L^{p}-ANALYSIS ON HOMOGENEOUS MANIFOLDS OF

REDUCTIVE TYPE AND REPRESENTATION THEORY

Toshiyuki KOBAYASHI

University of Tokyo

§1. Introduction.

\underline{1.1.} Let G be a real reductive linear Lie group, K a maximal compact subgroup of G, and θ the corresponding Cartan involution. Suppose that H is a closed θ-stable subgroup of G with finitely many connected components. Then the coset space G/H is called a homogeneous manifold of reductive type. Riemannian symmetric spaces (e.g. GL(n, ℝ)/O(n)), semisimple Lie group manifolds (e.g. GL(n, ℝ)/{e} ≃ GL(n, ℝ)×GL(n, ℝ)/ diag GL(n, ℝ)), or more generally, reductive symmetric spaces (e.g. GL(n, ℝ)/O(p, n−p)), and semisimple orbits in semisimple Lie algebras under the adjoint action (e.g. GL(n, ℝ)/GL(n_{1}, ℝ) × ⋯ × GL(n_{k}, ℝ) (∑ n_{j} = n)) are typical examples of homogeneous manifolds of reductive type. Various geometric structures of homogeneous manifolds of reductive type may be found in the survey [7] and references therein.

\underline{1.2.} If G/H is of reductive type, then there exists a G-invariant measure dμ on G/H, which is unique up to a scalar multiple. Then we have a continuous representation of G on the Banach space L^{p}(G/H; dμ) ≡ L^{p}(G/H) (p ≥ 1) by left translations. A fundamental problem in L^{p}-analysis on a homogeneous manifold G/H is to construct an irreducible representation of G in a closed G-invariant subspace of L^{p}(G/H). In particular, if p = 2, then G acts unitarily on the Hilbert space L^{2}(G/H) and an irreducible representation π of G is called a discrete series representation for G/H, provided π is realized in a closed G-invariant subspace of L^{2}(G/H). Discrete series representations are automatically unitary and we denote by Disc(G/H) the unitary equivalence classes of discrete series representations for G/H. By definition, Disc(G/H) ⊂ Ĝ, where Ĝ denotes the unitary dual of G.

\underline{1.3.} It is a celebrated work due to Harish-Chandra that Disc(G/\{e\}) ≠ ∅ iff rank G = rank K. Generalizing this, Flensted-Jensen, Matsuki and Oshima proved in [1] and [12] that Disc(G/H) ≠ ∅ iff

(1.3.1)  rank G/H = rank K/H ∩ K

Typeset by AMS-TEX

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

for a reductive symmetric space G/H (see §3.2 for definition).

\underline{1.4.} However, except for reductive symmetric spaces, our current knowledge about the existence of discrete series representations for G/H (or more generally, the existence of irreducible representations in L^{p}(G/H)) is very poor. This is partly because the known methods relied on

i) the commutativity of invariant differential operators on G/H, ii) a Cartan decomposition G = KAH ([1], §2), iii) the dual space G^{d}/H^{d} (see [2] for the notation). In our more general setting where G/H is of reductive type, (i) may fail, and neither (ii) nor (iii) always exist.

Thus, we need a new method to investigate the L^{p}-analysis on G/H for a general homogeneous manifold of reductive type. Our approach in this paper is based on the recent theory of the discrete decomposable restriction of unitary representations ([5], [6]), and on a comparison theorem of two homogeneous manifolds (Theorem 2.7) together with well-developed results on reductive symmetric spaces ([1], [2], [12]).

§2. Invariant measure on homogeneous manifolds of reductive type.

\underline{2.1.} For a reductive symmetric space G/H, there is a generalized Cartan decomposition “G = KAH” with A ≃ ℝ^{l}, which is an analog of the polar coordinate in the Euclidean space. Then, the corresponding integration formula ([1], Theorem 2.6) is a basic tool on harmonic analysis on a symmetric space G/H, because the L^{p}-estimate of functions on G/H can be studied by the decay condition along the abelian part A ≃ ℝ^{l}.

Unfortunately, there does not always exist an analog of a generalized Cartan decomposition G = KAH in our general setting where G/H is of reductive type, as one can easily observe by an argument of dimension (e.g. the case where dim H is very small).

Thus, in order to give an L^{p}-estimate of functions on G/H, we need a nice estimate of the invariant measure on G/H without using the KAH decomposition, which is the main goal of this section.

Let us fix some notation. Let 𝔤 = 𝔨 + 𝔭 be a Cartan decomposition of the Lie algebra 𝔤 of G corresponding to a Cartan involution θ of G, and we fix an Ad(G)-invariant non-degenerate symmetric bilinear form B on 𝔤 such that B|_{𝔨×𝔨} is negative definite, B|_{𝔭×𝔭} is positive definite and that 𝔨 and 𝔭 are orthogonal to each other (e.g. we can take B to be the Killing form if 𝔤 is semisimple). We write ∥X∥ (X ∈ 𝔭) for the induced norm on 𝔭.

\underline{2.2.} Suppose G/H is a homogeneous manifold of reductive type. We write 𝔥 for the Lie algebra of H. Then the restriction B|_{𝔥×𝔥} is non-degenerate. Let 𝔥^{⊥} be the orthogonal complementary subspace of 𝔥 in 𝔤. Then, we have a direct sum decomposition 𝔤 = 𝔥 + 𝔥^{⊥}.

Theorem 2.2. Let G/H be a homogeneous manifold of reductive type. Then there exists a non-negative function δ: 𝔥^{⊥} ∩ 𝔭 → ℝ such that

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

Here dk is the bi-invariant measure on K and dX is the Lebesgue measure on 𝔥^{⊥}∩𝔭. Furthermore, there exist constants ν_{G/H} > 0 and C > 0 such that

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

Remark 2.3. One can prove that there exists a constant ν ≡ ν(G) such that ν_{G/H} ≤ ν for any θ-stable closed subgroup H with finitely many connected components.

\underline{2.4.} Given ξ ∈ ℝ, we introduce a subspace of continuous functions on G/H:

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

There is an obvious inclusive relation C(G/H; ξ) ⊂ C(G/H; ξ′) for ξ > ξ′.

Corollary 2.4. Let G/H be a homogeneous manifold of reductive type. If 1 ≤ p ≤ ∞, then we have

C(G/H; ξ) ⊂ L^{p}(G/H) if p ξ > ν_{G/H}.

We put C^{∞}(G/H; ξ) := C(G/H; ξ) ∩ C^{∞}(G/H). We say that an admissible irreducible representation π of G is of decay type ξ for G/H if the underlying (𝔤, K)-module π_{K} can be realized as a subrepresentation of C^{∞}(G/H; ξ). If π is of decay type ξ for G/H with ξ > \frac{1}{2}ν_{G/H}, then π_{K} is unitarizable and Disc(G/H) ≠ ∅ by Corollary 2.4.

\underline{2.5.} Next, we give a comparison theorem of invariant measures of two homogeneous manifolds G′/H′ ⊂ G/H. Consider the following setting: G is a real reductive linear Lie group with a Cartan involution θ; both H and G′ are θ-stable closed subgroups of G with finitely many connected components. Let H′ := H ∩G′. Then G′/H′ is also of reductive type. We write ι: G′/H′ ↪ G/H for the natural embedding, and ι^{∗}: C(G/H) → C(G′/H′) for the pullback of continuous functions.

\underline{2.6.} In the setting of §2.5, we write 𝔤′ and 𝔥′ for the Lie algebras of G′ and H′, respectively. We put 𝔭′ := 𝔤′ ∩ 𝔭, and define 𝔥′^{⊥} to be the orthogonal complement of 𝔥′ in 𝔤′ with respect to B|_{𝔤′×𝔤′}. Then, we have direct sum decompositions 𝔤′ = 𝔥′ ⊕ 𝔥′^{⊥} and 𝔭′ = (𝔥′ ∩ 𝔭′) ⊕ (𝔥′^{⊥} ∩ 𝔭′). We define

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

Here, (0 ≤) ψ(𝔮′ ∩ 𝔭′, 𝔥 ∩ 𝔭) (≤ \frac{π}{2} ) is the angle between the subspaces of 𝔭, 𝔮′ ∩ 𝔭′ and 𝔥 ∩ 𝔭.

Lemma 2.6. If G′/H′ is noncompact, then b(G′/H′; G/H) > 0.

\underline{2.7.} Here is a comparison theorem that gives an estimate of the decay of a function on G/H restricted to a submanifold G′/H′.

Theorem 2.7. Let b := b(G′/H′; G/H) be a constant given in (2.6.1). Then

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

Corollary 2.8. Retain the setting of Theorem 2.7. Let ν_{G′/H′} be the constant given in Theorem 2.2. Then for any 1 ≤ p ≤ ∞, we have

ι^{∗}C(G/H; ξ) ⊂ L^{p}(G′/H′) if b p ξ > ν_{G′/H′}.

§3. Irreducible representations in L^{p}(G/H).

In this section, we investigate a sufficient condition for the existence of irreducible representations realized in a closed subspace of L^{p}(G/H). In particular, we construct new discrete series representations which play a fundamental role in the non-commutative harmonic analysis on G/H.

\underline{3.1.} Suppose G′ ⊂ G are real reductive Lie groups with maximal compact groups K′ ⊂ K. We say that π ∈ Ĝ is K′-admissible if

dim Hom_{K′}(τ, π|_{K′}) < ∞ for any τ ∈ K̂′.

Then we also say that the underlying (𝔤_{ℂ}, K)-module π_{K} is K′-admissible. Suppose we are in the setting 2.5 and 2.6. Let b = b(G′/H′; G/H). Here is a key lemma in the proof of Theorem 3.5, the main result in this section:

Lemma 3.1. Let π ∈ Ĝ be K′-admissible. If π is of decay type ξ for G/H, then there exists τ ∈ Ĝ′ that is of decay type bξ for G′/H′.

This lemma follows directly from an algebraic result in representation theory ([5], Part III, Proposition 1.6).

\underline{3.2.} Let 𝔱 be a Cartan subalgebra of 𝔨. We fix a positive system Δ^{+}(𝔨, 𝔱). Let τ be an involutive automorphism of G, and G′ an open subgroup of G^{τ} := {g ∈ G : τg = g}. Then (G, G′) is called a reductive symmetric pair. The homogeneous manifold G/G′ is called a reductive symmetric space (or a semisimple symmetric space if G is semisimple). We say that τ is in a standard position with respect to Δ^{+}(𝔨, 𝔱) if the following four conditions are satisfied: (3.2.1) τθ = θτ. (3.2.2) τ(𝔱) = 𝔱. (3.2.3) 𝔱^{−τ} is a maximal abelian subspace in 𝔨^{−τ}. (3.2.4) {α_{|𝔱^{−τ}} : α ∈ Δ^{+}(𝔨, 𝔱)}\{0} defines a positive system Σ^{+}(𝔨, 𝔱^{−τ}) of Σ(𝔨, 𝔱^{−τ}).

Here, we wrote 𝔪^{−τ} := {X ∈ 𝔪 : τX = −X} for a subspace 𝔪 of 𝔤.

\underline{3.3.} Given an element X ∈ \sqrt{−1}𝔱, we define a θ-stable parabolic subalgebra

𝔮 = 𝔩_{ℂ} + 𝔲 ≡ 𝔩(X)_{ℂ} + 𝔲(X) (⊂ 𝔤_{ℂ})

such that 𝔩_{ℂ} and 𝔲 are the sum of eigenspaces with 0 and positive eigenvalues of ad(X) ∈ End(𝔤_{ℂ}), respectively. Then 𝔩_{ℂ} is the complexification of the Lie algebra of L = Z_{G}(X), the centralizer of X in G. We write A_{𝔮}(λ) for the Zuckerman’s derived functor (𝔤, K)-module for a metaplectic L˜ -character ℂ_{λ} in the good range (see [3]).

We say that 𝔮 is in a standard position for a fixed positive system Δ^{+}(𝔨, 𝔱) if X sits inside a dominant chamber with respect to Δ^{+}(𝔨, 𝔱). We denote by ℝ_{+}⟨𝔲 ∩ 𝔭_{ℂ}⟩ the cone in \sqrt{−1}𝔱^{∗} defined by the ℝ_{+}-span of Δ(𝔲 ∩ 𝔭_{ℂ}, 𝔱).

Fact 3.3. Suppose (G, G′) is a reductive symmetric pair defined by an involution τ in a standard position with respect to Δ^{+}(𝔨, 𝔱). Suppose 𝔮 is a θ-stable parabolic subalgebra in a standard position with respect to Δ^{+}(𝔨, 𝔱). Then the following three conditions on (𝔮, G, G′) are equivalent:

1) ℝ_{+}⟨𝔲 ∩ 𝔭_{ℂ}⟩ ∩ \sqrt{−1}(𝔱^{−τ})^{∗} = \{0\}.

2) The restriction of A_{𝔮}(λ) to K′ is K′-admissible.

3) A_{𝔮}(λ) is decomposed into an algebraic sum of irreducible (𝔤′, K′)-modules.

Proof. See [5], Part I, Theorem 3.2 for (1) ⇒ (2); and [5], Part III, Theorem 4.2 for other implications. □

\underline{3.4.} Let (G, H) and (G, G′) be reductive symmetric pairs defined by involutive automorphisms σ and τ of G, respectively, which are in a standard position with respect to Δ^{+}(𝔨, 𝔱). We employ an analogous notation of §3.2 for σ. If (1.3.1) is satisfied, then 𝔱^{−σ} is a maximal abelian subspace in 𝔤^{−σ} We denote by W (𝔤, 𝔱^{−σ}) ⊃ W (𝔨, 𝔱^{−σ}) the Weyl groups of Σ(𝔤, 𝔱^{−σ}) ⊃ Σ(𝔨, 𝔱^{−σ}), respectively. Fix a positive system Σ^{+}(𝔤, 𝔱^{−σ}) which contains Σ^{+}(𝔨, 𝔱^{−σ}). A θ-stable parabolic subalgebra 𝔮 ≡ 𝔮(X) = 𝔩_{ℂ} + 𝔲 is attached by a strictly dominant element X ∈ \sqrt{−1}𝔱^{−σ} with respect to Σ^{+}(𝔤, 𝔱^{−σ}) (see §3.3). For each w ∈ W (𝔨, 𝔱^{−σ})\W (𝔤, 𝔱^{−σ}), we choose a representative m_{w} ∈ K such that Ad(m_{w})X is dominant with respect to Δ^{+}(𝔨, 𝔱), and define a θ-stable parabolic subalgebra 𝔮^{w} := Ad(m_{w})𝔮 = 𝔩_{ℂ} + 𝔲^{w}, where 𝔲^{w} := Ad(m_{w})𝔲.

\underline{3.5.} We write pr_{𝔤→𝔤′}: 𝔤_{ℂ}^{∗} → 𝔤′_{ℂ}^{∗} for the natural projection. We denote by Ass(π) ⊂ 𝔤′_{ℂ}^{∗} the associated variety of a (𝔤′_{ℂ}, K′)-module π of finite length.

Theorem 3.5. Let G/H be a reductive symmetric space satisfying (1.3.1). Retain the notation in §3.4. We assume that there exists w ∈ W (𝔨, 𝔱^{−σ})\W (𝔤, 𝔱^{−σ}) such that

ℝ_{+} ⟨𝔲^{w} ∩ 𝔭_{ℂ}⟩ ∩ \sqrt{−1} (𝔱^{−τ})^{∗} = \{0\}.

We put H′_{x} := G′ ∩ xHx^{−1} for x ∈ K. Then the following statements hold:

1) For any x ∈ K, there exists an irreducible (𝔤′_{ℂ}, K′)-module π such that

Ass(π) ⊃ pr_{𝔤→𝔤′} Ad(K_{ℂ}) (𝔲^{w} ∩ 𝔭_{ℂ}) , (3.5.1)

Hom_{(𝔤′_{ℂ},K′)} (π, C^{∞}(G/H) ∩ ⋂_{1≤p≤∞} L^{p}(G′/H′_{x})) ≠ {0}. (3.5.2)

In particular, Disc(G′/H′_{x}) ≠ ∅ for any x ∈ K.

2) Assume moreover that Z_{G}(𝔱^{−σ}) is compact. Then Disc(G′/H′_{x}) ∩ Disc(G′) ≠ ∅ for any x ∈ K.

The point here is that G′/H′_{x} gives different homogeneous manifolds of reductive type as x ∈ K varies. In general, G′/H′_{x} is non-symmetric. A recent study of the double coset decomposition G^{τ} \G/G^{σ} by Matsuki ([10], [11]) helps us to compute explicitly the isotropy subgroup H′_{x}.

Example 3.6. The assumptions of Theorem 3.5 are satisfied, if the triple of Lie groups (G, H, G′) ≡ (G, G^{σ}, G^{τ}) is one of the following cases:

(U(2p, 2q), Sp(p, q),  U(i, j) × U(2p − i, 2q − j)),
(O(p, q), O(m) × O(p − m, q), O(p, q − r) × O(r)),
(U(p, q), U(m) × U(p − m, q), U(p, q − r) × U(r)),
(Sp(p, q), Sp(m) × Sp(p − m, q), Sp(p, q − r) × Sp(r)),

where 0 ≤ i ≤ 2p, 0 ≤ j ≤ 2q, 0 ≤ 2m ≤ p and 0 ≤ r ≤ q. We note that σ does not commute with τ if i or j is odd in the first case.

Remark 3.7. The special case where dim H + dim G′ = dim G + dim(H ∩ G′) (and x = e) was studied in [5], Corollary 5.6, where we dealt with so-called the (non-symmetric) spherical homogeneous spaces.

Example 3.8. The homogeneous manifolds G/H = O(4m, n)/U(2m, j), (0 ≤ 2j ≤ n) admit discrete series representations. This was previously known when n = 2j and 2j − 1, where G/H is a semisimple symmetric space, and a nonsymmetric spherical homogeneous space, respectively. Other cases are new.

§4. Holomorphic discrete series representations.

\underline{4.1.} In this section, we investigate a nice subset of Disc(G/H), namely, “holomorphic discrete series representations for G/H”. As in Theorem 3.5, analogous results in this section hold for L^{p}-representations (1 ≤ p ≤ ∞), but we restrict to the case of p = 2 for simplicity.

We assume that G/K is an irreducible Hermitian symmetric space. Equivalently, G is simple and the center 𝔠(𝔨) of 𝔨 is one dimensional. Then we can take Z ∈ 𝔠(𝔨) so that

𝔤_{ℂ} = 𝔨_{ℂ} ⊕ 𝔭^{+} ⊕ 𝔭^{−}

are 0, \sqrt{−1} and −\sqrt{−1} eigenspaces of ad Z. For π ∈ Ĝ, we say that π is a highest weight module if there exists a non-zero vector in the underlying (𝔤, K)-module annihilated by 𝔭^{+}. We denote by Ĝ_{h.w.} (⊂ Ĝ) the unitary equivalence class of irreducible unitary highest weight modules. Then an element of Disc(G/H) ∩ Ĝ_{h.w.} is called a holomorphic discrete series representation for G/H. This terminology coincides with the usual one if H = {e}. Lowest weight modules and anti-holomorphic discrete series representations are defined similarly with 𝔭^{+} replaced by 𝔭^{−}.

Suppose τ is an involutive automorphism of G commuting with θ. Since τ𝔠(𝔨) = 𝔠(𝔨), there are two exclusive possibilities:

(4.1.1)  τZ = Z,
(4.1.2)  τZ = −Z.

We note that the induced action of τ on G/K is holomorphic for (4.1.1); antiholomorphic for (4.1.2).

\underline{4.2.} Retain the setting of §4.1. Let σ be an involutive automorphism of G satisfying σθ = θσ and σZ = −Z (see (4.1.2)), and x ∈ K. We consider the following two settings: Setting 1: L_{x} := {g ∈ G : xσ(g)x^{−1} = g}. Setting 2: Let τ be an involutive automorphism of G satisfying (4.1.1) and G′ := G^{τ}. We put H′_{x} := G^{τ} ∩ xG^{σ}x^{−1}. Here is an existence theorem of holomorphic discrete series representations for homogeneous manifolds of reductive type.

Theorem 4.2. For any x ∈ K, we have

♯ (Disc(G/L_{x}) ∩ Disc(G) ∩ Ĝ_{h.w.}) = ∞  in Setting 1,
♯ (Disc(G′/H′_{x}) ∩ Disc(G′) ∩ Ĝ′_{h.w.}) = ∞  in Setting 2.

\underline{4.3.} A very special case (i.e. x = e in Setting 1) leads to a new and elementary proof of the following result due to Ólafsson and Ørsted:

Corollary 4.3 (see [13]). There exist (infinitely many) holomorphic discrete series representations for a symmetric space of Hermitian type.

\underline{4.4.} Choosing x ∈ K, τ and σ, we can obtain a number of new holomorphic discrete series representations for homogeneous manifolds of reductive type (G/L_{x} and G′/H′_{x}). For instance, we have:

Example 4.4. The homogeneous manifolds

G/H = Sp(2n, ℝ)/Sp(n_{0}, ℂ) × GL(n_{1}, ℂ) × ⋯ × GL(n_{k}, ℂ) (∑ n_{j} = n)

admit holomorphic discrete series representations. We note that G/H is a semisimple symmetric space if and only if n_{1} = n_{2} = ⋯ = n_{k} = 0, which is previously the known case.

A detail proof will appear in [8] and [9].

References

[1] M. Flensted-Jensen, Discrete series for semisimple symmetric spaces, Ann. of Math. 111 (1980), 253–311.

[2] , Analysis on Non-Riemannian Symmetric Spaces, Conf. Board, 61, A. M. S., Providence R. I., 1986, ISBN 0-8218-0711-0.

[3] A. Knapp and D. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, 1995, ISBN 0-691-03756-6.

[4] T. Kobayashi, Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds U(p, q; 𝔽)/U(p − m, q; 𝔽), vol. 462, Memoirs of A.M.S., Providence R. I., 1992, ISBN 0-8218-2524-0.

[5] T. Kobayashi, Discrete decomposability of the restriction of A_{𝔮}(λ) with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), 181-205; Part II (preprint); III (to appear in Invent. Math.).

[6] T. Kobayashi, The Restriction of A_{𝔮}(λ) to reductive subgroups. I, II, Proc. Acad. Japan 69 (1993), 262-267; 71 (1995) 24-26.

[7] T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory (in Japanese), Sugaku of Math. Soc. Japan 46 (1994), 124-143, Amer. Math. Soc. Translations, Series 2, (in press).

[8] T. Kobayashi, Invariant measures on homogeneous manifolds of reductive type, Jour. reine und angew. Math. (to appear).

[9] T. Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. (to appear).

[10] T. Matsuki, Double coset decompositions of algebraic groups arising from two involutions I, J. Algebra 175 (1995), 865–925.

[11] , Double coset decompositions of reductive Lie groups arising from two involutions, preprint, 67 pages (1995).

[12] T.Matsuki and T.Oshima, A description of discrete series for semisimple symmetric spaces, Advanced Studies in Pure Math. 4 (1984), 331-390.

[13] G. Ólafsson and B. Ørsted, The holomorphic discrete series of an affine symmetric space, I, J. Funct. Anal. 81 (1988), 126–159.

印刷の方へ

次の文字はドイツ語の小文字です。特に 𝔨 と 𝔱 に御注意ください。

𝔤 𝔥 𝔨 𝔩 𝔭 𝔮 𝔱 𝔲
g h k l p q t u

𝔱^{−σ} はドイツ語小文字の t の右肩にマイナスシグマ (ギリシャ文字) がのったものです。同様に G^{τ} は英語大文字の G の右肩にタウ (ギリシャ文字) がのったものです。

ギリシャ文字の ω (オメガ) は使われていません。数式の中に w (英語小文字) はしばしば現れます。

ℂ_{λ} は, 大文字の C の黒板書体 ℂ (またはゴシック体 C) の右下にギリシャ文字 λ が添え字となっているものです。