INVARIANT MEASURES ON HOMOGENEOUS

MANIFOLDS OF REDUCTIVE TYPE

Toshiyuki KOBAYASHI

University of Tokyo

Abstract. We say that the homogeneous manifold G/H is of reductive type if G is a real reductive linear Lie group and if H is a connected closed subgroup which is reductive in G. Semisimple symmetric spaces (especially, Riemannian symmetric spaces and semisimple group manifolds) and semisimple orbits are of reductive type. In this paper, we give an upper estimate of the invariant measure on the homogeneous manifold G/H of reductive type. Furthermore, we also prove a comparison theorem of the measures of homogeneous submanifolds.

§1. Introduction

\underline{1.1.} Consider a real reductive linear Lie group G and a maximal compact subgroup K. The Cartan decomposition G = KAK is an analogue of the polar coordinate in a Euclidean space, where A ≃ ℝ^{k} with k := ℝ-rank G. Harmonic analysis on a group manifold G and on a Riemannian symmetric space G/K relied very much on the integration formula with respect to the Cartan decomposition G = KAK which was due to Harish-Chandra (e.g. [5], Chapter 1).

\underline{1.2.} Semisimple symmetric spaces are a wider class of homogeneous manifolds, including both a group manifold and a Riemannian symmetric space. Later on, harmonic analysis on semisimple symmetric spaces G/H has been developed largely in the last two decades (e.g. [2], [14], [3]), in which a generalized Cartan decomposition G = KBH with B ≃ ℝ^{l} (l := ℝ-rank G/H) plays also a fundamental role; for example, each joint eigenfunction f of G-invariant differential operators on G/H has a certain asymptotic behavior of exponential growth (or decay) along “Weyl chambers” of B ≃ ℝ^{l} if f is K-finite [13], which combined with the integration formula for G = KBH enables us to classify discrete series representations for semisimple symmetric spaces ([1], [12], [2]).

\underline{1.3.} Now, we consider a more general setting: We shall say that the homogeneous manifold G/H is of reductive type if H is a θ-stable closed subgroup of G with at most finite connected components. Semisimple symmetric spaces are of reductive type. Semisimple orbits G/Z_{G}(X) ≃ Ad(G)·X are also of reductive type, where X is a semisimple element of the Lie algebra 𝔤 of G. Although homogeneous spaces of

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 22E46; Secondary 43A85, 22E15, 22E30.

reductive type naturally arise in various fields such as the geometric construction of unitary representations in the philosophy of the Kirillov-Kostant orbit method and the study of automorphic forms, few results on L^{2}-harmonic analysis have been known except for semisimple symmetric spaces (see [9] and references therein). One of the difficulties is the lack of a structure theory based on some “root system”. For example, there is no analogue of the Cartan decomposition G = KBH (this is easily observed by the fact that dim G can be larger than dim K + rank G + dim H) except for semisimple symmetric spaces (and some other few cases). In particular, we cannot expect an integration formula based on a “Cartan decomposition” for a general homogeneous space of reductive type.

\underline{1.4.} In this paper, we study the invariant measure on the homogeneous manifold G/H of reductive type and give an explicit upper estimate. This will be useful especially if we want to prove that given functions on G/H belong to L^{p}(G/H) (see Theorem 3.7). Difficulties arising from the lack of an analogue of the “Cartan decomposition” are overcome by using a natural indefinite-Riemannian metric on a certain principal fiber bundle over G/H instead of the G-invariant measure on G/H itself (see §2). Some inequality in linear algebra will be presented in §4.

Another object of our study is the comparison of the measures of G′/H′ and G/H if G′/H′ ⊂ G/H is an embedding of a homogeneous manifold of reductive type (see Theorem 5.6). The idea of the proof is to employ the Riemannian geometry with non-positive sectional curvature. Our comparison theorem is basic in the study of the restriction of irreducible representations of G (realized in the space of sections of equivariant vector bundles over G/H) with respect to G′.

\underline{1.5.} I have omitted in this paper all applications to representation theory and non-commutative harmonic analysis. One of the applications will be given in a subsequent paper [10] which deals with new discrete series representations for nonsymmetric homogeneous manifolds. Here, combined with the knowledge of the restriction of irreducible unitary representations, the comparison theorem of the invariant measure obtained in this paper will be a main tool.

We also note that symmetric assumptions were required in the unitarization of geometric construction of Vogan-Zuckerman’s derived functor modules [19] and non-zero harmonic forms associated to modular symbols defined by arithmetic quotients of Riemannian symmetric spaces [15], where it seems desirable to drop the symmetric assumptions. We hope to return these problems in future.

⟨ Acknowledgement ⟩

The final manuscript was written while the author was a guest at the Institut Mittag-Leffler of the Royal Swedish Academy of Sciences. He expresses his sincere gratitude to the staff of the Institute and to the organizers of the special year “Analysis on Lie Groups” which presented a warm and wonderful atmosphere of research.

§2. Invariant measure on a homogeneous manifold of reductive type.

\underline{2.1.} First we review the notion of invariant connection of a reductive homogeneous manifold (cf. [6]).

Let L be a Lie group with Lie algebra 𝔩 and π: P → M a smooth principal Lbundle (on the right). A connection on P → M is a splitting of the tangent bundle T P → P into an L-equivariant Whitney sum T P = V er(P) ⊕ Hor(P), where V er(P) = Ker(dπ: T P → T M) is the tangent bundle along fibers, and Hor(P) is so called a horizontal subbundle. The right action of L on P induces a vector field X̃ on P for each X ∈ 𝔩 given by X̃_{p} := \frac{d}{dt}_{|t=0}p · exp(tX) ∈ V er(P)_{p} ⊂ T P_{p}, which

is called the fundamental vector field on P. We note that 𝔩 → V er(P)_{p}, X ↦ X̃_{p} is a bijection for each p ∈ P. The connection form is the 𝔩-valued 1-form on P, denoted by α ∈ E^{1}(P, 𝔩), which is defined by the composition of

T P → V er(P),

the first projection of the splitting T P = V er(P) ⊕ Hor(P), and V er(P) → 𝔩,

the inverse of 𝔩 ⟶^{∼} V er(P)_{p}, X ↦ X̃_{p}.

\underline{2.2.} In the setting of §2.1, we suppose that M is equipped with a pseudo-Riemannian metric g and that L is a real reductive Lie group. There is a nondegenerate symmetric Ad(L)-invariant bilinear form B_{𝔩} on 𝔩. Then we can define naturally a pseudo-Riemannian metric G ≡ G(α, B_{𝔩}, g) on P by

(2.2.1) G_{p}(u, v) := g_{π(p)}(dπ(u), dπ(v)) + B_{𝔩}(α(u), α(v)) (u, v ∈ T P_{p}).

Lemma 2.2. The pseudo-Riemannian metric G is L-invariant.

Proof. We write u = u_{1} + u_{2} ∈ T P_{p} = V er(P)_{p} ⊕ Hor(P)_{p} and X := α(u) ∈ 𝔩. By the definition of α, we have u_{1} = \frac{d}{dt}_{|t=0}p · exp(tX). Fix l ∈ L. Since T P = V er(P) ⊕ Hor(P) is an L-equivariant splitting, we have u · l = u_{1} · l + u_{2} · l ∈ V er(P)_{p·l} ⊕ Hor(P)_{p·l}. Hence we have α(u · l) = Ad(l)^{−1}α(u) because

u_{1} · l = \frac{d}{dt}_{|t=0}p · exp(tX)l = \frac{d}{dt}_{|t=0}p · l exp(t Ad(l)^{−1}X) = { \atop (Ad(l)^{−1}X)}_{p·l}.

In light of dπ(u · l) = dπ(u) ∈ T M_{π(p·l)} = T M_{π(p)} and the L-invariance of B_{𝔩}, we have G_{p·l}(u · l, v · l) = g_{π(p·l)}(dπ(u · l), dπ(v · l)) + B_{𝔩}(α(u · l), α(v · l)) = G_{p}(u, v). □

We write dl, dμ(x) and dσ(p) for the measures on L, M and P which are induced from B_{𝔩}, g and G_{α,B_{𝔩},g}, respectively. We identify dl with a non-zero element of ∧^{dim L}𝔩^{∗} by left translations. Because L is unimodular, the right translation yields the same measure. The connection form α ∈ E^{1}(P, 𝔩) induces a linear map α̃: 𝔩^{∗} → E^{1}(P), and then ∧^{dim L}𝔩^{∗} → E^{dim L}(P), which we also denote by α̃. Then we have

(2.2.2)  dσ(p) = π^{∗}(dμ(x)) α̃(dl).

Let us prove that the measure dσ(p) does not depend on the choice of the connection. To see this, it suffices to show that the restriction of α̃(ω) ∈ E^{1}(T P) with respect to the vertical submanifold π^{−1}(x) (x ∈ M) is independent of α for each ω ∈ 𝔩^{∗}. Fix p ∈ π^{−1}(x) ⊂ P and v ∈ V er(P)_{p}. We find a unique element X ∈ 𝔩 such that v = X̃_{p} ∈ V er(P)_{p} ⊂ T P_{p}. Then we have ⟨v, α̃(ω)_{p}⟩ = ⟨α(v), ω⟩ = ⟨X, ω⟩, which shows the independence of the connection of the principal bundle.

Lemma 2.3. With the setting in §2.2, we assume that L is compact.

1) For any Borel measurable function f on M, we have

(2.3.1)  ∫_{P} (π^{∗}f)(p)dσ(p) = volume(L) ∫_{M} f(x)dμ(x).

2) An L-invariant measure on P satisfying (2.3.1) is unique.

Proof. The first statement is clear from (2.2.2). Because L is compact, an Linvariant measure on P is determined by the integral of all L-invariant functions on P, namely, all the functions of the form π^{∗}f. Hence the second statement follows. □

\underline{2.4.} Now we fix a main setting throughout this paper:

Setting 2.4. Let G be a real reductive linear Lie group, K a maximal compact subgroup, θ the corresponding Cartan involution and 𝔤 = 𝔨 + 𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G given by θ. We fix a non-degenerate symmetric Ad(G)-invariant bilinear form B on 𝔤 such that B is positive definite on 𝔭 and negative definite on 𝔨 and that 𝔨 and 𝔭 is orthogonal. Suppose H is a θ-stable closed subgroup with finitely many connected components. Then H is also a real reductive linear Lie group and we say that the homogeneous manifold G/H is of reductive type. We write o := eH ∈ G/H. Let 𝔮 be the orthogonal complement of 𝔥 in 𝔤 with respect to B. Then we have a direct sum decomposition 𝔤 = 𝔥 + 𝔮 because the restriction B_{|𝔥} is also a non-degenerate symmetric bilinear form.

\underline{2.5.} Suppose we are in the setting 2.4. We recall that G/H has a K-equivariant fiber bundle structure with the base K/H ∩ K and fibers 𝔮 ∩ 𝔭 (cf. [7], Lemma 2.7), by the map

π: K × (𝔮 ∩ 𝔭) ∋ (k, X) ↦ k exp(X) · o ∈ G/H.

That is, we have a diffeomorphism

(2.5.1)  G/H ≃ (K/H ∩ K) ×_{Ad_{|H∩K}} (𝔮 ∩ 𝔭).

We apply results in §2.2 by putting

P := K × (𝔮 ∩ 𝔭),  M := G/H,  L := H ∩ K.

Then L = H ∩ K acts on P freely from the right by (k, X) ↦ (kh, Ad(h^{−1})X) for h ∈ H ∩ K. Then (2.5.1) means that the quotient map

π: P → M

is a principal H ∩ K-bundle. Let (k, X) ∈ P = K × (𝔮 ∩ 𝔭). By left translations, the tangent spaces are identified with Lie algebras as follows:

(2.5.2)  L_{k∗} ⊕ L_{X∗}: 𝔨 ⊕ (𝔮 ∩ 𝔭) ⟶^{∼} T K_{k} ⊕ T (𝔮 ∩ 𝔭)_{X} = T P_{(k,X)},
(2.5.3)  L_{ke^{X}∗}  : 𝔤/𝔥  ⟶^{∼} T (G/H)_{ke^{X}·o} = T M_{π(k,X)},

where L_{k}: K → K, x ↦ kx, L_{X}: 𝔮 ∩ 𝔭 → 𝔮 ∩ 𝔭, Z ↦ X + Z and L_{g}: G/H → G/H, x · o ↦ gx · o. With these identifications, the differential π_{(k,X)∗}: T P_{(k,X)} → T M_{π(k,X)} is given by (see [4], Chapter II, Theorem 1.7)

β ≡ β_{X}: 𝔨 ⊕ (𝔮 ∩ 𝔭) → 𝔤/𝔥,
(2.5.4)  (Y, Z)  ↦ e^{− ad(X)}Y + \frac{1 − e^{− ad(X)}}{ad(X)} Z mod 𝔥.

\underline{2.6.} With notation in (2.5.2), we define a connection of the principal bundle P = K × (𝔮 ∩ 𝔭) → M = G/H by putting a horizontal subbundle

(2.6.1) Hor(P)_{(k,X)} := L_{k∗}(𝔮 ∩ 𝔨) ⊕ L_{X∗}(𝔮 ∩ 𝔭) ⊂ T P_{(k,X)} = T K_{k} ⊕ T (𝔮 ∩ 𝔭)_{X}.

Lemma 2.6. With the identification (2.5.2), the connection form α ∈ E^{1}(P, 𝔥∩𝔨) (see §2.1) is given by

(2.6.2)  α: 𝔨 ⊕ (𝔮 ∩ 𝔭) → 𝔥 ∩ 𝔨, (Y, Z) ↦ Y_{+},

which is independent of (k, X) ∈ P. Here we write Y = Y_{+} + Y_{−} if Y ∈ 𝔨 according to the direct sum 𝔨 = (𝔥 ∩ 𝔨) ⊕ (𝔮 ∩ 𝔨).

Proof. The fundamental vector field Ỹ_{+} on P induced by Y_{+} ∈ 𝔥 ∩ 𝔨 is given by

Ỹ_{+}_{(k,X)} = \frac{d}{dt}|_{t=0}(k exp(tY_{+}), Ad(exp(−tY_{+}))X)

= (L_{k∗} ⊕ L_{X∗})(Y_{+}, [X, Y_{+}]) ∈ V er(P)_{(k,X)}, (2.6.3)

for (k, X) ∈ K × (𝔮 ∩ 𝔭) = P. Because [X, Y_{+}] ∈ [𝔮 ∩ 𝔭, 𝔥 ∩ 𝔨] ⊂ 𝔮 ∩ 𝔭, we have (L_{k∗} ⊕ L_{X∗})(Y_{−}, Z − [X, Y_{+}]) ∈ Hor(P)_{(k,X)} by (2.6.1). Therefore, we have

(L_{k∗} ⊕ L_{X∗})(Y, Z) = (L_{k∗} ⊕ L_{X∗})(Y_{+}, [X, Y_{+}]) + (L_{k∗} ⊕ L_{X∗})(Y_{−}, Z − [X, Y_{+}])
= Ỹ_{+}_{(k,X)} + (L_{k∗} ⊕ L_{X∗})(Y_{−}, Z − [X, Y_{+}])
∈ V er(P)_{(k,X)} ⊕ Hor(P)_{(k,X)}.

Hence the vertical projection T P_{(k,X)} → V er(P)_{(k,X)} is given by

(L_{k∗} ⊕ L_{X∗})(Y, Z) ↦ Ỹ_{+}_{(k,X)}.

By the definition of α, this formula implies α(Y, Z) = Y_{+} via the identification (2.5.2). □

\underline{2.7.} Let us fix X ∈ 𝔮 ∩ 𝔭. Combining two maps (2.5.4) and (2.6.2), we define a linear map φ ≡ φ_{X} = β_{X} ⊕ α: 𝔨 ⊕ (𝔮 ∩ 𝔭) → (𝔤/𝔥) ⊕ (𝔥 ∩ 𝔨) by

(2.7.1) (Y, Z) = (Y_{+} + Y_{−}, Z) ↦ (e^{− ad(X)}Y + \frac{1 − e^{− ad(X)}}{ad(X)} Z mod 𝔥, Y_{+}).

We write

(2.7.2) N_{H} := dim(𝔨 ⊕ (𝔮 ∩ 𝔭)) = dim((𝔤/𝔥) ⊕ (𝔥 ∩ 𝔨)) = dim 𝔤 − dim(𝔥 ∩ 𝔭).

The bilinear form B (see §2.4) induces non-degenerate symmetric bilinear forms on 𝔨, 𝔮∩𝔭, 𝔤/𝔥 and 𝔥∩𝔨, which we denote by B_{𝔨}, B_{𝔮∩𝔭}, B_{𝔤/𝔥} and B_{𝔥∩𝔨}, respectively. In particular, they define invariant (pseudo-)Riemannian structures on K, 𝔮 ∩ 𝔭, G/H and H ∩ K, respectively, by left translations. We denote the corresponding invariant measures by dk, dX, dμ and dh, respectively. Note that B_{𝔮∩𝔭} is positive definite and dX is the Lebesgue measure. The induced pseudo-Riemannian metric on the principal bundle P = K × (𝔮 ∩ 𝔭) from those on the base space G/H and the fiber H ∩ K is given by using the identification (2.5.2) as follows:

G_{k,X}(Y_{1} + Z_{1}, Y_{2} + Z_{2})

= B_{𝔤/𝔥}(β_{X}(Y_{1}, Z_{1}), β_{X}(Y_{2}, Z_{2})) + B_{𝔥∩𝔨}(α(Y_{1}, Z_{1}), α(Y_{2}, Z_{2})), (2.7.3)

for (k, X) ∈ K × (𝔮 ∩ 𝔭) = P and for Y_{i} + Z_{i} ∈ 𝔨 ⊕ (𝔮 ∩ 𝔭) (i = 1, 2). Now, we are ready to describe explicitly the measure dσ on P ≃ K × (𝔮 ∩ 𝔭) induced from the indefinite-Riemannian metric defined in §2.2. We fix an orthonormal base {u_{j}} of 𝔨 ⊕ (𝔮 ∩ 𝔭) with respect to the symmetric (indefinite) bilinear form B_{𝔨} ⊕ B_{𝔮∩𝔭} such that u_{j} belongs to one of 𝔥 ∩ 𝔨, 𝔮 ∩ 𝔨 or 𝔮 ∩ 𝔭. We define a non-negative function δ: 𝔮 ∩ 𝔭 → ℝ by

(2.7.4)  δ(X) := |det ((B_{𝔤/𝔥} ⊕ B_{𝔥∩𝔨})(φ_{X}(u_{i}), φ_{X}(u_{j}))_{1≤i,j≤N_{H}} |^{\frac{1}{2}} .

In light of (2.7.3), we have the following Proposition:

Proposition 2.7. With notation as above, the measure on the principal bundle P = K × (𝔮 ∩ 𝔭) is given by dσ(k, X) = δ(X)dkdX.

§3. Upper estimate of the invariant measure.

\underline{3.1.} We retain the setting §2.4. We give an upper estimate of the natural measure on P = K × (𝔮 ∩ 𝔭) by using the function δ(X) defined in (2.7.4).

\underline{3.2.} Let 𝔞_{𝔭} be a maximally abelian subspace of 𝔭. For 𝔞_{𝔭}^{∗} = Hom_{ℝ}(𝔞_{𝔭}, ℝ), we define 𝔤(𝔞_{𝔭}; λ) := {X ∈ 𝔤 : [H, X] = λ(H)X for H ∈ 𝔞_{𝔭}}. The finite set Σ(𝔤, 𝔞_{𝔭}) := {λ ∈ 𝔞_{𝔭}^{∗} \ {0} : 𝔤(𝔞_{𝔭}; λ) ≠ 0} is said to be the restricted root system of the pair (𝔤, 𝔞_{𝔭}). Let B̄ be a positive definite bilinear form on 𝔤 given by B̄(X, Y) := −B(X, θY). B̄ coincides with B on 𝔭. Denote by ∥ · ∥ the norm on 𝔤 defined by B̄. For a subspace V of 𝔤, we write ∥ · ∥_{V^{∗}} and ∥ · ∥_{𝔤/V} for the induced norms on V^{∗} and 𝔤/V, respectively. We put

(3.2.1)  Λ ≡ Λ(G) := max{∥λ∥_{𝔞_{𝔭}^{∗}} : λ ∈ Σ(𝔤, 𝔞_{𝔭})}.

With the notation in §2.4, we write the projection

𝔤 → 𝔮, Y ↦ Y_{−}

with respect to the direct sum 𝔤 = 𝔥 ⊕ 𝔮. Then ∥Y ∥_{𝔤/𝔥}^{2} = ∥Y_{−}∥^{2} by definition. We note that Y ↦ Y_{−} induces the projection 𝔭 → 𝔮 ∩ 𝔭. We regard 𝔞_{𝔭}^{∗} ⊂ 𝔭^{∗} ≃ 𝔭 by the bilinear form B̄_{𝔭×𝔭} and write 𝔭^{∗} → (𝔮 ∩ 𝔭)^{∗}, λ ↦ λ_{−} for the projection. We define

(3.2.2) Λ_{H} ≡ Λ_{H}(G) := max{∥ (Ad^{∗}(k)λ)_{−} ∥_{(𝔮∩𝔭)^{∗}} : k ∈ K, λ ∈ Σ(𝔤, 𝔞_{𝔭})},
(3.2.3) ν_{G/H} ≡ ν_{H}(G) := N_{H}Λ_{H},  ν ≡ ν(G) := Λ · dim 𝔤.

Here we recall N_{H} = dim 𝔤 − dim(𝔥 ∩ 𝔭).

Lemma 3.3. For X, Y ∈ 𝔤.

1) Λ_{H} ≤ Λ and ν_{G/H} ≤ ν.

2) For any X ∈ 𝔮 ∩ 𝔭, and any eigenvalue μ of ad(X) on 𝔤 we have |μ| ≤ Λ_{H}∥X∥.

3) For any X ∈ 𝔮 ∩ 𝔭 and Y ∈ 𝔤, ∥ exp(− ad(X))Y ∥ ≤ exp(Λ_{H}∥X∥)∥Y ∥.

4) For any X ∈ 𝔮 ∩ 𝔭 and Y ∈ 𝔤, ∥\frac{1−exp(− ad(X))}{ad(X)}Y ∥ ≤ exp(Λ_{H}∥X∥)∥Y ∥.

Proof. 1) Clear.

2) We take k ∈ K such that X′ := Ad(k)X ∈ 𝔞_{𝔭}. If Y ∈ 𝔤(𝔞_{𝔭}; λ) and if we put Z := Ad(k^{−1})Y then ad(X)Z = Ad(k^{−1}) (ad(X′)Y) = λ(X′)Z. Thus the set of eigenvalues of ad(X) is given by {λ(Ad(k)X) : λ ∈ Σ(𝔤; 𝔞_{𝔭}) ∪ {0}}. Then we have

|λ(Ad(k)X)| = |(Ad^{∗}(k^{−1})λ)(X)| = |(Ad^{∗}(k^{−1})λ)_{−}(X)| ≤ Λ_{H}∥X∥,

where the last inequality is from the definition (3.2.1). Hence we have proved (2).

3) ad(X) is a symmetric operator on 𝔤 with respect to the positive definite form B̄. Let Ker(ad(X) − μ) be the eigenspace of ad(X) with eigenvalue μ ∈ ℝ. We take an orthonormal basis Y_{μ,i} of 𝔤 with respect to B̄ such that Y_{μ,i} belongs to the eigenspace Ker(ad(X) − μ) (1 ≤ i ≤ dim Ker(ad(X) − μ)). We write Y = ∑_{μ,i} a_{μ,i}Y_{μ,i}. Then we have ad(X)Y = ∑_{μ,i} a_{μ,i}μY_{μ,i} and therefore

∥e^{− ad(X)}Y ∥^{2} = ∑_{μ,i} |a_{μ,i}|^{2}e^{−2μ} ≤ e^{2Λ_{H}∥X∥} ∑_{μ,i} |a_{μ,i}|^{2} = e^{2Λ_{H}∥X∥}∥Y ∥^{2}.

4) The proof is similar to (3) because |\frac{1−exp(−μ)}{μ}| ≤ exp |μ| for μ ∈ ℝ. □

\underline{3.4.} As is similar to B in §2.7, B̄ induces non-degenerate symmetric bilinear forms on 𝔨, 𝔮∩𝔭, 𝔤/𝔥 and 𝔥∩𝔨, which we denote by B̄_{𝔨}, B̄_{𝔮∩𝔭}, B̄_{𝔤/𝔥} and B̄_{𝔥∩𝔨}, respectively.

Lemma 3.4. Let u_{j} (1 ≤ j ≤ N_{H}) be as in §2.7. Then we have

(B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨})(φ_{X}(u_{j}), φ_{X}(u_{j})) ≤ 2 exp(Λ_{H}∥X∥), 1 ≤ j ≤ N_{H}.

8 TOSHIYUKI KOBAYASHI

Proof. First we note that ∥u_{j}∥ = 1 because B(u_{j}, u_{j}) = ±1 and because u_{j} belongs to 𝔥 ∩ 𝔨, 𝔮 ∩ 𝔨 or 𝔮 ∩ 𝔭. We divide into three cases:

1) If u_{j} ∈ 𝔥 ∩ 𝔨 then φ_{X}(u_{j}) = (e^{− ad(X)}u_{j}, u_{j}) ∈ (𝔤/𝔥) ⊕ (𝔥 ∩ 𝔨). Hence we have

(B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨})(φ_{X}(u_{j}), φ_{X}(u_{j})) = ∥(e^{− ad(X)}u_{j})_{−}∥^{2} + ∥u_{j}∥^{2}
≤ ∥e^{− ad(X)}u_{j}∥^{2} + ∥u_{j}∥^{2}
≤ (e^{2Λ_{H}∥X∥} + 1)∥u_{j}∥^{2}
= e^{2Λ_{H}∥X∥} + 1
≤ 2e^{2Λ_{H}∥X∥}.

Here we have used Lemma 3.3 (3).

2) If u_{j} ∈ 𝔮 ∩ 𝔨 then φ_{X}(u_{j}) = (e^{− ad(X)}u_{j}, 0). As in the above proof, we have (B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨})(φ_{X}(u_{j}), φ_{X}(u_{j})) ≤ e^{2Λ_{H}∥X∥}.

3) If u_{j} ∈ 𝔮 ∩ 𝔭 then φ_{X}(u_{j}) = (\frac{1−e^{− ad(X)}}{ad(X)} u_{j}, 0). By using Lemma 3.3 (4), we have (B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨})(φ_{X}(u_{j}), φ_{X}(u_{j})) ≤ e^{2Λ_{H}∥X∥}. Therefore, we have proved Lemma. □ Lemma 3.5. Suppose v_{i} ∈ (𝔤/𝔥) ⊕ (𝔥 ∩ 𝔨) for 1 ≤ i ≤ N_{H}. Then we have

|det ((B_{𝔤/𝔥} ⊕ B_{𝔥∩𝔨})(v_{i}, v_{j}))_{1≤i,j≤N_{H}} | ≤ det ((B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨})(v_{i}, v_{j}))_{1≤i,j≤N_{H}} . We shall postpone the proof of this lemma to the next section (see §4.1).

Lemma 3.6. Retaining the notation, we have

(3.6.1) |det ((B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨})(φ_{X}(u_{i}), φ_{X}(u_{j})))_{1≤i,j≤N_{H}} | ≤ 2^{N_{H}} exp(ν_{G/H}∥X∥).

Proof. We recall an elementary fact that if A is a positive semi-definite symmetric

n × n matrix and if V := ( v_{1}^{→}, …, v_{n}^{→} ) ∈ M(n, ℝ) with v_{j}^{→} ∈ ℝ^{n} (1 ≤ j ≤ n), then

det^{t}V AV ≤ ∏_{j=1}^{n}(Av_{j}^{→}, v_{j}^{→}). In fact, the volume of the parallelotope spanned by \sqrt{A}v_{1}^{→}, …, \sqrt{A}v_{n}^{→} is less than ∏_{j=1}^{n} ∥\sqrt{A}v_{j}^{→}∥. This implies that

det(^{t}V AV) = det(\sqrt{A}V)^{2} ≤ ∏_{j=1}^{n} ∥\sqrt{A}v_{j}^{→}∥^{2} = ∏_{j=1}^{n}(v_{j}^{→}, Av_{j}^{→}).

Since B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨} is a positive definite symmetric bilinear form, we have

(the left side of (3.6.1)) ≤ ∏_{j=1}^{N_{H}} (B̄_{𝔤/𝔥} ⊕ B̄_{𝔥∩𝔨})(φ_{X}(u_{j}), φ_{X}(u_{j}))
≤ (2 exp(Λ_{H}∥X∥))^{N_{H}} = (the right side of (3.6.1)),

where the second inequality follows from the previous lemma. □

Theorem 3.7. Let G/H be a homogeneous manifold of reductive type and dμ(x) a G-invariant measure. Then there exists a non-negative function δ: 𝔮∩𝔭 → ℝ such that

∫_{G/H} f(x)dμ(x) = ∫_{K} ∫_{𝔮∩𝔭} f(ke^{X} · o) δ(X) dkdX for any f ∈ C_{c}(G/H).

Furthermore, there exist constants ν_{G/H} > 0 and C > 0 such that

δ(X) ≤ C exp(ν_{G/H}∥X∥) for any X ∈ 𝔮 ∩ 𝔭.

The precise formula of δ is given in (2.7.4).

We note that with the normalization of ∥·∥ induced from a fixed norm on 𝔭, there exists a constant ν ≡ ν(G) such that ν_{G/H} ≤ ν for any θ-stable closed subgroup H with finitely many components.

Proof. The first statement follows from Lemma 2.3 and Proposition 2.7. The upper estimate of δ(X) is deduced from (2.7.4), Lemma 3.5 and Lemma 3.6. The last statement was proved in Lemma 3.3 with the constant ν ≡ ν(G) given in (3.3.3). □

\underline{3.8.} Given ξ ∈ ℝ, we introduce a subspace of continuous functions

C(G/H; ξ) := {f ∈ C(G/H) : sup_{k∈K} sup_{X∈𝔭∩𝔮} f(k exp X) exp(ξ∥X∥) < ∞}.

We note that C(G/H; ξ) ⊂ C(G/H; ξ′) if ξ > ξ′.

Corollary 3.9. Let G/H be a homogeneous manifold of reductive type and dμ a G-invariant measure. If 1 ≤ p ≤ ∞ then we have

C(G/H; ξ) ⊂ L^{p}(G/H; dμ) for any ξ > \frac{ν_{G/H}}{p} .

Proof. If p = ∞ then any function belonging to C(G/H; ξ) (ξ > 0) is bounded by definition and there is nothing to prove. We assume 1 ≤ p < ∞. Suppose f ∈ C(G/H; ξ) for ξ > \frac{ν_{G/H}}{p}. Then we find a constant C′ > 0 such that

|f(k exp(X))| ≤ C′ exp(−ξ∥X∥).

It follows from Theorem 3.7 that

∫_{G/H} |f(x)|^{p}dμ(x) = ∫_{K} ∫_{𝔮∩𝔭} |f(k exp(X) · o)|^{p}δ(X) dkdX
≤ (C′)^{p}C ∫_{K} ∫_{𝔮∩𝔭} exp((ν_{G/H} − pξ)∥X∥) dkdX < ∞.

□

§4. Some elementary linear algebra.

\underline{4.1.} The purpose of this section is to prove an inequality of the determinants of symmetric matrices:

Lemma 4.1. Suppose A, B ∈ M(n, ℝ) are symmetric and positive semi-definite matrices. Let τ ∈ GL(n, ℝ) satisfy τ^{2} = 1 and ^{t}τBτ = B. Then we have

| det(A + Bτ)| ≤ det(A + B).

Admitting Lemma 4.1, we complete the proof of Lemma 3.5.

Proof of Lemma 3.5. We fix a base {e_{1}, …, e_{n}} (n := N_{H}) of (𝔤/𝔥) ⊕ (𝔥 ∩ 𝔨) so that each e_{j} belongs to one of 𝔨/(𝔥 ∩ 𝔨), 𝔭/(𝔥 ∩ 𝔭) or 𝔥 ∩ 𝔨. We put A := B̄_{𝔥∩𝔨} and B := B̄_{𝔤/𝔥}, which are identified with positive semi-definite symmetric matrices via the above base. Because 𝔤 = 𝔨 + 𝔭 is orthogonal with respect to the bilinear form B on 𝔤, we have θ ∈ O(B) by Lemma 4.2 (see below). Then the induced bilinear form B̄_{𝔤/𝔥} on 𝔤/𝔥 also satisfies ^{t}θB̄_{𝔤/𝔥}θ = B̄_{𝔤/𝔥} where θ is regarded as an involution on (𝔤/𝔥) ⊕ (𝔥 ∩ 𝔨) ≃ ℝ^{n}. Since B̄_{𝔥∩𝔨} = −B_{𝔥∩𝔨} and B̄_{𝔤/𝔥}θ = −B_{𝔤/𝔥}, we have from Lemma 4.1 that

| det (B_{𝔥∩𝔨} + B_{𝔤/𝔥}) | = | det (B̄_{𝔥∩𝔨} + B̄_{𝔤/𝔥}θ) | ≤ det (B̄_{𝔥∩𝔨} + B̄_{𝔤/𝔥}) .

Hence we have Lemma 3.5. □

The proof of Lemma 4.1 is elementary, but we shall give the proof for the sake of completeness.

\underline{4.2.} Let V be a finite dimensional vector space over ℝ and T a symmetric bilinear form on V which is possibly degenerate. We define the orthogonal group with respect to T by

O(T) := {g ∈ GL_{ℝ}(V) : T (u, v) = T (gu, gv) for any u, v ∈ V }.

Suppose σ ∈ GL_{ℝ}(V) is an involution, that is, σ^{2} = 1. Then we have a direct sum decomposition V = V_{+} + V_{−}, where V_{+} and V_{−} are the eigenspaces for +1 and −1 eigenvalues of σ, respectively.

Lemma 4.2. σ ∈ O(T) if and only if V_{+} is orthogonal to V_{−} with respect to T.

Proof. If σ ∈ O(T), then T (u + σu, v − σv) = (T (u, v) − T (σu, σv)) + (T (σu, v) − T (u, σv)) = 0 for u, v ∈ V. Because V_{±} = {u ± σu : u ∈ V }, we have V_{+} ⊥ V_{−}. Conversely, assume V_{+} ⊥ V_{−}. Let u = u_{+} + u_{−}, v = v_{+} + v_{−} be arbitrary elements of V = V_{+}+V_{−}. Then T (σu, σv) = T (u_{+}−u_{−}, v_{+}−v_{−}) = T (u_{+}, v_{+})+T (u_{+}, u_{−}) = T (u, v), which shows σ ∈ O(T). □

Definition 4.3. Under the equivalent conditions in Lemma 4.2, we define a bilinear form by

T_{σ}(u, v) := T (σu, v) = T (u, σv), for u, v ∈ V.

\underline{4.4.} Suppose g ∈ GL_{ℝ}(V) and σ ∈ GL_{ℝ}(V). We define a bilinear form by T^{g} := ^{t}gT g, that is, T^{g}(u, v) = T (gu, gv) (u, v ∈ V). Let σ^{g} := g^{−1}σg. Then the following is obvious from definition:

Lemma 4.4. σ ∈ O(T) if and only if σ^{g} ∈ O(T^{g}).

\underline{4.5.} We fix a base of V and identify End(V) with M(n, ℝ) where n = dim_{ℝ} V. We also identify an n × n symmetric matrix T with a symmetric bilinear form on V and write O(T) for the orthogonal group with respect to T as in §4.2. O(I_{n}) will be denoted by O(n) as usual. Denote by E_{ij} the matrix unit. Fix m (≤ n) and we put T_{m} := ∑_{i=1}^{m} E_{ii} ∈ M(n, ℝ).

Lemma 4.5. Suppose σ = ( \begin{matrix}P & Q \\ R & S \end{matrix} ), where P ∈ M(m, ℝ), Q ∈ M(m, n −

m; ℝ), R ∈ M(n − m, m; ℝ) and S ∈ M(n − m, ℝ).

1) σ ∈ O(T_{m}) if and only if P ∈ O(m) and Q = 0. Then, T_{m}σ = ( \begin{matrix}P & O \\ O & O \end{matrix} ).

2) If σ ∈ O(T_{m}) and if σ^{2} = 1, then there exist k ∈ O(m) and ε_{j} = ±1 (1 ≤ j ≤ m) such that P = k diag(ε_{1}, …, ε_{m})k^{−1}.

Proof. 1) Since ^{t}σT_{m}σ = ( \begin{matrix}^{t}P P & ^{t}P Q \\ ^{t}QP & ^{t}QQ \end{matrix} ), we have ^{t}σT_{m}σ = T_{m} if and only if P ∈

O(m) and Q = 0. The last statement follows from the formula T_{m}σ = ( \begin{matrix}P & Q \\ O & O \end{matrix} ).

2) Since σ ∈ O(T_{m}), σ is of the form ( \begin{matrix}P & O \\ R & S \end{matrix} ) with P ∈ O(m) from (1). Since

σ^{2} = 1, P ∈ O(m) satisfies P^{2} = I_{m}, whence the second statement. □

\underline{4.6.} Given C_{j} > 0 for 1 ≤ j ≤ m (≤ n), we define a symmetric matrix by T := diag(C_{1}, …, C_{m}, 0, …, 0) ∈ M(n, ℝ).

Lemma 4.6. Suppose σ ∈ O(T) satisfies σ^{2} = 1. Then there exist k ∈ O(m) and ε_{j} = ±1 (1 ≤ j ≤ m) such that

T σ = g ( \begin{matrix}kεk^{−1} & O \\ O & O \end{matrix} ) g,

where ε := diag(ε_{1}, …, ε_{m}) ∈ O(m) and g := diag(\sqrt{C_{1}}, …, \sqrt{C_{m}}, 1, …, 1) ∈ M(n, ℝ).

Proof. From Lemma 4.4 we have σ^{g^{−1}} ∈ O(T^{g^{−1}}). We note that

T^{g^{−1}} = ^{t}g^{−1}T g^{−1} = ∑_{i=1}^{m} E_{ii} ≡ T_{m}.

Then we can find k ∈ O(m) and ε_{j} = ±1 (1 ≤ j ≤ m) such that T^{g^{−1}}σ^{g^{−1}} =

( \begin{matrix}kεk^{−1} & O \\ O & O \end{matrix} ) by Lemma 4.5. The left side equals ^{t}g^{−1}T σg^{−1}, whence Lemma. □

Lemma 4.7. Let u_{1}, …, u_{n} ∈ ℝ^{n} and ∥ · ∥ be the Euclidean norm in ℝ^{n}. Then

| det (u_{1}, …, u_{n}) | ≤ ∏_{j=1}^{n} ∥u_{j}∥.

Proof. Clear from the Gram-Schmidt orthogonalization. □

\underline{4.8. Proof of Lemma 4.1} First we assume det(A + B) = 0. Then we find a nonzero vector u ∈ ℝ^{n} such that ^{t}u(A + B)u = 0. Because both A and B are positive semi-definite, we have ^{t}uAu = ^{t}uBu = 0. By using the positive semi-definiteness again, we have Au = Bu = 0. Therefore (A + Bτ)u = (A + ^{t}τB)u = 0. Hence det(A + Bτ) = 0, proving | det(A + Bτ)| ≤ det(A + B).

Second we assume det(A + B) ≠ 0. Then A + B is positive definite, and we find g ∈ GL(n, ℝ) such that^{t}g(A+B)g = I_{n}. Since^{t}gAg is positive semi-definite, we find k ∈ O(n) such that ^{t}k(^{t}gAg)k = diag(1, …, 1, a_{1}, …, a_{m}) where a_{1}, …, a_{m} are nonnegative numbers other than 1. We put h := gk. Then ^{t}hBh = ^{t}h(A+B)h−^{t}hAh = diag(0, …, 0, 1 − a_{1}, …, 1 − a_{m}). Since B is positive semi-definite, we have a_{j} ≤ 1 for all j with 1 ≤ j ≤ m. Together with the previous condition 0 ≤ a_{j} ≠ 1, we have

0 ≤ a_{j} < 1 (1 ≤ j ≤ m).

We put

b_{j} := 1 − a_{j}  (1 ≤ j ≤ m),
b := diag(1, …, 1, \sqrt{b_{1}}, …, \sqrt{b_{m}}) ∈ M(n, ℝ),
σ := h^{−1}τh  ∈ GL(n, ℝ),
T := ^{t}hBh = diag(0, …, 0, b_{1}, …, b_{m}) ∈ M(n, ℝ).

Then a simple computation yields σ^{2} = 1 and ^{t}σT σ = T. From Lemma 4.6, there exist k ∈ O(m) and ε_{j} = ±1 (1 ≤ j ≤ m) such that

T σ = ^{t}hBhσ = b ( \begin{matrix}O & O \\ O & kεk^{−1} \end{matrix} ) b,

where ε := diag(ε_{1}, …, ε_{m}) ∈ O(m). Then

det(^{t}hAh +^{t}hBhσ)

= det (diag(1, …, 1, a_{1}, …, a_{m}) + b ( \begin{matrix}O & O \\ O & kεk^{−1} \end{matrix} ) b)

= b_{1} ⋯ b_{m} det (diag(\frac{a_{1}}{b_{1}} , …, \frac{a_{m}}{b_{m}}) + kεk^{−1})

= b_{1} ⋯ b_{m} det (diag(\frac{a_{1}}{b_{1}} , …, \frac{a_{m}}{b_{m}})k + kε)

= det (diag(a_{1}, …, a_{m})k + diag(b_{1}, …, b_{m})kε)

= det (((a_{i} + b_{i}ε_{j})k_{ij}))_{1≤i,j≤m} .

Here we write k = (k_{ij})_{1≤i,j≤m}. In view of |a_{i} +b_{i}ε_{j}| ≤ 1 for any i, j and k ∈ O(m), the column vector satisfies

∥^{t}((a_{1} + b_{1}ε_{j})k_{1j}, …, (a_{m} + b_{m}ε_{j})k_{mj})∥ ≤ 1,

for any 1 ≤ j ≤ m. Hence | det (((a_{i} + b_{i}ε_{j})k_{ij})_{1≤i,j≤m}) | ≤ 1 by Lemma 4.7. In light of ^{t}h(A + Bτ)h = det(^{t}hAh +^{t}hBhσ) and (det h)^{2} = (det g)^{2} = det(A + B)^{−1}, we have

| det(A + Bτ)| = | det(^{t}h(A + Bτ)h)| det(h)^{−2} ≤ 1 · det(A + B) = det(A + B),

proving Lemma 4.1. □

§5. Estimates on functions on homogeneous submanifolds.

\underline{5.1.} In this section we give an estimate of the asymptotic behavior of a function on G/H when restricted to a homogeneous submanifold G′/H′. This result plays a fundamental role of harmonic analysis on the orbit space defined by two involutions ([10]).

The key idea is to reduce the problem to the Riemannian symmetric space K\G instead of a non-Riemannian manifold G/H and then to employ an estimate in Riemannian manifolds with non-positive sectional curvatures.

\underline{5.2.} Let W be a finite dimensional vector space over ℝ with an inner product. Given non-zero vector subspaces V_{1} and V_{2}, we define the angle of V_{1} and V_{2} by

(5.2.1)  ψ(V_{1}, V_{2}) := min{the angle of l_{1} and l_{2}}

Here, the minimum is taken over all (l_{1}, l_{2}) such that l_{j} are lines in V_{j} going through the origin for j = 1, 2. We note that 0 ≤ ψ(V_{1}, V_{2}) ≤ \frac{π}{2} and that

(5.2.2)  ψ(V_{1}, V_{2}) = 0 if and only if V_{1} ∩ V_{2} ≠ {0}.

\underline{5.3.} Suppose we are in the setting (2.4). Let G/H be a homogeneous manifold of reductive type. It is well known that the mappings

(5.3.1)  (𝔮 ∩ 𝔭) ⊕ (𝔥 ∩ 𝔭) → K\G, (X, Y) ↦ K exp X exp Y,
(5.3.2)  𝔭 → K\G,  Z ↦ K exp Z

are surjective diffeomorphisms. In particular, a smooth map γ: 𝔭 → 𝔮∩𝔭 is uniquely defined with the relation

(5.3.3)  exp Z ∈ K exp(γ(Z)) exp(𝔥 ∩ 𝔭).

\underline{5.4.} Here is a crucial step in L^{p}-estimates of the function restricted to a submanifold.

Lemma 5.4. Let V be a non-zero subspace of 𝔭. Then we have:

∥Z∥ ≥ ∥γ(Z)∥ ≥ sin ψ(V, 𝔥 ∩ 𝔭) ∥Z∥ for any Z ∈ V.

Proof. The Ad(G)-invariant symmetric bilinear form B on 𝔤 induces a norm ∥ · ∥ on 𝔭 and a Riemannian metric on the right coset space K\G by the right translation of G. Then K\G is a simply connected Riemannian symmetric space with non-positive sectional curvatures. We write exp Z = k exp(γ(Z)) exp Y with the uniquely determined elements k ∈ K and Y ∈ 𝔥 ∩ 𝔭 according to(5.3.3). Let O := K · e ∈ K\G and we write A := O · exp Y and B := O · exp Z ∈ K\G. The curves

{O · exp(tZ) : 0 ≤ t ≤ 1},
{O · exp(tY) : 0 ≤ t ≤ 1},
{O · exp(t γ(Z)) exp Y : 0 ≤ t ≤ 1}

are the geodesics forming the sides of the triangle, namely, OB, OA and AB respectively. We write |OB|, |OA| and |AB| for the geodesic distances, which equal ∥Z∥, ∥Y ∥ and ∥γ(Z)∥, respectively. Denote by ψ_{0} the angle of OA and OB. It follows from the definition (5.2.1) that

ψ_{0} ≥ ψ(V, 𝔥 ∩ 𝔭)

because Z ∈ V and Y ∈ 𝔥 ∩ 𝔭. Then we have (cf. [4], Chapter I, Corollary 13.2)

|AB|^{2} ≥ |OA|^{2} +|OB|^{2} −2|OA||OB| cos ψ_{0} = (|OA|−|OB| cos ψ_{0})^{2} +|OB|^{2} sin^{2} ψ_{0}.

Hence |AB| ≥ |OB| sin ψ_{0}. This means that

∥γ(Z)∥ = |AB| ≥ |OB| sin ψ_{0} = ∥Z∥ sin ψ_{0} ≥ sin ψ(V, 𝔥 ∩ 𝔭) ∥Z∥.

Similarly, we have |OB| ≥ |AB| sin ∠OAB. We translate A = O · exp(Y) into O by the right action of exp(−Y). Put A′ := O · exp(−Y) and B′ := B · exp(−Y) = O·exp(γ(Z)). Then ∠OAB = ∠A′OB′, which equals ∠OAB = \frac{π}{2} because Y ∈ 𝔥∩𝔭 and γ(Z) ∈ 𝔮 ∩ 𝔭 and because (𝔥 ∩ 𝔭) ⊥ (𝔮 ∩ 𝔭). Therefore we have |OB| ≥ |AB|, namely, ∥Z∥ ≥ ∥γ(Z)∥. This completes the proof. □

\underline{5.5.} Suppose both H and G′ are θ-stable closed subgroups of G with finitely many connected components. We put H′ := G′ ∩ H. Then G′/H′ ⊂ G/H are homogeneous manifolds of reductive type. Let 𝔤′ and 𝔥′ be the Lie algebras of G′ and H′, respectively. We put 𝔭′ := 𝔤′ ∩ 𝔭, and define 𝔮′ to be the orthogonal complement of 𝔥′ in 𝔤′ with respect to the non-degenerate symmetric bilinear form B_{𝔤′}. Then we have direct sum decompositions 𝔤′ = 𝔥′⊕𝔮′ and 𝔭′ = (𝔥′∩𝔭′)⊕(𝔮′∩𝔭′). We define

(5.5.1)  b(G′/H′; G/H) := sin ψ(𝔮′ ∩ 𝔭′, 𝔥 ∩ 𝔭).

Lemma 5.5. If G′/H′ is noncompact then b(G′/H′; G/H) > 0.

Proof. It suffices to show (𝔮′ ∩ 𝔭′) ∩ (𝔥 ∩ 𝔭) = {0} by (5.2.2). We have

(𝔮′ ∩ 𝔭′) ∩ (𝔥 ∩ 𝔭) = (𝔮′ ∩ 𝔭′) ∩ (𝔥 ∩ 𝔭′) = (𝔮′ ∩ 𝔭′) ∩ (𝔥′ ∩ 𝔭′) ⊂ 𝔮′ ∩ 𝔥′ = {0}.

Hence lemma. □

\underline{5.6.} Now we are ready to give an estimate of the decay of a function on G/H restricted to a submanifold G′/H′.

Theorem 5.6. Suppose that G is a real reductive linear Lie group with a Cartan involution θ and that both H and G′ are θ-stable closed subgroups of G with finitely many connected components. Let H′ := H ∩ G′. We write ι: G′/H′ ↪ G/H for the natural embedding, and ι^{∗}: C(G/H) → C(G′/H′) for the pullback of functions. Let b := b(G′/H′; G/H) be a constant given in (5.5.1). Then

ι^{∗}C(G/H; ξ) ⊂ C(G′/H′; b ξ) for any ξ ≥ 0.

Proof. Let f ∈ C(G/H; ξ). It follows from definition that there exists C > 0 such that

|f(k exp Z · o)| ≤ C exp(−ξ∥Z∥) for any k ∈ K and Z ∈ 𝔮 ∩ 𝔭,

where o = eH ∈ G/H. Let k′ ∈ K′, Z′ ∈ 𝔮′ ∩ 𝔭′. According to the decomposition (5.3.3), we find k′′ ∈ K and Y ∈ 𝔥 ∩ 𝔭 such that exp Z′ = k′′ exp(γ(Z′)) exp Y. It follows from Lemma 5.4 that ∥γ(Z′)∥ ≥ b∥Z′∥. Therefore

|f(k′ exp Z′ · o)| = |f(k′k′′ exp(γ(Z′)) exp Y · o)|
= |f(k′k′′ exp(γ(Z′)) · o)|
≤ C exp(−ξ ∥γ(Z′)∥)
≤ C exp(−ξ b∥Z′∥).

This shows that ι^{∗}f ∈ C(G′/H′; bξ). □

\underline{5.7.} The following Corollary will play a basic role in the forthcoming paper on harmonic analysis of the double coset space associated to two involutive automorphisms, of which the geometry has been studied recently by T. Matsuki.

Corollary 5.7. Retain the setting of Theorem 5.6. Let dμ be a G′-invariant measure on G′/H′. Let ν_{G′/H′} and b(G′/H′; G/H) be the constants given in (3.2.3) and in (5.5.1), respectively. Then for any 1 ≤ p ≤ ∞, we have

ι^{∗}C(G/H; ξ) ⊂ L^{p}(G′/H′; dμ) for any ξ > \frac{ν_{G′/H′}}{b(G′/H′; G/H) p}.

Proof. Corollary is immediate from Corollary 3.9 and Theorem 5.6. □

Remark 5.8. Neither Theorem 5.6 nor Corollary 5.7 is true in general if we assume only G′, H and H′ := G′ ∩ H are reductive in G. For example, let G :=

SL(2, ℝ), a(t) := ( \begin{matrix}exp(t) & 0 \\ 0 & exp(−t) \end{matrix} ) ∈ G, H := \{a(t) : t ∈ ℝ\} ⊂ G, n :=

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ G, and G′ := nHn^{−1}. Then H, G′ and H′ := G′ ∩ H = \{e\} are

reductive in G. Then

lim_{t→−∞} na(t)n^{−1}H = n lim_{t→−∞} a(t)n^{−1}a(−t)H = nH in G/H.

The natural imbedding ι: G′ ≃ G′/H′ → G/H is given by na(t)n^{−1} ↦ na(t)n^{−1}H. Thus we have

lim_{t→−∞} ι^{∗}f(a(t)) = f(nH),

for f ∈ C(G/H). In particular, if f(nH) ≠ 0, then lim_{t→−∞} ι^{∗}f(a(t)) ≠ 0. Therefore, ι^{∗}f ∉ C(G′/H′; ξ′) for ξ′ > 0 and ι^{∗}f ∉ L^{p}(G′/H′; dμ) for 1 ≤ p < ∞. Hence,

ι^{∗}C(G/H; ξ) ⊄ C(G′/H′; c ξ) for any ξ > 0 and for any c > 0,
ι^{∗}C(G/H; ξ) ⊄ L^{p}(G′/H′; dμ) for any ξ > 0 and for any 1 ≤ p < ∞.

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Graduate School of Mathematical Sciences University of Tokyo Komaba, Meguro, Tokyo 153 Japan

E-mail address: toshi@ms.u-tokyo.ac.jp