THE RESTRICTION OF A_{𝔮}(λ) TO REDUCTIVE SUBGROUPS

Toshiyuki KOBAYASHI

University of Tokyo

1. Discrete decomposability with respect to symmetric pairs.

Let G be a real reductive linear Lie group and Ĝ the unitary dual of G. Suppose G′ is a reductive subgroup of G. The representation π ∈ Ĝ is called G′-admissible if the restriction π_{|G′} splits into a discrete sum of irreducible representations of G′ with finite multiplicity. It may well happen that the restriction π_{|G′} contains continuous spectrum (even worse, with infinite multiplicity) which is sometimes difficult to analyse. Thus, the notion of admissibility is emphasized here to single out a very nice pair (π, G′) for the study of the restriction π_{|G′}. Here are famous examples where the restriction π_{|G′} is G′-admissible.

(1.1)(a) If G′ is a maximal compact subgroup of G, then any π ∈ Ĝ is G′-admissible (Harish-Chandra). An explicit decomposition formula is known as a generalized Blattner formula if π = A_{𝔮}(λ) (attached to elliptic orbits in the sense of orbit method; see [1], [7] Theorem 6.3.12). (1.1)(b) A restriction formula of a holomorphic discrete series π is found with respect to some reductive subgroups G′ (eg. [3], [4]). Also the restriction of the Segal-Shale-Weil representation π with respect to dual reductive pair with one factor compact is intensively studied (Howe’s correspondence).

We remark that G′ is compact in the case (1.1)(a), while π ∈ Ĝ is a highest weight module in (1.1)(b). On the other hand, in some special settings, explicit restriction formulas have been found where π ∈ Ĝ do not belong to unitary highest weight modules but is G′-admissible for noncompact G′ ⊂ G, such as (G, G′) = (SO(4, 2), SO(4, 1)) and π is non-holomorphic discrete series ([5] Example 3.4.2), (G, G′) = (SO(4, 3), G_{2}(ℝ)) and π is in some family of derived functor modules (Kobayashi-Uzawa, 1989 at Math. Soc. Japan, see Theorem 3.6), and a recent work of Howe and Tan [2]. In this section we shall find a more general but still good framework to study the restriction π_{|G′}.

Let θ be a Cartan involution of G. Write 𝔤_{0} for the Lie algebra of G, 𝔤 = 𝔤_{0} ⊗_{ℝ} ℂ for its complexification,

K = G^{θ} for the fixed point group of θ, and 𝔤_{0} = 𝔨_{0} + 𝔭_{0} for the corresponding Cartan decomposition. Take a fundamental Cartan subalgebra 𝔥_{0}^{c} (⊂ 𝔤_{0}). Then 𝔱_{0}^{c} : = 𝔥_{0}^{c} ∩ 𝔨_{0} is a Cartan subalgebra of 𝔨_{0}. A θ-stable parabolic subalgebra 𝔮 ≡ 𝔮(λ) = 𝔩(λ) + 𝔲(λ) ⊂ 𝔤 and a Levi part L(λ) ⊂ G are given by an element

λ ∈ \sqrt{−1}(𝔱_{0}^{c})^{∗} (see [7] Definition 5.2.1). Let R_{𝔮}^{j} ≡ (R_{𝔮}^{𝔤})^{j} (j ∈ ℕ) be the Zuckerman’s derived functor from the category of metaplectic (𝔩, (L ∩ K)^{∼})-modules to that of (𝔤, K)-modules. In this paper, we follow the normalization in [8] Definition 6.20 and some terminologies such as weakly fair in [9] Definition 2.5.

Let σ be an involutive automorphism of G. If G′ is an open subgroup of the fixed points of σ, (G, G′) is called a reductive symmetric pair. Choose a Cartan involution θ of G so that σθ = θσ. Then K′ := K ∩ G′ is a maximal compact subgroup of G′. We write 𝔨_{0±} := {X ∈ 𝔨_{0} : σ(X) = ±X}. Fix a σ-stable Cartan subalgebra 𝔱_{0}^{c} of 𝔨_{0} such that 𝔱_{0−}^{c} := 𝔱_{0}^{c} ∩ 𝔨_{0−} is a maximal abelian subspace in 𝔨_{0−}. Choose a positive system

Typeset by AMS-TEX

Σ^{+}(𝔨, 𝔱_{−}^{c}) of the restricted root system Σ(𝔨, 𝔱_{−}^{c}) and a positive system Δ^{+}(𝔨, 𝔱^{c}) which is compatible with Σ^{+}(𝔨, 𝔱_{−}^{c}). Let 𝔮 = 𝔮(μ) = 𝔩 + 𝔲 be a θ-stable parabolic subalgebra of 𝔤 given by an element μ ∈ \sqrt{−1}(𝔱_{0}^{c})^{∗}, which we can assume to be dominant with respect to Δ^{+}(𝔨, 𝔱^{c}) without loss of generality. Define a closed cone in \sqrt{−1}(𝔱_{0}^{c})^{∗} by

ℝ_{+} ⟨𝔲 ∩ 𝔭⟩ := { ∑_{β∈Δ(𝔲∩𝔭,𝔱^{c})} n_{β} β : n_{β} ≥ 0}.

Theorem 1.2. In the setting as above, if ℝ_{+} ⟨𝔲∩𝔭⟩ ∩ \sqrt{−1} (𝔱_{0−}^{c})^{∗} = \{0\}, then \overline{R_{𝔮}^{S}(ℂ_{λ})} is K′-admissible for

any metaplectic unitary character ℂ_{λ} of L̃ in the weakly fair range. In particular, \overline{R_{𝔮}^{S}(ℂ_{λ})} is G′-admissible.

Remark 1.3. In Proposition 4.1.3 in [6], we have established a different type of admissibility in the case where 𝔨 has a direct sum decomposition 𝔨 = 𝔨_{1} ⊕ 𝔨_{2}, G′ ⊃ K_{1} and 𝔮 = 𝔮(μ) such that μ_{|𝔱^{c}∩𝔨_{2}} = 0.

2. Discrete series for homogeneous spaces of reductive type.

Let G be a Lie group and G′, H are its closed subgroups. Then G′ naturally acts on X = G/H from the left. Given x ∈ G/H, we write the isotropy subgroup H′ ≡ G′_{x} := {g ∈ G′ : g · x = x} and put X′ = G′/H′. As a representation theoretic counterpart of an embedding f: X′ ↪ X we consider the restriction of representations of G with respect to G′ which arises as the pullback of function spaces f^{∗}: Γ(X) → Γ(X′).

If H is a reductive algebraic subgroup of a real reductive linear Lie group G, we call the homogeneous space G/H of reductive type. An irreducible unitary representation π ∈ Ĝ is called discrete series for L^{2}(G/H) if π can be realized as a closed subspace of the regular representation L^{2}(G/H). The totality of discrete series for L^{2}(G/H) is denoted by Disc(G/H) (⊂ Ĝ). We also write 𝔻isc(G/H) for the multiset of Disc(G/H) counted with multiplicity occurring in L^{2}(G/H). Analogous notation is used for L^{2}-sections of G-equivariant vector bundles over G/H associated to a unitary representation of H. On the other hand, given (π, V) ∈ Ĝ, we write Disc(π_{|H}) (⊂ Ĝ) for the set of irreducible discrete summands of the restriction π_{|H}, and 𝔻isc(π_{|H}) for the corresponding multiset counted with multiplicity.

Theorem 2.1. Suppose G is a real reductive linear group and G′, H are reductive subgroups stable under θ simultaneously. Let H′ := H ∩ G′. Assume there exists a minimal parabolic subgroup P′ of G′ such that

(2.1)(a) dim H + dim G′ = dim G + dim (H ∩ G′), (2.1)(b) dim H′ + dim P′ = dim G′ + dim (H′ ∩ P′). Then we have a bijection between multisets ⋃_{π∈𝔻isc(G/H)} 𝔻isc(π_{|G′}) = 𝔻isc(G′/H′). In particular, Disc(G′/H′) = ∅ if and only if either Disc(G/H) = ∅ or π_{|G′} is decomposed into only continuous spectrum for any π ∈ Disc(G/H). Moreover, if discrete series for G′/H′ is multiplicity free, then the discrete part of the restriction of π_{|G′} is multiplicity free for all π ∈ Disc(G/H) ⊂ Ĝ.

An abundant theory on the harmonic analysis on G/H has been developed in these fifteen years when G/H is a semisimple symmetric space, while very little has been studied when it is non-symmetric. We note that if one knows Disc(G/H) and the restriction formula π_{|G′} for π ∈ Disc(G/H), then Theorem (2.1) gives a construction and exhaustion of discrete series for G′/H′. More weakly, only a combination of Theorem (1.3) and Theorem (2.1) gives new results on the existence of discrete series of some non-symmetric spherical homogeneous spaces such as Corollary 2.2.

1) Disc(SO(2p − 1, 2q)/U(p − 1, q)) ≠ ∅ if and only if pq ∈ 2ℤ.

2) Disc(SU(2p − 1, 2q)/Sp(p − 1, q)) ≠ ∅ for any p, q.

3) Disc(SO(4, 3)/G_{2}(ℝ)) ≠ ∅, Disc(G_{2}(ℝ)/SL(3, ℝ)) ≠ ∅, Disc(G_{2}(ℝ)/SU(2, 1)) ≠ ∅.

Let relax the assumption (2.1)(a). In the setting at the beginning of §2, we say f: G′/H′ ⊂ G/H regular if there exists a submanifold I of G/H such that G′_{y} = H′ for any y ∈ I and that φ: G′/H′×I → G/H, (g, y) ↦ g · y is an open embedding.

Example 2.3 (group manifolds). If H′ = H = {e}, then G′ ⊂ G is regular. We can take I to be a local section of the principal bundle G → G/G′.

Example 2.4 (semisimple orbits in symmetric spaces). Let σ, τ be commutative involutive automorphisms of G, (G, G′) and (G, H) the corresponding symmetric pairs. Fix a maximally abelian semisimple subspace 𝔞 in {X ∈ 𝔤_{0} : σ(X) = τ(X) = −X} and define M′ := {g ∈ G′ ∩ H : Ad(g)X = X for X ∈ 𝔞}. Then G′/M′ ⊂ G/H is regular. The regular semisimple orbit in G under the adjoint action of G is a typical example.

Theorem 2.5. In the setting of Theorem (2.1), suppose φ_{j}: G′ × H′_{j} × I_{j} → G/H (j ∈ J) define regular orbits such that the disjoint union of φ_{j}(G′/H′_{j} × I_{j}) is open dense in G/H. Then we have ⋃_{π∈Disc(G/H)} Disc(π_{|G′}) ⊂ ⋃_{j} Disc(G′/H′_{j}). In particular, if Disc(G′/H′_{j}) = ∅, then either Disc(G/H) = ∅ or Disc(π_{|G′}) = ∅ for any π ∈ Disc(G/H). Moreover, if π ∈ Disc(G/H) is K′-admissible, then Disc(π_{|G′}) ⊂ ⋂_{j} Disc(G′/H′_{j}).

A very special case corresponding to Example (2.4) yields

Corollary 2.6. Suppose π = \overline{A_{𝔮}} ∈ Ĝ is a (Harish-Chandra’s) discrete series for G. If π is G′-admissible, then π_{|G′} is decomposed into discrete series for G′. In particular, if rankG′ > rankK′ and rankG = rankK, then π_{|G′} is decomposed into only continuous spectrum.

Remark 2.7. In general, if π ∈ Disc(G), then π_{|G′} is supported on tempered representations of G′ by Mackey-Anh’s reciprocity theorem.

3. Examples of decomposition formulas.

In this section we present explicit branching formula of discrete series for indefinite Stiefel manifolds U(p, q; 𝔽)/U(p − 1, q; 𝔽) with respect to U(p, s; 𝔽) × U(q − s; 𝔽) (𝔽 = ℝ, ℂ, ℍ).

Let G = SO_{0}(p, q) ⊃ K = SO(p) × SO(q) (p ≥ 1, q ≥ 0). We take a (standard) basis \{f_{i}\} of \sqrt{−1}(𝔱_{0}^{c})^{∗} as in [6] §2.5 and define θ-stable parabolic subalgebras by 𝔮_{+} := 𝔮(f_{1}) = 𝔩 + 𝔲_{+}, 𝔮_{−} := 𝔮(−f_{1}) = 𝔩 + 𝔲_{−} (p ≥ 2). Then L(f_{1}) = L(−f_{1}) ≃ 𝕋 × SO_{0}(p − 2, q). Put Q := \frac{1}{2}(p + q) − 2. For λ ∈ ℤ + Q, we write ℂ_{λf_{1}} for the metaplectic representation of L̃ corresponding to λf_{1} ∈ \sqrt{−1}(𝔱_{0}^{c})^{∗}. If λ ∈ ℤ + Q and λ ≥ 0 (moreover if λ ≥ \frac{1}{2}p − 1 when q = 0), we define (𝔤, K)-modules by

U_{+}(λ) ≡ U_{+}^{SO_{0}(p,q)}(λ) := (R_{𝔮_{+}}^{𝔤})^{p−2} (ℂ_{λf_{1}}), U_{−}(λ) ≡ U_{−}^{SO_{0}(p,q)}(λ) := (R_{𝔮_{−}}^{𝔤})^{p−2} (ℂ_{−λf_{1}}).

Then U_{±}(λ) are non-zero irreducible (𝔤, K)-modules and U_{±}(λ) ∈ Disc(SO_{0}(p, q)/SO_{0}(p − 1, q)) if λ > 0.

Next, let G′ = U(p, q) ⊃ K′ = U(p)×U(q). We represent the root system of 𝔨′ as Δ(𝔨′, 𝔱′^{c}) = {±(e_{i} − e_{j}) : 1 ≤ i < j ≤ p or p + 1 ≤ i < j ≤ p + q}. We define θ-stable parabolic subalgebras of 𝔤′ by I) Let p ≥ 1, q ≥ 1. Fix A ≫ B ≫ 0. 𝔮′_{+} := 𝔮(Ae_{1} + Be_{p+1}) and 𝔮′_{−} := 𝔮(−Ae_{p} − Be_{p+q}). II) Let p ≥ 2, q ≥ 0. 𝔮′_{0} := 𝔮(e_{1} − e_{p}).

For λ ∈ ℕ_{+}, l ∈ ℤ such that l ≡ λ + p + q + 1 mod 2, we define (𝔤′, K′)-modules by:

V_{+}(λ, l) ≡ V_{+}^{U(p,q)}(λ, l) := (R_{𝔮′_{+}}^{𝔤′})^{p+q−2} (ℂ_{\frac{λ+l}{2} e_{1}+ \frac{−λ+l}{2} e_{p+1}})  if l > λ > 0, pq ≥ 1,
V_{0}(λ, l) ≡ V_{0}^{U(p,q)}(λ, l) := (R_{𝔮′_{0}}^{𝔤′})^{2p−4} (ℂ_{\frac{λ+l}{2} e_{1}+ \frac{−λ+l}{2} e_{p}})  if λ ≥ |l|, p ≥ 2,
V_{−}(λ, l) ≡ V_{−}^{U(p,q)}(λ, l) := (R_{𝔮′_{−}}^{𝔤′})^{p+q−2} (ℂ_{\frac{−λ+l}{2} e_{p}+ \frac{λ+l}{2} e_{p+q}})  if − l > λ > 0, pq ≥ 1.

We list up some properties of V_{+}(λ, l), V_{0}(λ, l), V_{−}(λ, l) in the case q ≥ 1 (cf. [6] Theorem 2): (3.1)(a) V_{+}(λ, l), V_{0}(λ, l), V_{−}(λ, l) are non-zero and irreducible (𝔤′, K′)-modules with Z(𝔤′)-infinitesimal character (\frac{λ+l}{2}, \frac{−λ+l}{2}, Q′, Q′ − 1, …, −Q′) in the Harish-Chandra parametrization, where Q′ := \frac{p+q−3}{2}. (3.1)(b) V_{+}(λ, l) ≃ V_{−}(λ, −l)^{∨} (l > λ > 0), V_{0}(λ, l) ≃ V_{0}(λ, −l)^{∨} (λ ≥ |l|). (3.1)(c) 𝔻isc(U(p, q))/U(1) × U(p − 1, q); χ_{l}) (q ≥ 1, l ∈ ℤ) are given by,

\{ \begin{matrix}\{V_{ε}(λ, l) : |l| > λ > 0\} ∪ \{V_{0}(λ, l) : λ ≥ |l|\} & (p ≥ 2, l ≠ 0, ε = sgn l), \\ \{V_{0}(λ, l) : λ > 0\} & (p ≥ 2, l = 0), \\ \{V_{ε}(λ, l) : |l| − q ≥ λ > 0\} & (p = 1, |l| > q, ε = sgn l), \\ ∅ & (p = 1, |l| ≤ q), \end{matrix}

Here, χ_{l} is a character of U(1) and λ runs over λ ∈ 2ℤ + l + p + q + 1 (resp. λ ∈ 2ℤ + l + q).

Third, let G′′ = Sp(p, q) ⊃ K′′ = Sp(p) × Sp(q), and represent the root system of 𝔨′′ as Δ(𝔨′′, 𝔱′′^{c}) = {±(h_{i} − h_{j}), ±2h_{l} : 1 ≤ i < j ≤ p or p + 1 ≤ i < j ≤ p + q, 1 ≤ l ≤ p + q}. Fix A ≫ B ≫ 0 and we define θ-stable parabolic subalgebras of 𝔤′′ by I) Suppose p ≥ 1, q ≥ 1. 𝔮′′_{+} := 𝔮(Ah_{1} + Bh_{p+1}), L′′_{+} ≃ 𝕋 × Sp(p − 1, q − 1) × 𝕋. II) Suppose p ≥ 2, q ≥ 0. 𝔮′′_{0} := 𝔮(Ah_{1} + Bh_{2}). L′′_{0} ≃ 𝕋^{2} × Sp(p − 2, q).

For λ ∈ ℕ_{+}, j ∈ ℕ such that j ≡ λ + 1 mod 2 we define (𝔤′′, K′′)-modules by:

W_{+}(λ, j) ≡ W_{+}^{Sp(p,q)}(λ, j) := (R_{𝔮′′_{+}}^{𝔤′′})^{2p+2q−2} (ℂ_{\frac{λ+j+1}{2} h_{1}+ \frac{−λ+j+1}{2} h_{p+1}})  if j + 1 > λ > 0, pq ≥ 1,
W_{0}(λ, j) ≡ W_{0}^{Sp(p,q)}(λ, j) := (R_{𝔮′′_{0}}^{𝔤′′})^{4p−4} (ℂ_{\frac{λ+j+1}{2} h_{1}− \frac{−λ+j+1}{2} h_{2}})  if λ ≥ j + 1 > 0, p ≥ 2.

We list up some properties of W_{+}(λ, j), W_{0}(λ, j) in the case q ≥ 1 (cf. [6] Theorem 1): (3.2)(a) W_{+}(λ, j), W_{0}(λ, j) are non-zero and irreducible (𝔤′′, K′′)-modules with Z(𝔤′′)-infinitesimal character (\frac{λ+j+1}{2}, \frac{−λ+j+1}{2}, Q′′, Q′′ − 1, …, 1) in the Harish-Chandra parametrization, where Q′′ := p + q − 2. (3.2)(b) Disc(Sp(p, q)/Sp(1) × Sp(p − 1, q); σ_{j}) (q ≥ 1 and j ∈ ℕ) are given by,

\{ \begin{matrix}\{W_{0}(λ, j) : λ > j\} ∪ \{W_{+}(λ, j) : j > λ > 0\} & (p ≥ 2), \\ \{W_{+}(λ, j) : j − 2q + 1 ≥ λ > 0\}, & (p = 1, j ≥ 2q), \\ ∅, & (p = 1, j < 2q). \end{matrix}

Here, σ_{j} is the irreducible j + 1 dimensional representation of Sp(1). In (3.2)(b), λ runs over λ ∈ 2ℤ + j + 1 and the multiplicity of discrete series is uniformly j + 1 or 0.

We write H^{k}(ℝ^{p}) for spherical harmonics on S^{p−1} of degree k (k ∈ ℕ), which is isomorphic to U_{+}^{SO(p)}(k +

\frac{1}{2}p − 1) if p ≥ 3 or (p, k) = (2, 0) to U_{+}^{SO(p)}(k) ⊕ U_{−}^{SO(p)}(k) if p = 2 and k ≥ 1. If p = 1, we put H^{k}(ℝ^{1}) := ℂ for k = 0, 1 and := 0 for k ≥ 2. Next, we write spherical harmonics of degree (α, β) (α, β ∈ ℕ) as H^{α,β}(ℂ^{p}) = V_{0}^{U(p)}(α + β + p − 1, α − β) ⊂ H^{α+β}(ℝ^{2p}) p ≥ 2. In the case p = 1, it is non-zero only if αβ = 0. Finally, we write F^{Sp(p)}(x, y) (x ≥ y ≥ 0) for the irreducible representation of Sp(p) with an extremal weight xf_{1} + yf_{2}. In the case p = 1, it is non-zero only if y = 0.

Theorem 3.3 (SO_{0}(p, q) ↓ SO_{0}(p, s) × SO(q − s)). Let p ≥ 2, s ≥ 1, q − s ≥ 1, λ ∈ ℤ + \frac{1}{2}(p + q), λ > 0.

U_{+}^{SO_{0}(p,q)}(λ)_{|SO_{0}(p,s)×SO(q−s)} ≃ ⨁_{a,k∈ℕ} U_{+}^{SO_{0}(p,s)}(λ + \frac{1}{2}(q − s) + a + 2k) ⊠ H^{a}(ℝ^{q−s}).

Theorem 3.4 (U(p, q) ↓ U(p, s) × U(q − s)). Let s ≥ 1, q − s ≥ 1, λ ∈ ℕ_{+}, l ∈ 2ℤ + λ + p + q + 1. For convenience, we define an irreducible representation of U(p, s) × U(q − s) by

V_{δ}(α, β, k; λ, l) := V_{δ}^{U(p,s)}(λ + q − s + α + β + 2k, l − α + β) ⊠ H^{α,β}(ℂ^{q−s}).

1)(i) Suppose p ≥ 2, l > λ + q − s.

V_{+}^{U(p,q)}(λ, l)_{|U(p,s)×U(q−s)} ≃ ⨁_{\begin{matrix}α,β,k∈ℕ \\ α+k< \frac{1}{2} (l−λ−q+s) \end{matrix}} V_{+}(α, β, k; λ, l) ⊕ ⨁_{\begin{matrix}α,β,k∈ℕ \\ α+k≥ \frac{1}{2} (l−λ−q+s) \end{matrix}} V_{0}(α, β, k; λ, l).

(ii) Suppose p ≥ 2. We put δ = + if λ + q − s ≥ l > λ and δ = 0 if λ ≥ l ≥ −λ in the left side.

V_{δ}^{U(p,q)}(λ, l)_{|U(p,s)×U(q−s)} ≃ ⨁_{α,β,k∈ℕ} V_{0}(α, β, k; λ, l).

We omit the cases −λ > l ≥ −λ − q + s or −λ − q + s > l because of the duality (3.1)(b).

2) Suppose p = 1, l ≥ λ + q. (We omit the case −l ≥ λ + q because of the duality (3.1)(b).)

V_{+}^{U(1,q)}(λ, l)_{|U(1,s)×U(q−s)} ≃ ⨁_{\begin{matrix}α,β,k∈ℕ \\ α+k≤ \frac{1}{2} (l−λ−q) \end{matrix}} V_{+}(α, β, k; λ, l).

Theorem 3.5 (Sp(p, q) ↓ Sp(p, s) × Sp(q − s)). Let s ≥ 1, q − s ≥ 1, λ ∈ ℕ_{+}, j ∈ ℕ, j ∈ 2ℤ + λ + 1. For convenience, we define an irreducible representation of Sp(p, s) × Sp(q − s) by

W_{δ}(y, v, k, t; λ, j) := W_{δ}^{Sp(p,s)}(λ + 2q − 2s + 2y + 2k + t + v, j + v − t) ⊠ F^{Sp(q−s)}(y + t + v, y).

1)(i) Suppose p ≥ 2, j + 1 > λ + 2q − 2s.

W_{+}^{Sp(p,q)}(λ, j)_{|Sp(p,s)×Sp(q−s)} ≃ ⨁_{\begin{matrix}y,v,k,t∈ℕ \\ 0≤t≤j \\ y+k+t< \frac{1}{2} (j+1−λ)−q+s \end{matrix}} W_{+}(y, v, k, t; λ, j)⊕  ⨁_{\begin{matrix}y,v,k,t∈ℕ \\ 0≤t≤j \\ y+k+t≥ \frac{1}{2} (j+1−λ)−q+s \end{matrix}} W_{0}(y, v, k, t; λ, j).

(ii) Suppose p ≥ 2. We put δ = + if λ + 2q − 2s ≥ j + 1 > λ and δ = 0 if p ≥ 2, λ ≥ j + 1 in the left side.

W_{δ}^{Sp(p,q)}(λ, j)_{|Sp(p,s)×Sp(q−s)} ≃ ⨁_{\begin{matrix}y,v,k,t∈ℕ \\ 0≤t≤j \end{matrix}} W_{0}(y, v, k, t; λ, j).

2) Suppose p = 1, j ≥ λ + 2q − 1.

W_{+}^{Sp(1,q)}(λ, j)_{|Sp(1,s)×Sp(q−s)} ≃ ⨁_{\begin{matrix}y,v,k,t∈ℕ \\ y+k+t≤ \frac{1}{2} (j+1−λ)−q \end{matrix}} W_{+}(y, v, k, t; λ, j).

Finally, we give some examples that the restriction π_{|G′} is still irreducible. Let G = SO_{0}(4, 3) and represent the root system of 𝔨 as Δ(𝔨, 𝔱^{c}) = {±(f_{1} ± f_{2}), ±f_{3}} as in a previous example. We define θ-stable parabolic subalgebras by 𝔮_{1} := 𝔮(f_{3}) and 𝔮_{2} := 𝔮(f_{1}). Next, let G′ = G_{2}(ℝ) be a non-compact real form of an exceptional simple Lie group G_{2}(ℂ) with K′ ≃ SO(4). We choose a compact short root α and a non-compact short root β. We define θ-stable parabolic subalgebras by 𝔮′_{1} := 𝔮(α) and 𝔮′_{2} := 𝔮(β). Then we have

Theorem 3.6 (joint work with T.Uzawa).

(R_{𝔮_{1}}^{𝔤})^{1} (ℂ_{λf_{3}})_{|G_{2}(ℝ)} = (R_{𝔮′_{1}}^{𝔤′})^{1} (ℂ_{λα}),  (R_{𝔮_{2}}^{𝔤})^{2} (ℂ_{λf_{1}})_{|G_{2}(ℝ)} = (R_{𝔮′_{2}}^{𝔤′})^{2} (ℂ_{λβ}), (λ ∈ ℕ + \frac{1}{2}),
𝔻isc(G_{2}(ℝ)/SU(2, 1)) = \{(R_{𝔮′_{1}}^{𝔤′})^{1} (ℂ_{λα}) : λ ∈ ℕ + \frac{1}{2}\},
𝔻isc(G_{2}(ℝ)/SL(3, ℝ)) = \{(R_{𝔮′_{2}}^{𝔤′})^{2} (ℂ_{λβ}) : λ ∈ ℕ + \frac{1}{2}\}.

The irreducibility of the (𝔤′, K′)-modules (R_{𝔮′_{1}}^{𝔤′})^{1} (ℂ_{λα}) and (R_{𝔮′_{2}}^{𝔤′})^{2} (ℂ_{λβ}) at λ = \frac{1}{2} in Theorem (3.6) is

not known to the author. A detailed proof of this article is to appear elsewhere.

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Department of Mathematical Sciences, University of Tokyo, Meguro, Komaba, 153, Tokyo, Japan E-mail address: toshi@tansei.cc.u-tokyo.ac.jp