraut, J., Quelques applications de la convexité à l’étude des fonctions sphériques sur un espace symétrique, in this volume.

[3] Faraut, J., Hilgert, J., Olafsson, G., Analysis on ordered symmetric spaces, in preparation.

[4] Harish-Chandra, Representations of semi-simple Lie groups, IV, Amer. J. Math. 77(1955), 743–777.

[5] Harish-Chandra, Representations of semi-simple Lie groups, VI, Amer. J. Math. 78(1956), 564–628.

[6] Helgason, S., “Differential geometry, Lie groups, and symmetric spaces”, Acad. Press, London, 1978.

[7] Helgason, S., “Groups and geometric analysis”, Acad. Press, London, 1984.

[8]  Hilgert, J., Olafsson, G., Causal symmetric spaces, in prep..

[9] Loos, O., “Symmetric spaces I : General Theory”, Benjamin, New York, Amsterdam, 1969.

[10] Neeb, K.–H., A convexity theorem for semisimple symmetric spaces, submitted.

[11] Olafsson, G., and B. Ørsted, The holomorphic discrete series for affine symmetric spaces I. J. Funct. Anal. 81(1988), 126–159.

[12] Ol’shanskiĭ, G. I., Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. and Appl. 15, 275–285 (1982).

Analyse Complexe et Géométrie Département de Mathématiques Université Pierre et Marie Curie 4, place Jussieu F-75252 Paris Cedex 05, France and Fachbereich Mathematik Technische Hochschule Darmstadt D-6100 Darmstadt, Germany

Received July 9, 1991 and in final form September 24, 1991