THE REPRESENTATION RING OF A

COMPACT LIE GROUP REVISITED

by Bob Oliver

In his 1968 paper, Segal [Seg] used elliptic operators to construct induction homomorphisms R(H) ⟶ R(G) for an arbitrary pair H ⊆ G of compact Lie groups, and then applied this to prove (among other things) a detection result for when a class function on G is a character. In this paper, we give new proofs of these results, but in the reverse order. We begin in Section 1 by showing that a class function on G is a character if its restrictions to all finite subgroups of G are characters. Then, in Section 2, we first define induction homomorphisms Cl(H) ⟶ Cl(G) for class functions, and afterwards apply the results of Section 1 to show that they send characters to characters and hence define induction maps between the representation rings. This gives a construction of the induction homomorphisms which is more elementary than that of Segal (though also less elegant), in that it only assumes the standard theory of representations of a compact connected Lie group.

It is the results in Section 3 which, while more technical, provided the original motivation for this work. Let S_{P}(G) be the family of all p-toral subgroups of G (for all primes p), where a group is called p-toral if it is an extension of a torus by a finite p-group. Let R_{P}(G) be the inverse limit of the representation rings R(P) for all P ∈ S_{P}(G), where the limit is taken with respect to restriction and conjugation in G. This group R_{P}(G) was shown in [JO, Theorem 1.8] to be isomorphic to the Grothendieck group 𝕂(BG) of the monoid of vector bundles over BG; and the “restriction” homomorphism

rs_{G} : R(G) ⟶ R_{P}(G) = \underleftarrow{lim}_{P ∈S_{P} (G)} R(P)

is isomorphic to the natural homomorphism R(G) ⟶ 𝕂(BG) which sends a representation V to the bundle (EG×_{G}V)↓BG.

The main result of Section 3 is a description, for arbitrary G, of the cokernel of the homomorphism rs_{G}. In particular, we show that it is onto whenever G is finite or π_{0}(G) is a p-group; but that it is not surjective in general. Precise necessary and sufficient conditions for rs_{G} to be onto are given in Theorem 3.10, and several simpler sufficient conditions are given in Corollary 3.11. Note that rs_{G} is surjective if and only if bundles over BG have the following property: for each ξ↓BG there exist G-representations V, V′ such that ξ ⊕ (EG×_{G}V′) ≅ (EG×_{G}V) (since by [JO, Theorem 1.8], every bundle over BG is a summand of a bundle coming from a Grepresentation).

Typeset by AMS-TEX

In the above discussion, we have for simplicity dealt only with the complex representation rings. But most of the results are shown below for real as well as complex representations.

I would like to thank in particular Stefan Jackowski for his comments and suggestions about this work. Originally, Sections 1 and 3 were intended to go into our joint paper [JO], but then they grew to the point where we decided to publish them separately.

Section 1 Detection of characters

Throughout this section, G denotes a fixed compact Lie group, and G_{0} its identity connected component. Fix a maximal torus T ⊆ G, let W_{G} = N_{G}(T)/T denote its Weyl group, and let 𝔱 ⊆ 𝔤 denote the Lie algebras of T and G. For any Weyl chamber C ⊆ 𝔱, define

N_{G}(T, C) = {g ∈ N_{G}(T) | Ad(g)(C) = C},

and

N_{G}(T, ±C) = {g ∈ N_{G}(T) | Ad(g)(±C) = (±C)}.

Here, Ad(g) denotes the adjoint (conjugation) action of g on 𝔱 and 𝔤. We will see in Proposition 1.1 that N_{G}(T, C) has exactly one connected component for each connected component of G, and that every element of G is conjugate to an element of N_{G}(T, C). Then, in Proposition 1.2 below, we show that a (continuous) class function f ∈ Cl(G) is a character of G if and only if f|N_{G}(T, C) is a character, and that f is a real character of G (i.e., the character of a virtual ℝG-representation) if and only if f|N_{G}(T, ±C) is a real character. At the same time, we construct (Proposition 1.4) a one-to-one correspondence between the irreducible representations of G, and those irreducible representations of N_{G}(T, C) whose weights lie in the dual Weyl chamber C^{∗}. This generalizes the standard relationship, for a connected compact Lie group G, between the irreducible representations of G and those of T.

Afterwards, the detection result is extended to show that an element f ∈ Cl(G) is a character (real character) if and only if f|H is a character (real character) of H for each finite subgroup H ⊆ G. The classical theorem of Brauer for detecting characters on finite groups can then be applied to further restrict the class of finite subgroups of G which have to be considered.

We first recall the definition and basic properties of the Weyl chambers of a compact connected Lie group G. The set of irreducible representations (or irreducible characters) of T will be identified here with T^{∗} =^{def} Hom(T, S^{1}); which will in turn be regarded as a lattice in 𝔱^{∗} = Hom(𝔱, ℝ).

The roots of G (or of G_{0}) are the characters of the nontrivial irreducible summands of the adjoint representation of T on ℂ⊗_{ℝ}𝔤. They occur in pairs ±θ. Let R ⊆ T^{∗} ⊆ 𝔱^{∗} denote the set of roots of G. Any element x_{0} ∈ 𝔱 such that θ(x_{0}) ≠ 0 for all θ ∈ R determines a choice of positive roots

R_{+} = {θ ∈ R | θ(x_{0}) > 0}.

And this in turn determines a Weyl chamber

C = {x ∈ 𝔱 | θ(x) ≥ 0 ∀θ ∈ R_{+}} ⊆ 𝔱

and a dual Weyl chamber

C^{∗} = {x ∈ 𝔱^{∗} | ⟨θ, x⟩ ≥ 0 ∀θ ∈ R_{+}} ⊆ 𝔱^{∗}.

Here, in the definition of C^{∗}, ⟨−, −⟩ denotes any G-invariant inner product on 𝔤^{∗}. Note that C^{∗} is independent of the choice of inner product, since a G-invariant inner product is uniquely defined up to scalar on each simple component of G.

Proposition 1.1. Fix a maximal torus T ⊆ G and a Weyl chamber C ⊆ 𝔱, and set N = N_{G}(T, C). Then N ∩ G_{0} = T, N·G_{0} = G, and hence N/T ≅ G/G_{0}. Also, any element of G is conjugate to an element of N.

Proof. Recall that the Weyl group W_{G_{0}} = N_{G_{0}}(T)/T of G_{0} permutes the Weyl chambers of T simply and transitively (cf. [Ad, Lemma 5.13]). Hence each coset of N_{G_{0}}(T) in N_{G}(T) contains exactly one connected component of N = N_{G}(T, C); and so N ∩ G_{0} = T, N·G_{0} = G, and N/T ≅ G/G_{0}.

By [Bo, §5.3, Theorem 1(b)], any automorphism of G_{0} leaves invariant some maximal torus and some Weyl chamber in G_{0}. Hence, any element g∈G is contained in N_{G}(T′, C′) for some maximal torus T′ and some Weyl chamber C′ ⊆ T′; and T′ and T are conjugate in G_{0} (cf. [Ad, Corollary 4.23]). Since N_{G_{0}}(T)/T permutes the Weyl chambers for T transitively, there is a ∈ G_{0} such that T = aT′a^{−1} and C = aC′a^{−1}; and aga^{−1} ∈ N = N_{G}(T, C). □

When dealing with real representations, we need to distinguish between the different types of irreducible representations and characters. As usual, we say that a G-representation V (over ℂ) has real type if it has the form V ≅ ℂ⊗_{ℝ}V′ for some ℝG-representation V′; and that V has quaternion type if it is the restriction of an ℍG-representation. If V is irreducible and its character is real-valued, then V has real or quaternion type, but not both [Ad, Proposition 3.56]. By a real character will be meant the character of a virtual representation of real type (i.e., the difference of two representations of real type).

Proposition 1.2. Fix a maximal torus T ⊆ G and a Weyl chamber C ⊆ 𝔱. Then a continuous class function f : G ⟶ ℂ is a character of G if and only if f|N_{G}(T, C) is a character of N_{G}(T, C). And a continuous class function f : G ⟶ ℝ is a real character of G if and only if f|N_{G}(T, ±C) is a real character.

The proof of Proposition 1.2 will be given after that of Proposition 1.4 below. We first note some elementary conditions for f to be a character or a real character. In the following lemma, we adopt the usual notation, and write ⟨φ, ψ⟩ = ∫_{G} φ(g)ψ(g) for any pair of continuous functions φ, ψ : G ⟶ ℂ (where the integral is the Haar integral on G with measure 1).

Lemma 1.3. (a) A class function f ∈ Cl(G) is a character of G if and only if ⟨f, χ⟩ ∈ ℤ for each character χ of G.

(b) A class function f : G → ℝ is a real character of G if and only if f is a character, and ⟨f, χ_{V} ⟩ ∈ 2ℤ for each G-representation V of quaternion type.

(c) A class function f : G → ℝ is a real character of G if f is a character, and f|H is a real character of H for some H ◁ G of finite odd index.

Proof. For any pair W, V of complex G-representations,

⟨χ_{W} , χ_{V} ⟩ = dim_{ℂ}((W^{∗} ⊗_{ℂ} V)^{G}) = dim_{ℂ}(Hom_{ℂG}(W, V)) ∈ ℤ.

(Recall that χ_{W^{∗}}(g) = χ_{W} (g) for all g ∈ G.) Also, if W has real type and V has quaternion type, then Hom_{ℂG}(W, V) is a quaternion vector space, and so its complex dimension is even. This proves the “only if” parts of points (a) and (b).

Conversely, assume that f ∈ Cl(G) is such that ⟨f, χ⟩ ∈ ℤ for each character χ of G. Since the irreducible characters form an orthonormal set, we know that ⟨f, f⟩ ≥ ∑_{i=1}^{k} ⟨f, χ_{i}⟩^{2} for any set χ_{1}, …, χ_{k} of distinct irreducible characters. Since each ⟨f, χ⟩ ∈ ℤ, this shows that ⟨f, χ⟩ = 0 for all but finitely many irreducible chararcters χ; and so f = ∑_{χ} ⟨f, χ⟩·χ is a character of G by the Peter-Weyl theorem (cf. [Ad, Theorem 3.47]).

We now consider conditions for a real valued character to be a real character; or equivalently for a self-adjoint representation to be of real type. An irreducible G-representation (over ℂ) is of complex type if its character is not real valued; i.e., if V ≅ V̸^{∗}. It follows from [Ad, Theorem 3.57] that a G-representation V (over ℂ) is of real type if and only if it is a sum of irreducible representations of real type, of representations ℂ⊗_{ℝ}W ≅ W ⊕W^{∗} for W irreducible of complex type, and of representations ℂ⊗_{ℝ}W ≅ W ⊕W for W irreducible of quaternion type. If v = ∑_{i=1}^{k} n_{i}[V_{i}] ∈ R(G) has real valued character, where the V_{i} are distinct irreducible G-representations, then ∑_{i=1}^{k} n_{i}[V_{i}] = ∑_{i=1}^{k} n_{i}[(V_{i})^{∗}], and so each pair V_{i}, (V_{i})^{∗} occurs with the same multiplicity. Hence v has real type if 2|n_{i} for each i such that V_{i} has quaternion type. Since n_{i} = ⟨χ_{v}, χ_{V_{i}}⟩, this proves point (b).

It remains to prove point (c): that an element v ∈ R(G) with real valued character has real type if v|H has real type for some normal subgroup H ◁ G of finite odd index; we may assume that v is the class of an actual ℂ[G]-representation V. Since all irreducible ℂ[G/H]-representations, aside from the trivial one, have complex type (cf. [Ser, Exercise 13.12]), we can write ℂ[G/H] ≅ ℂ⊕W ⊕W^{∗} for some representation W. Since by assumption, V |H has real type and V^{∗} ≅ V, the isomorphism

Ind_{H}^{G}(V |H) ≅ ℂ[G/H] ⊗_{ℂ} V ≅ V ⊕ (W ⊗_{ℂ}V) ⊕ (W ⊗_{ℂ}V)^{∗} ≅ V ⊕ ℂ⊗_{ℝ}(W ⊗_{ℂ}V)

shows that V has real type. □

By a weight of the compact Lie group G is meant an element of the lattice T^{∗} ⊆ 𝔱^{∗}, regarded as an irreducible character of T. If V is any representation of G, then the set of “weights of V ” is defined to be the set of characters of irreducible components of V |T. Consider the partial ordering of the weights of G, where ϕ_{1} ≤ ϕ_{2} if ϕ_{1} is contained in the convex hull of the W_{G}-orbit of ϕ_{2} (cf. [Ad, Definition 6.23]). One of the basic theorems of representation theory says that if G is connected, then any irreducible G-representation V has a unique W_{G}-orbit of highest (maximal) weights, each of which occurs with multiplicity one. Furthermore, distinct irreducible representations have distinct orbits of higher weights, and every weight of G can be realized as the highest weight of some irreducible G-representation. Thus, the irreducible representations of any connected G are in one-to-one correspondence with the W_{G}-orbits of weights of G. And since any given dual Weyl chamber C^{∗} ⊆ 𝔱^{∗} contains exactly one element in each W_{G} orbit in 𝔱^{∗} (cf. [Ad, Corollary 5.16]), the irreducible representations of G are in one-to-one correspondence with the weights in C^{∗}. For more detail, see, e.g., [Ad, Theorem 6.33] or [BtD, Section VI.2].

Now assume that G is not connected. If V is an irreducible G-representation, and if V_{0} is any irreducible component of V |G_{0}, then V is an irreducible summand of Ind_{G_{0}}^{G}(V_{0}). Hence each irreducible summand of V |G_{0} is obtained from V_{0} by conjugation by some element of π_{0}(G); and there is still a uniquely defined W_{G}-orbit of highest weights for V. In this case, however, the highest weights can occur with multiplicity greater than one; and there can be several irreducible G-representations with the same orbit of highest weights.

In the next proposition, Irr(G) will denote the set of irreducible representations of G. Also, if N = N_{G}(T, C) (for any maximal torus T ⊆ G and any Weyl chamber C ⊆ 𝔱), then Irr(N, C^{∗}) denotes the set of irreducible representations of N whose weights all lie in the dual Weyl chamber C^{∗} of C. For any V ∈ Irr(G), mx_{C^{∗}}(V) ⊆ C^{∗} ∩ T^{∗} denotes the set of those maximal weights of the irreducible summands of V |G_{0} which lie in C^{∗}. And for any N-invariant set of weights Φ ⊆ T^{∗}, V ⟨Φ⟩ denotes the sum of all irreducible summands of V |T with weights in Φ, regarded as an N-representation.

Proposition 1.4. Fix a maximal torus T ⊆ G and a Weyl chamber C ⊆ 𝔱, and set N = N_{G}(T, C). Then there is a bijection

β_{G} : Irr(G) ⟶^{≅} Irr(N, C^{∗})  defined by  β_{G}([V]) = [V ⟨ mx_{C^{∗}}(V)⟩] (1)

and an isomorphism

β̄_{G} : R(G) ⟶^{≅} R(N, C^{∗})  defined by  β̄_{G}([V]) = [V ⟨C^{∗}⟩].  (2)

In particular, β_{G}(V) is always an irreducible summand of V |N having multiplicity one.

Proof. Fix an irreducible G_{0}-representation V_{0}, and let ϕ be the maximal weight of V_{0} lying in C^{∗}. Set Φ = (N/T)·ϕ ⊆ C^{∗}, the N/T -orbit of ϕ. Let (V_{0}) ⊆ Irr(G_{0}) denote the G/G_{0}-orbit of V_{0}, and let Irr(G, (V_{0})) denote the set of all irreducible G-representations with support in (V_{0}); i.e., the set of those irreducible G-representations V such that all irreducible summands of V |G_{0} lie in (V_{0}).

Let V_{ϕ} denote the (1-dimensional) irreducible representation with weight (character) ϕ; regarded as a subspace of V_{0}. Since G/G_{0} ≅ N/T, the uniqueness of maximal weights in C^{∗} shows that each irreducible component of (Ind_{G_{0}}^{G}(V_{0}))|G_{0} contains exactly one weight in Φ = (N/T)·ϕ (and with multiplicity one). Thus,

V_{0}⟨Φ⟩ = V_{ϕ}  and  Ind_{G_{0}}^{G}(V_{0})⟨Φ⟩ = Ind_{T}^{N}(V_{ϕ}).  (3)

So for any G-representation V′ with support in (V_{0}) (i.e., for any [V′] ∈ Irr(G, (V_{0}))), there is a commutative diagram

Hom_{G_{0}}(V_{0}, V′) ⟶_{≅}^{r_{1}} Hom_{T} (V_{ϕ}, V′⟨Φ⟩)^{F_{1}}↓≅^{F_{2}}↓≅
Hom_{G}(Ind_{G_{0}}^{G}(V_{0}), V′) ⟶^{r_{2}} Hom_{N}(Ind_{T}^{N}(V_{ϕ}), V′⟨Φ⟩)

(4)

where F_{1} and F_{2} are the Frobenius reciprocity isomorphisms, and r_{1} and r_{2} are defined by restriction to summands with weights in Φ. The one-to-one correspondence between irreducible G_{0}-representations and highest weights contained in C^{∗} shows that r_{1} is an isomorphism, and thus that r_{2} is also an isomorphism.

Now assume that V and V′ are two irreducible G-representations with support in (V_{0}). By Frobenius reciprocity again (Hom_{G}(Ind_{G_{0}}^{G}(V_{0}), V) ≅ Hom_{G_{0}}(V_{0}, V) ≠ 0), V is a summand of Ind_{G_{0}}^{G}(V_{0}). So by (3) and (4), for any [V], [V′] ∈ Irr(G, (V_{0})),

Hom_{N}(V ⟨Φ⟩, V′⟨Φ⟩) ≅ Hom_{G}(V, V′) ≅ \{ \begin{matrix}ℂ & if V ≅ W \\ 0 & if V ≅ W̸ . \end{matrix}

This shows that V ⟨Φ⟩ is N-irreducible for any [V] ∈ Irr(G, (V_{0})), and that V ⟨Φ⟩ ≅ V′⟨Φ⟩ if and only if V ≅ V′. And finally, any irreducible N-representation with support in Φ is a summand of Ind_{T}^{N}(V_{ϕ}) ≅ Ind_{G_{0}}^{G}(V_{0})⟨Φ⟩, and hence has the form V ⟨Φ⟩ for some V ∈ Irr(G_{0}, (V_{0})).

We have now shown that β_{Φ} : Irr(G, (V_{0})) ⟶^{≅} Irr(N, Φ), defined by setting β_{Φ}([V]) = [V ⟨Φ⟩], is a well defined bijection. Since the restriction to G_{0} of any irreducible G-representation is a sum of representations in just one G/G_{0}-orbit of irreducible G_{0}-representations, β_{G} : Irr(G) ⟶ Irr(N, C^{∗}) is the disjoint union of the β_{Φ} taken over all N/T -orbits Φ ⊆ (C^{∗}∩T^{∗}) and hence also a bijection. This proves point (1). At the same time, it shows that the homomorphism β̄_{G} : R(G) ⟶ R(N, C^{∗}) of (2) is an isomorphism, since its matrix with respect to the bases of irreducible representations is triangular with 1’s along the diagonal. □

We are now ready to prove that a class function is a (real) character if its restriction to N_{G}(T, C) (N_{G}(T, ±C)) is a (real) character.

Proof of 1.2. Complex case: Fix a continuous class function f : G ⟶ ℂ such that f|N is a character of N. We must show that f is a character of G. Let v_{0} ∈ R(N) be such that χ_{v_{0}} = f|N, let χ be the character of β̄_{G}^{−1}(v_{0}⟨C^{∗}⟩) ∈ R(G) (Proposition 1.4(2)), and set f′ = f−χ. By construction, f′|N is the character of an element v ∈ R(N) such that v⟨C^{∗}⟩ = 0. We will show that v = 0. It then follows that f′ = 0 (since every element of G is conjugate to an element of N), and hence that f = χ is a character of G.

Fix any ϕ ∈ T^{∗}, and let N_{ϕ} ⊆ N denote the subgroup of elements fixing ϕ. Choose any ψ ∈ interior(C^{∗})^{N} (N/T acts linearly on 𝔱^{∗} and leaves the dual Weyl chamber C^{∗} invariant). Then ϕ+ℝψ is not contained in the wall of any dual Weyl chamber (since ψ is not); and so there is a dual Weyl chamber C_{1}^{∗} such that ϕ+ϵψ ∈ interior(C_{1}^{∗}) for small ϵ > 0. Let w ∈ W_{G} be any element such that w(C_{1}^{∗}) = C^{∗} (W_{G_{0}} permutes the Weyl chambers transitively). Then wϕ ∈ C^{∗}, since ϕ ∈ C_{1}^{∗}. Also, for any a ∈ N_{ϕ}, a(ψ) = ψ and a(ϕ) = ϕ by assumption, so a leaves C_{1}^{∗} invariant, and hence waw^{−1} leaves C^{∗} = w(C_{1}^{∗}) invariant. Thus wN_{ϕ}w^{−1} ⊆ N; and so v⟨wϕ⟩ = 0 ∈ R(wN_{ϕ}w^{−1}) since v⟨C^{∗}⟩ = 0 ∈ R(N). Since χ_{v} is constant on G-conjugacy classes (it is the restriction of a class function on G), it now follows that v⟨ϕ⟩ = 0 ∈ R(N_{ϕ}).

Let ϕ_{1}, …, ϕ_{k} ∈ T^{∗} be N/T -orbit representatives for the support of v, and write N_{i} = N_{ϕ_{i}} (the subgroup of elements which fix ϕ_{i}). Then v = ∑_{i=1}^{k} Ind_{N_{i}}^{N}(v⟨ϕ_{i}⟩). We have just seen that v⟨ϕ_{i}⟩ = 0 ∈ R(N_{i}) for each i, and hence v = 0.

Real case: Write N_{±} = N_{G}(T, ±C), for short. Fix a class function f : G ⟶ ℂ such that f|N_{±} is a real character. Then f is a character by Step 2, and f(G) ⊆ ℝ since any element of G is conjugate to an element of N ⊆ N_{±} (Proposition 1.1). By Lemma 1.3(b), we can assume (after replacing f by its sum with an appropriate real character) that f = χ_{V}, where V = ∑_{i=1}^{k} V_{i}, the V_{i} are distinct irreducible G-representations of quaternion type, and V |N_{±} is a representation of real type. We claim that V = 0 (i.e., that k = 0).

Assume otherwise: that k > 0. Choose a W_{G}-orbit Ψ of maximal weights in one of the V_{i} — say V_{1} — which does not occur in any of the others except possibly as maximal weights. Set Φ = Ψ∩C^{∗} and Φ_{±} = Ψ∩(±C^{∗}). By Proposition 1.4 (and the original assumption on Ψ), V_{1}⟨Φ⟩ is irreducible as an N-representation, and does not occur as a summand of V_{i}|N for any i ≠ 1. So the N_{±}-representation V′_{1} =^{def} V_{1}⟨Φ_{±}⟩ is irreducible — since

V′_{1}|N ≅ V_{1}⟨Φ⟩ ⊕ V_{1}⟨Φ_{±}∖Φ⟩

— and V′_{1} does not occur as a summand of V_{i}|N_{±} for any i ≠ 1. Also, since V_{1} is selfconjugate, the elements of Ψ, and hence of Φ_{±}, occur in pairs ±ϕ. This shows that V′_{1} = V_{1}⟨Φ_{±}⟩ is invariant under the conjugate linear automorphism j : V_{1} ⟶ V_{1}, and hence that it also has quaternion type. Thus, V |N_{±} contains with multiplicity one the irreducible summand V′_{1} of quaternion type, and this contradicts the assumption that V |N_{±} is a representation of real type. □

It remains to extend this criterium to a result which detects characters by restriction to finite subgroups of G. As usual, a finite group is called elementary if it is the product of a p-group (for some prime p) and a cyclic group. A finite group G is called ℝ-elementary if it is elementary, or if it contains a normal cyclic subgroup C ◁ G of 2-power index with the property that for any g ∈ G, conjugation by g acts on C via the identity or via (x ↦ x^{−1}).

Proposition 1.5. For any class function f : G ⟶ ℂ, f is a character of G if and only if its restriction to any finite elementary subgroup of G is a character; and f is a real character of G if and only if its restriction to each finite ℝ-elementary subgroup of G is a real character.

Proof. When G is finite, the proposition holds by the classical Brauer theorems for detecting characters of finite groups (cf. [Ser, Theorem 21 and Proposition 36]). So it will suffice to show that f is a (real) character of G if and only if its restrictions to all finite subgroups of G are (real) characters. By Proposition 1.2, it suffices to prove this when the connected component G_{0} of G is a torus.

Assume now that G_{0} = T is a torus. We can choose a sequence H_{1} ⊆ H_{2} ⊆ H_{3} ⊆ … of subgroups of G such that each H_{i} intersects all connected components of G, and such that the union of the H_{i} is dense in G. The simplest way to see this is to set n = |G/T |, let _{n}T ⊆ T denote the n-torsion subgroup, and note that the homomorphism H^{2}(G/T;_{n}T) ⟶ H^{2}(G/T; T) is surjective since n·H^{2}(G/T; T) = 0. Hence there is a subgroup H_{0} ⊆ G such that H_{0} ∩ T = _{n}T and ⟨H_{0}, T ⟩ = G; and we can define H_{k} = ⟨H_{0}, _{n·2^{k}}T ⟩ for each k > 0.

Let f ∈ Cl(G) be any class function whose restriction to each H_{i} is a character. For each character χ of G,

⟨f, χ⟩_{G} =^{def} ∫_{G} f·χ = lim_{i⟶∞}(\frac{1}{|H_{i}|} ∑_{g∈H_{i}} f(g)·χ(g)) = lim_{i⟶∞} ⟨f, χ⟩_{H_{i}}

(by definition of the Riemann integral); and ⟨f, χ⟩_{H_{i}} ∈ ℤ for each i since f|H_{i} is a character of H_{i}. Thus, ⟨f, χ⟩_{G} ∈ ℤ for each χ, and so f is a character of G by Lemma 1.3(a). And if f|H_{i} is a real character for each i, then f is real valued (the union of the H_{i} being dense in G), ⟨f, χ⟩_{G} = lim_{i⟶∞} ⟨f, χ⟩_{H_{i}} ∈ 2ℤ for each character χ of quaternion type by Lemma 1.3(b), and so f is a real character by Lemma 1.3(b). □

Section 2 Induction for representations of compact Lie groups

Again, throughout the section, G denotes a fixed compact Lie group. We construct an induction homomorphism R(H) ⟶ R(G), for an arbitrary closed subgroup H ⊆ G, by first defining it between the groups of class functions, and then using the results of Section 1 to show that it sends characters to characters.

The following lemma is useful for constructing continuous functions on G, and on certain closed subsets of G.

Lemma 2.1. Let F be any set of closed subgroups of G, closed under conjugation and closed in the space of all subgroups (with the Hausdorff topology). Set G_{F} = ∪_{H∈F}H: the union of the subgroups in F. Then for any function f : G_{F} ⟶ ℂ invariant under conjugation, f is continuous on G_{F} if f|H is continuous for all H ∈ F.

Proof. Fix any conjugation invariant function f : G_{F} ⟶ ℂ such that f|H is continuous for all H ∈ F. It will suffice to show, for any sequence g_{i} → g in G_{F}, that some subsequence of the f(g_{i}) converges to f(g). Since if f is not continuous at g, then there is ϵ > 0 and a sequence {g_{i}} in G_{F} converging to g such that |f(g_{i}) − f(g)| > ϵ for all i.

Fix such g_{i} and g; and for each i choose H_{i} ∈ F such that g_{i} ∈ H_{i}. Since F is closed in the space of closed subgroups of G, and since this space is compact (cf. [tD, Proposition IV.3.2(i)]), we can replace the g_{i} by a subsequence and assume that the H_{i} converge to some subgroup H ∈ F. By [tD, Theorem I.5.9], there exist elements a_{i} → e such that a_{i}H_{i}a_{i}^{−1} ⊆ H for i sufficiently large. And hence

lim_{i→∞} f(g_{i}) = lim_{i→∞} f(a_{i}g_{i}a_{i}^{−1}) = f(g)

since f|H is continuous. □

The next lemma is also rather technical, and will be used later to show that the induction homomorphism we define for class functions is well defined.

Lemma 2.2. Fix a closed subgroup H ⊆ G and an element g ∈ G, and let (G/H)^{g} be the fixed point set of the action of g on G/H. A coset aH ∈ G/H lies in (G/H)^{g} if and only if a^{−1}ga ∈ H. And if a_{1}H and a_{2}H lie in the same connected component of (G/H)^{g}, then a_{2}H = xa_{1}H for some x ∈ C_{G}(g). In particular, in this situation, a_{1}^{−1}ga_{1} is conjugate in H to a_{2}^{−1}ga_{2}.

Proof. For any a ∈ G, aH ∈ (G/H)^{g} if and only if gaH = aH, if and only if a^{−1}ga ∈ H. Also, if a_{2}H = xa_{1}H for any a_{1}, a_{2} ∈ G and any x ∈ C_{G}(g), then a_{1}^{−1}ga_{1} and a_{2}^{−1}ga_{2} are conjugate by an element of H.

Now fix an element aH ∈ (G/H)^{g}. Let C_{G}(g)_{0} be the identity connected component of the centralizer of g. We must show that the connected component of aH in (G/H)^{g} is C_{G}(g)_{0}·aH. Equivalently, via translation by a^{−1}, we must show that the connected component of eH in (G/H)^{a^{−1}ga} is C_{G}(a^{−1}ga)·eH. So upon replacing a^{−1}ga by g, we are reduced to the case where a = e and g ∈ H.

Let 𝔥 ⊆ 𝔤 denote the Lie algebras of H ⊆ G. For all x ∈ G, xH ∈ (G/H)^{g} if and only if xH = gxH = gxg^{−1}H. In particular, C_{G}(g)·H ⊆ (G/H)^{g}; and the tangent plane at eH to the manifold (G/H)^{g} is (𝔤/𝔥)^{Ad(g)} (the fixed point set of the adjoint action of g on 𝔤/𝔥). Also, the projection of 𝔤 onto 𝔤/𝔥 is split, equivariantly with respect to the action of the compact group H, and so 𝔤^{Ad(g)} surjects onto (𝔤/𝔥)^{Ad(g)}. Since 𝔤^{Ad(g)} is the Lie algebra of C_{G}(g), this shows that the two submanifolds C_{G}(g)_{0}·H ⊆ (G/H)^{g} have the same dimension, and hence that C_{G}(g)_{0}·H is the connected component of eH in (G/H)^{g}. □

We can now define the induction homomorphism for class functions, motivated by the formula given by Segal [Seg, p. 119].

Propositition 2.3. Let H ⊆ G be any closed subgroup. Then there is a homomorphism

Ind_{H}^{G} : Cl(H) ⟶ Cl(G)

determined (uniquely) by the following formula. Fix any g ∈ G, let F_{1}, …, F_{k} be the connected components of (G/H)^{g}, and choose elements a_{i}H ∈ F_{i}. Then for any f ∈ Cl(H),

Ind_{H}^{G}(f)(g) = ∑_{i=1}^{k} χ(F_{i})·f(a_{i}^{−1}ga_{i}).  (1)

Proof. Fix any f ∈ Cl(H). By Lemma 2.2, for each g ∈ G, Ind_{H}^{G}(f)(g) is independent of the choice of representatives a_{i}H for the components of (G/H)^{g}. Also, Ind_{H}^{G}(f) is conjugation invariant by definition; and it only remains to check that it is continuous.

Let F be the family of abelian subgroups of G. Clearly, F is closed in the Hausdorff topology, and its union is all of G. By Lemma 2.1, it will suffice to show that f|A is continuous for each A ∈ F. Let X be a connected component of some subgroup A ∈ F; we can assume that X generates π_{0}(A). For any g ∈ X, A/⟨g⟩ is connected (where ⟨g⟩ is the closure of the subgroup generated by g); and hence is a torus (or trivial). If (G/H)^{g} = ∐_{i=1}^{k} F_{i}, where the F_{i} are connected components, then (G/H)^{A} = ∐_{i=1}^{k}(F_{i})^{A/⟨g⟩} and χ((F_{i})^{A/⟨g⟩}) = χ(F_{i}) for each i. Thus, if we write (G/H)^{A} = ∐_{j=1}^{m} E_{j} (where the E_{j} are the connected components), and choose elements b_{j}H ∈ E_{j}, then

Ind_{H}^{G}(f)(g) = ∑_{j=1}^{m} χ(E_{j})·f(b_{j}^{−1}gb_{j}).

This formula holds for all g ∈ X, and shows that Ind_{H}^{G}(f) is continuous on X. □

The following double coset formula for induction and restriction of class functions is analogous to that shown by Feshbach [Fe] for equivariant cohomology theories. It was shown for representations by Snaith [Sn, Theorem 2.4], using Segal’s definition. We prove it here for class functions, using directly the definition in Proposition 2.3.

Lemma 2.4. Fix closed subgroups H, K ⊆ G, and write

K\G/H = ∐_{i=1}^{k} U_{i}

where each U_{i} is a connected component of one orbit type for the action of K on G/H. Fix elements a_{1}, …, a_{k} ∈ G such that Ka_{i}H ∈ U_{i}. For each i, let φ_{i} : Cl(H) ⟶ Cl(K) denote the composite

φ_{i} : Cl(H) ⟶^{Res} Cl(a_{i}^{−1}Ka_{i} ∩ H) ⟶^{conj(a_{i}^{−1})} Cl(K ∩ a_{i}Ha_{i}^{−1}) ⟶^{Ind} Cl(K).

Then, as functions from Cl(H) to Cl(K),

Res_{K}^{G} ∘ Ind_{H}^{G} = ∑_{i=1}^{k} χ^{♯}(U_{i})·φ_{i};  (1)

where for each i,

χ^{♯}(U_{i}) = χ(U_{i}, U_{i} ∖ U_{i}) = χ(U_{i}) − χ(U_{i} ∖ U_{i}).

Proof. Fix elements f ∈ Cl(H) and g ∈ K. We will compare the two maps in (1), when evaluated on a given class function f and a given element g.

Let Ũ_{i} ⊆ G/H denote the inverse image of U_{i} under the projection to K\G/H. Let F_{1}, …, F_{m} be the connected components of (G/H)^{g}. Thus, G/H = ∐_{i=1}^{k} Ũ_{i} and (G/H)^{g} = ∐_{j=1}^{m} F_{j}. For each i, j, set

V_{ij} = (K·a_{i}H) ∩ F_{j} ⊆ Ũ_{i} ∩ F_{j} ⊆ G/H

(note that V_{ij} need not be connected). Then the V_{ij} ⟶ Ũ_{i}∩F_{j} ⟶ U_{i} are fibration sequences, and so

χ(F_{j}) = ∑_{i=1}^{k} χ^{♯}(Ũ_{i} ∩ F_{j}) = ∑_{i=1}^{k} χ(V_{ij})·χ^{♯}(U_{i}).

for each j. Fix elements b_{ij} ∈ K, for each i, j, such that b_{ij}a_{i}H ∈ V_{ij}. Then by definition of the induction map (and Lemma 2.1),

(Res_{K}^{G} ∘ Ind_{H}^{G})(f)(g) = Ind_{H}^{G}(f)(g) = ∑_{i=1}^{k} ∑_{j=1}^{m} χ(V_{ij})·χ^{♯}(U_{i})·f(a_{i}^{−1}b_{ij}^{−1}gb_{ij}a_{i})

= ∑_{i=1}^{k} χ^{♯}(U_{i})· ∑_{j=1}^{m} χ(V_{ij})·(f ∘ conj(a_{i}^{−1}))(b_{ij}^{−1}gb_{ij}).

And for each i, if we set K_{i} = K∩a_{i}Ha_{i}^{−1} (the isotropy subgroup of the action of K on a_{i}H ∈ G/H), then (K/K_{i})^{g} ≅ (K·a_{i}H)^{g} = ∐_{j=1}^{m} V_{ij} (⊆ G/H); and so

∑_{j=1}^{m} χ(V_{ij})·(f ∘ conj(a_{i}^{−1}))(b_{ij}^{−1}gb_{ij}) = Ind_{K_{i}}^{K}(f ∘ conj(a_{i}^{−1}))(g) = φ_{i}(f)(g). □

When G is finite, the formula given in Proposition 2.3 is just the usual formula for the induction of characters (cf. [Ser, Theorem 12]). Hence by the double coset formula in Lemma 2.4, for each character (real character) χ of H and each finite subgroup K ⊆ G, (Ind_{H}^{G}(χ))|K is a character (or real character) of K. The detection result of Proposition 1.5 now applies to show:

Theorem 2.5. The homomorphism Ind_{H}^{G} of Proposition 2.3 sends characters to characters, and sends real characters to real characters. It thus restricts to homomorphisms

Ind_{H}^{G} : R(H) ⟶ R(G)  and  Ind_{H}^{G} : RO(H) ⟶ RO(G).  □

We leave it as an exercise to check, using the formula for induced characters given in [Seg, p. 119], that this induction homomorphism is the same as that defined by Segal in [Seg].

It is not hard to prove Frobenius reciprocity for induction and restriction of representations, using the definition given here. And that in turn implies, for example, that the induction map Ind_{N(T)}^{G} : R(N(T)) ⟶ R(G) is always surjective, and split by the restriction map. See also [Sn, Section 2.3] for the proofs of these results using Segal’s definition of induction.

Section 3 Representations supported by p-toral subgroups

Again, throughout this section, G will be a fixed compact Lie group, and G_{0} will denote its identity connected component. Let S_{P}(G) denote the family of p-toral subgroups of G, for all primes p. We now consider the groups

R_{P}(G) = \underleftarrow{lim}_{P ∈S_{P} (G)} R(P)  and  RO_{P}(G) = \underleftarrow{lim}_{P ∈S_{P} (G)} RO(P),

where the limits are taken with respect to inclusion and conjugation; and the natural “restriction” maps

rs_{G}^{U} : R(G) ⟶ R_{P}(G)  and  rs_{G}^{O} : RO(G) ⟶ RO_{P}(G).

These groups were shown in [JO] to be naturally isomorphic to the Grothendieck groups 𝕂(BG) and 𝕂𝕆(BG), respectively, of vector bundles over BG (and rs_{G}^{U} and rs_{G}^{O} are isomorphic to the natural homomorphisms R(G) ⟶ 𝕂(BG) and RO(G) ⟶ 𝕂𝕆(BG)).

The homomorphisms rs_{G} are shown here to split as a direct sums of homomorphisms between finitely generated groups, one for each G/G_{0}-orbit of irreducible G_{0}- representations, and the cokernel of each summand is computed (Theorem 3.9). In particular, this yields necessary and sufficient conditions for rs_{G}^{U} to be onto (Theorem 3.10 and Corollary 3.11). The orthogonal case seems to be much more complicated; but we do at least show that rs_{G}^{O} is onto whenever G is finite or π_{0}(G) has prime power order (Propositions 3.2 and 3.4), and then give some examples which show that rs_{G}^{O} can fail to be onto even when rs_{G}^{U} is onto.

It will be useful to define the “character” of an element of R_{P}(G). For any compact Lie group G, let G_{P} denote the union of the connected components in G of prime power order in π_{0}(G). Let Cl(G_{P}) denote the space of continuous functions G_{P} ⟶ ℂ invariant on conjugacy classes (i.e., the “class functions” on G_{P}).

Lemma 3.1. There is a (unique) character homomorphism

χ : R_{P}(G) ⟶ Cl(G_{P}),

such that for any v = (v_{P} )_{P ∈S_{P} (G)} ∈ R_{P}(G), χ(v)|P = χ_{v_{P}} for all P in S_{P}(G). Also, χ sends R_{P}(G) (RO_{P}(G)) isomorphically to the subgroup of those class functions on G_{P} whose restriction to each p-toral subgroup P ⊆ G, for all primes p, is a character of P (a real character of P).

Proof. Let F be the set of p-toral subgroups of G (for all primes p), whose identity connected component is a maximal torus of G. Clearly, F is closed in the Hausdorff topology (note that for P ∈ F, the order of π_{0}(P) is bounded by |N_{G}(T)/T |). And by Proposition 1.1, G_{P} is the union of the P ∈ F.

Now, for any v = (v_{P} )_{P ∈S_{P} (G)} ∈ R_{P}(G), define χ(v) : G_{P} → ℂ to be the union of the characters χ_{v_{P}}. This is well defined, and invariant under conjugation, by definition of the inverse limit. Also, χ(v) is continuous by Lemma 2.1, applied to the family F; and so χ(v) ∈ Cl(G_{P}).

The character homomorphism χ is clearly a monomorphism, and the descriptions of the images of R_{P}(G) and RO_{P}(G) are immediate from the construction. □

We are now ready to study the groups R_{P}(G) and RO_{P}(G), beginning with the following case.

Proposition 3.2. If π_{0}(G) has prime power order, then

rs_{G}^{U} : R(G) ⟶^{≅} R_{P}(G)  and  rs_{G}^{O} : RO(G) ⟶^{≅} RO_{P}(G)

are isomorphisms. Furthermore, for any G,

R_{P}(G) ≅ \underleftarrow{lim}_{P ∈F_{P} (G)} R(P)  and  RO_{P}(G) ≅ \underleftarrow{lim}_{P ∈F_{P} (G)} RO(P);  (1)

where F_{P}(G) denotes the family of subgroups H ⊆ G of finite index such that H/G_{0} has prime power order (and the limits are taken with respect to inclusion and conjugation).

Proof. If π_{0}(G) is a p-group for any prime p, then G = G_{P}, and so rs_{G}^{U} and rs_{G}^{O} are both monomorphisms by Lemma 3.1. To prove that they are isomorphisms, we must show that a class function f ∈ Cl(G) is a (real) character if its restriction to all p-toral subgroups of G is a (real) character.

Fix a maximal torus T and a Weyl chamber C ⊆ 𝔱, and set N = N_{G}(T, C). Then N is p-toral by Proposition 1.1; and by Proposition 1.2 a class function f ∈ Cl(G) is a character of G if f|N is a character of N. This shows that rs_{G}^{U} is an isomorphism. If π_{0}(G) is a 2-group, then N_{G}(T, ±C) is 2-toral, and the same argument shows that rs_{G}^{O} is an isomorphism. Finally, if π_{0}(G) is a p-group for an odd prime p, then for any v ∈ RO_{P}(G), χ(v) is a real valued character of G whose restriction to G_{0} is a real character of G_{0} (since π_{0}(G_{0}) = 1 is a 2-group); and so χ(v) is a real character of G by Lemma 1.3(c).

This finishes the proof of the first statement above. The formulas in (1) now follow immediately (by the transitivity of inverse limits). □

The importance of the formulas in (1) above is that they show that the groups R_{P}(G) and RO_{P}(G), and also the maps rs_{G}^{U} and rs_{G}^{O}, split as sums of groups and maps indexed by the irreducible representations of the identity component G_{0}. This will be made more explicit in Theorem 3.9 below.

The next proposition describes how standard induction techniques apply to study R_{P}(G) and rs_{G}^{U}. Recall that a finite group Γ is p-elementary if it is a product of a p-group and a cyclic group, and is elementary if it is p-elementary for some prime p. Also, Γ is 2-ℝ-elementary if it contains a normal cyclic subgroup C_{m} of 2-power index such that any element of G either centralizes C_{m} or acts on it via (a ↦ a^{−1}); and is ℝ-elementary if it is elementary or 2-ℝ-elementary.

Proposition 3.3. (a) For any subgroup H ⊆ G of finite index, there is an induction homomorphism

Ind_{H}^{G} : R_{P}(H) ⟶ R_{P}(G),

with the property that for any v ∈ R_{P}(H) and any g ∈ G_{P},

χ(Ind_{H}^{G}(v))(g) = (Ind_{H}^{G}(χ(V)))(g) = ∑_{aH∈(G/H)^{g}} χ(v)(a^{−1}ga).  (1)

(b) Let E(G) and E_{ℝ}(G) denote the sets of subgroups E ⊆ G of finite index such that E/G_{0} is elementary or ℝ-elementary, respectively. Then restriction induces isomorphisms

Coker(rs_{G}^{U}) ⟶^{≅} \underleftarrow{lim}_{E∈E(G)} Coker(rs_{E}^{U}) and Coker(rs_{G}^{O}) ⟶^{≅} \underleftarrow{lim}_{E∈E_{ℝ}(G)} Coker(rs_{E}^{O});

where the limits are taken with respect to inclusion and conjugation in G.

Proof. We regard Ind_{H}^{G} as a homomorphism Cl(H_{P}) ⟶ Cl(G_{P}), defined via formula (1). Note that this is just the restriction to G_{P} of the formula given in Proposition 2.3 (though only in the case where [G:H] < ∞). In particular, the double coset formula of Lemma 2.4 applies in this situation.

(a) Fix any v ∈ R_{P}(H), and let χ = χ(v) ∈ Cl(H_{P}) be its character. We must show that Ind_{H}^{G}(χ) is the character of an element of R_{P}(G); or equivalently (by Lemma 3.1) that Ind_{H}^{G}(χ)|P is a character for all p-toral subgroups P ⊆ G (for all primes p). And for any such P, gP g^{−1} ∩ H is p-toral for each g ∈ G, so χ|(gP g^{−1} ∩ H) is a character, and hence Ind_{H}^{G}(χ)|P is a character of P by the double coset formula.

(b) Let F(G) be the class of subgroups of G of finite index. The functor H ↦ R(H/G_{0}) satisfies the double coset formula and Frobenius reciprocity relations for induction and restriction, and hence is a Green ring over F(G) in the sense of Dress [Dr]. Also, the double coset formula of Lemma 2.4 says that H ↦ R_{P}(H) and H ↦ Coker(rs_{H}^{U}) are both Mackey functors over F(G) (again in the sense of Dress); and both are modules over R(−/G_{0}) satisfying Frobenius reciprocity. Since R(G/G_{0}) is generated by induction from the R(E/G_{0}) for E ∈ E(G) [Ser, §10.5, Theorem 19], the “fundamental theorem” of Mackey functors and Green rings says that F (G) ≅ \underleftarrow{lim}_{E∈E(G)}(F (E)) for any such module over R(−/G_{0}). This is shown in [Dr, Propositions 1.1′ and 1.2], and a more direct proof is given in [Ol, Theorem 11.1].

Similarly, H ↦ RO_{P}(H) and H ↦ Coker(rs_{H}^{O}) are Mackey functors over F(G), and modules over RO(−/G_{0}) satisfying Frobenius reciprocity. Since RO(G/G_{0}) is generated by induction from the RO(E/G_{0}) for E ∈ E_{ℝ}(G) [Ser, §12.6, Theorem 27], the same argument applies to show that Coker(rs_{G}^{O}) ≅ \underleftarrow{lim}_{E∈E_{ℝ}(G)} Coker(rs_{E}^{O}). □

In fact, the induction map Ind_{H}^{G} : R_{P}(H) ⟶ R_{P}(G) is defined for any closed subgroup H ⊆ G, using the formula for induction of characters in Proposition 2.3. To see this, one must check, for any f ∈ Cl(H), that Ind_{H}^{G}(f)|G_{P} = 0 if f|H_{P} = 0. This would be immediate if we knew that H∩G_{P} ⊆ H_{P}; but that is not the case in general. The existence of the induction map is thus slightly more tricky than in the case where [G : H] < ∞, but is not difficult.

We now turn to the case of finite groups.

Proposition 3.4. If G is finite, then rs_{G}^{U} and rs_{G}^{O} are both surjective.

Proof. By Proposition 3.3(b), it suffices to show that rs_{G}^{U} is onto when G is elementary, and that rs_{G}^{O} is onto when G is ℝ-elementary. We do this in the orthogonal case only; the unitary case is similar (but simpler).

Assume that G is ℝ-elementary, and fix an element v = (v_{P} )_{P ∈S_{P} (G)} ∈ RO_{P}(G). In other words, v_{P} ∈ RO(P) for each p-subgroup P ⊆ G (for each prime p||G|); and by subtracting a constant character we can assume that χ_{v_{P}} (1) = 0 for each P. For each p||G|, write v_{p} = v_{Syl_{p}(G)} ∈ RO(Syl_{p}(G)). It will suffice to show that each v_{p} extends to an element v′_{p} ∈ R(G) whose character vanishes on all elements of order prime to p (then v = rs_{G}^{O}(∑ v′_{p})). This is clear if Syl_{p}(G) has a normal complement, since in that case v′_{p} can be taken to be the composite of v_{p} with a surjection G ↠ Syl_{p}(G).

The only case left to consider is that where p is odd, G is 2-ℝ-elementary, and Syl_{p}(G) has no normal complement. Set p^{k} = | Syl_{p}(G)|; then there is a surjection G ↠ D(2p^{k}), where D(2p^{k}) is dihedral of order 2p^{k}. One easily checks that any v_{p} ∈ RO(C_{p^{k}}) such that χ_{v_{p}}(1) = 0 extends to an element v′′_{p} ∈ RO(D(2p^{k})) such that χ_{v′′_{p}}(g) = 0 for all g of order prime to p. And hence if v′_{p} ∈ RO(G) is the composite of v′′_{p} with the surjection G ↠ D(2p^{k}), then v′_{p}| Syl_{p}(G) = v_{p} and χ_{v′_{p}} vanishes on all elements of order prime to p. □

Recall that for any torus T, we let 𝔱 denote the Lie algebra of T, and regard the group T^{∗} = Hom(T, S^{1}) of irreducible characters of T as a lattice in 𝔱^{∗} = Hom(𝔱, ℝ). The following definitions establish some of the notation which will be used when dealing with irreducible characters and representations of groups with torus identity component.

Definition 3.5. If G is a compact Lie group with identity component T, then the support of a G-representation V is the (G/T -invariant) subset Supp(V) ⊆ T^{∗} of all characters of irreducible summands of V |T. More generally, for any v ∈ R(G), Supp(v) ∈ T^{∗} is the union of the supports of the irreducible G-representations which occur in the decomposition of v. For any G/T -invariant subset Φ ⊆ T^{∗}, Irr(G, Φ) denotes the set of irreducible G-representations with support in Φ, and R(G, Φ) ⊆ R(G) denotes the subgroup of elements with support in Φ. For ϕ ∈ T^{∗}, we write (ϕ) for the G/T -orbit of ϕ (and write Irr(G, ϕ), etc., if ϕ is G/T -invariant). Finally, if V is any G-representation, then V ⟨Φ⟩ and V ⟨ϕ⟩ denote the largest summands of V with support in Φ or ϕ, respectively.

The descriptions of Coker(rs_{G}^{U}) in Lemma 3.8 and Theorem 3.9 below will be given in terms of a certain function δ(G), defined for compact Lie groups whose identity component is a torus and central.

Definition 3.6. Assume that G lies in a central extension 1 ⟶ T ⟶ G ⟶ Γ ⟶ 1, where T is a torus and Γ is a finite group. For each ϕ∈T^{∗}, define

δ(G, ϕ) = gcd{dim(V) | V ∈ Irr(G, ϕ)};

and set

δ(G) = lcm{δ(G, ϕ) | ϕ∈T^{∗}}.

The next lemma gives a partial description of this function, independantly of representations; and also lists some of its more technical properties which will be needed in later proofs.

Lemma 3.7. Assume that G_{0} ⊆ Z(G); i.e., that G lies in a central extension 1 ⟶ T ⟶ G ⟶ Γ ⟶ 1, where T is a torus and Γ is finite. Set e = expt(T ∩ [G, G]). For each prime p||Γ|, let G_{p} be a maximal p-toral subgroup of G: the extension of T by a Sylow p-subgroup of Γ. Then

(a) δ(G) = 1 if and only if e = 1, if and only if G ≅ T × Γ

(b) e|δ(G) and δ(G)^{2}||Γ|

(c) δ(G) = ∏_{p||Γ|} δ(G_{p}), and δ(G, ϕ) = ∏_{p||Γ|} δ(G_{p}, ϕ) for all ϕ∈T^{∗}

(d) δ(G, ϕ′) = δ(G, ϕ) for all ϕ′, ϕ ∈ T^{∗} with ϕ′ ≡ ϕ (mod e)

(e) δ(G, nϕ) = δ(G, ϕ) for all ϕ∈T^{∗}, and all n ∈ ℤ with (n, e) = 1.

Proof. Note first that for any H ⊆ G of finite index, and any ϕ ∈ T^{∗},

δ(H, ϕ)|δ(G, ϕ)|[G : H]·δ(H, ϕ).  (1)

The first relation holds since each G-representation with support in ϕ can be regarded as an H-representation; and the second since Ind_{H}^{G}(V) has support in ϕ for any Hrepresentation V with support in ϕ.

(b) Fix any ϕ ∈ T^{∗}, and choose a ∈ T ∩[G, G] such that ϕ(a) generates ϕ(T ∩[G, G]). Then for any G-representation V with support in ϕ, a acts on V via multiplication by ϕ(a); and since a ∈ [G, G], ϕ(a)· Id_{V} has determinant ϕ(a)^{dim(V)} = 1. Thus, |ϕ(a)|| dim(V) for all such V, and so

|ϕ(a)| = |ϕ(T ∩ [G, G])||δ(G, ϕ).  (2)

In particular, e = expt(T ∩ [G, G]) divides δ(G).

Now fix any ϕ ∈ T^{∗}, and let V_{ϕ} be the 1-dimensional irreducible T -representation with character ϕ. Let V_{1}, …, V_{k} be the irreducible G-representations with support in ϕ. For each i, the multiplicity of V_{i} in Ind_{T}^{G}(V_{ϕ}) is

dim_{ℂ}(Hom_{G}(Ind_{T}^{G}(V_{ϕ}), V_{i})) = dim_{ℂ}(Hom_{T} (V_{ϕ}, V_{i})) = dim_{ℂ} V_{i}.

Thus, |Γ| = dim(Ind_{T}^{G}(V_{ϕ})) = ∑_{i=1}^{k} dim(V_{i})^{2}. And so δ(G, ϕ), the greatest common divisor of the dim(V_{i}), is such that δ(G, ϕ)^{2}||Γ|.

(a) We prove here the slightly more general equivalence that

δ(G, ϕ) = 1 ⟺ ϕ(T ∩ [G, G]) = 1 ⟺ G/ Ker(ϕ) ≅ T/ Ker(ϕ) × Γ. (3)

The third statement clearly implies the first, and the first implies the second by (2).

By the universal coefficient theorem, H^{2}(Γ; T) ≅ Hom(H_{2}(Γ), T); and T ∩ [G, G] is the image of the homomorphism η_{G} : H_{2}(Γ) ⟶ T which corresponds to [G] as an element if H^{2}(Γ; T). So G ≅ T × Γ if T ∩ [G, G] = 1, and G/ Ker(ϕ) ≅ T/ Ker(ϕ) × Γ if ϕ(T ∩ [G, G]) = 1.

(c) This formula follows immediately from (1), and the fact that δ(G_{p}, ϕ)||G_{p}/T | is a power of p for each p.

(d) If ϕ ≡ 0 (mod e), then ϕ(T ∩[G, G]) = 1, and so δ(G, ϕ) = 1 by (3). If ϕ′ ≡ ϕ ≢ 0 (mod e), then the two composites

H_{2}(Γ) ⟶^{η_{G}} T ⟶_{ϕ′}^{ϕ} S^{1}

are equal. Hence (G/ Ker(ϕ), ϕ) ≅ (G/ Ker(ϕ′), ϕ′) as pairs, and δ(G, ϕ) = δ(G, ϕ′).

(e) For any n ∈ ℤ and any G-representation V with support ϕ, ψ^{n}(V) is a virtual representation with support nϕ: since χ_{ψ^{n}V} (gt) = χ_{V} (g^{n}t^{n}) = χ_{ψ^{n}V} (g)·ϕ(t)^{n} for any g∈G and t∈T. Cf. [Ad, Lemma 3.61] for details. Also, V and ψ^{n}(V) have the same (virtual) dimension, and hence δ(G, nϕ)|δ(G, ϕ). So by (d), δ(G, nϕ) = δ(G, ϕ) if n is invertible mod e. □

Ian Leary has pointed out to me that δ(G) is the greatest common divisor of the indices [G : H] of those subgroups H ⊆ G of finite index such that H splits as a product H ≅ T × (H/T).

Whenever G_{0} = T is a torus, R(G) splits as the direct sum, taken over all G/T - orbits (ϕ) ⊆ T^{∗}, of the subgroups R(G, (ϕ)) of finite rank. In a similar fashion, rs_{G}^{U} splits as the direct sum over all (ϕ) ⊆ T^{∗} of homomorphisms

rs_{G,(ϕ)} : R(G, (ϕ)) ⟶ R_{P}(G, (ϕ)).

We are now ready to describe the cokernel of each of these summands for such G. The key case to consider is that when T = G_{0} is central and ϕ is faithful.

Lemma 3.8. Assume that G lies in a central extension 1 ⟶ T ⟶ G ⟶^{σ} Γ ⟶ 1, where T ≅ S^{1}, and where Γ is finite. Fix a faithful (injective) character ϕ ∈ T^{∗}. Let S be the set of all conjugacy classes of elements g ∈ Γ such that no two elements in σ^{−1}g are conjugate; and let S_{P} ⊆ S be the set of conjugacy classes of elements of prime power order. For each g ∈ S_{P}, let η(g) be the largest divisor of δ(C_{G}(g), ϕ) which is prime to the order of g. Then

R(G, ϕ) ≅ ℤ^{|S|}, R_{P}(G, ϕ) ≅ ℤ^{|S_{P} |}, and Coker(rs_{G,ϕ}) ≅ ⨁_{1≠g∈S_{P}} ℤ/η(g).

Proof. A character χ of G has support in ϕ if and only if it satisfies the relation χ(gt) = χ(g)ϕ(t) for all g ∈ G and t ∈ T. In particular, since ϕ is injective, χ(g) = 0 for any g which is conjugate to gt for some 1 ≠ t ∈ T. Thus, Cl(G, ϕ) is a complex vector space of dimension |S|; and by the Peter-Weyl theorem (and the independence of irreducible characters) R(G, ϕ) is a free abelian group of rank |S|. Also, R_{P}(G, ϕ) is torsion free (it is detected by characters defined on G_{P}), and Ker(rs_{G,ϕ}) is the set of elements of R(G, ϕ) whose characters vanish on G_{P}. So the image of rs_{G,ϕ} is free of rank |S_{P}|; and once we have shown that rs_{G,ϕ} has finite cokernel it will follow that R_{P}(G, ϕ) is a free abelian group of the same rank.

The computation of the cokernel of rs_{G,ϕ} will be carried out in two steps.

Step 1 Assume first that Γ is p-elementary for some prime p. Then we can write G = C_{n} × P, where C_{n} is cyclic of order n prime to p, and where P is p-toral. In particular, R(G) ≅ R(C_{n})⊗R(P) and R(G, ϕ) ≅ R(C_{n})⊗R(P, ϕ). Let IR(−) denote the augmentation ideal of R(−), and similarly for IR_{P}(−). Consider the following commutative diagram with split short exact rows:

0 ⟶ IR(C_{n}) ⊗ R(P, ϕ) ⟶ R(G, ϕ) ⟶ R(P, ϕ) ⟶ 0
rs_{Cn} ⊗augm.↓  ^{rs_{G,ϕ}}↓  ^{=}↓
0 ⟶ IR_{P}(C_{n}) ⊗ ℤ ⟶ R_{P}(G, ϕ) ⟶ R(P, ϕ) ⟶ 0.

Here, IR_{P}(C_{n}) is the product of the IR(Syl_{q}(C_{n})) for q|n, and any v ∈ IR(Syl_{q}(C_{n})) lifts to an element of IR(C_{n}) whose character vanishes on other Sylow subgroups. Hence IR(C_{n}) surjects onto IR_{P}(C_{n}), and so

Coker(rs_{G,ϕ}) ≅ Coker(rs_{C_{n}} ⊗ augm.) ≅ IR_{P}(C_{n}) ⊗ Coker[R(P, ϕ) ⟶^{augm.} ℤ].

The cokernel of this augmentation map is by definition ℤ/δ(P, ϕ), and so

Coker(rs_{G,ϕ}) ≅ IR_{P}(C_{n}) ⊗ (ℤ/δ(P, ϕ)).  (1)

Step 2 Now assume that G is arbitrary. Let E(G) be the set of subgroups of G of finite index such that E/T is elementary, and (for each prime p||Γ|) let E_{p}(G) be the set of those E ∈ E(G) such that E/T is p-elementary. By Proposition 3.3, Coker(rs_{G,ϕ}) is the inverse limit of the groups Coker(rs_{E,ϕ}), taken over all E ∈ E(G). By (1), Coker(rs_{E,ϕ}) is a finite p-group for all E ∈ E_{p}(G). Hence Coker(rs_{G,ϕ}) is finite; and (for each p) Coker(rs_{G,ϕ})_{(p)} is the inverse limit of the Coker(rs_{E,ϕ}) for E ∈ E_{p}(G).

Fix a prime p||Γ|; we want to determine the p-power torsion in Coker(rs_{G,ϕ}). If K′ ⊆ K are finite cyclic subgroups of order prime to p, then the composite

IR(K′)_{(p)} ⟶^{Ind} IR(K)_{(p)} ⟶^{Res} IR(K′)_{(p)}  (2)

is multiplication by [K:K′], and hence an isomorphism. Thus, if K is cyclic of order prime to p, we can split

IR_{P}(K)_{(p)} = ⨁_{q||K|} IR(Syl_{q}(K)) ≅ ⨁_{1≠K′⊆K_{P}} ĨR(K′)_{(p)}

(i.e., taking the second sum over subgroups of prime power order). Here, ĨR(K′) ⊆ IR(K′) is the kernel of the map given by restriction to the subgroup of prime index, and is free with rank equal to the number of generators of K′.

For each n||Γ| prime to p, let Cyc_{n} be the set of all cyclic subgroups K ⊆ Γ of order n if n is a prime power, and set Cyc_{n} = ∅ otherwise. By Lemma 3.7(c), for any maximal p-toral subgroup P ⊆ H, δ(P, ϕ) is the largest power of p dividing δ(H, ϕ). So with the help of (1) we now get

Coker(rs_{G,ϕ})_{(p)} ≅ \underleftarrow{lim}_{E∈E_{p}(G)} Coker(rs_{E,ϕ})
≅ ⨁_{p∤n||Γ|}( \underleftarrow{lim}_{K∈Cyc_{n}}(ĨR(K) ⊗ ℤ/δ(σ^{−1}(C_{Γ}(K)), ϕ))_{(p)}).  (3)

For each n = q^{k} (where q ≠ p is prime), set

Cyc′_{n} = {K=⟨g⟩∈ Cyc_{n} | no two elts. in σ^{−1}g conjugate in G}.

Fix some K ∈ Cyc_{q^{k}} ∖ Cyc′_{q^{k}}, and let K′ ⊆ K be the subgroup of index q. Then there exists x ∈ N_{G}(σ^{−1}K) such that for each g ∈ σ^{−1}(K∖K′), xgx^{−1} = gt for some 1 ≠ t ∈ T. The character of any element v ∈ ĨR(K) ≅ ĨR(σ^{−1}K, ϕ) vanishes on σ^{−1}K′; and hence (since χ_{v}(gt) = χ_{v}(g)·ϕ(t)) v is fixed by the action of x only if v = 0. Thus, x acts on ĨR(K) with trivial fixed point set; and in particular such terms contribute nothing to the limit in (3).

Formula (3) thus reduces to a sum, over conjugacy class representatives for all K ∈ Cyc′_{n}, of the groups

H^{0}(N_{G}(σ^{−1}K); ĨR_{P}(K)) ⊗ (ℤ/δ(C_{G}(σ^{−1}K), ϕ))_{(p)}.

The first factor here is free of rank equal to the number of Γ-conjugacy classes of generators of K. The formula for Coker(rs_{G,ϕ}) now follows upon taking the product over all primes p||Γ|. □

As an example, consider the group G = C_{n} × (S^{1} ×_{C_{2}} Q(8)), where n is odd, Q(8) is a quaternion group of order 8, and the second product is taken while identifying the central elements of order 2 in S^{1} and Q(8). By Lemma 3.8, if ϕ ∈ T^{∗} is a generator, then rs_{G,kϕ} is onto for k even, while Coker(rs_{G,kϕ}) ≅ ℤ/2 ⊗ IR_{P}(C_{n}) ≠ 0 if k is odd.

The groups dealt with in Lemma 3.8 seem quite specialized, but we are now ready to show that the general case — for an arbitrary compact Lie group G — can always be reduced to the cases handled there.

Theorem 3.9. Let G be any compact Lie group. Fix a maximal torus T ⊆ G and a Weyl chamber C ⊆ 𝔱, and set N = N(T, C) ⊆ G. Then rs_{G}^{U} splits as a direct sum of homomorphisms

rs_{G,(V_{0})} : R(G, (V_{0})) ⟶ R_{P}(G, (V_{0})),

taken over all G/G_{0}-orbits (V_{0}) ⊆ Irr(G_{0}).

For any V_{0} ∈ Irr(G_{0}), let ϕ be the maximal weight of V_{0} in C^{∗}, let N_{ϕ} ⊆ N be the subgroup of elements which fix ϕ, and set K_{ϕ} = Ker(ϕ) ⊆ T. Then the assignment ([V] ↦ [V ⟨ϕ⟩]) induces isomorphisms

R(G, (V_{0})) ⟶^{≅} R(N_{ϕ}/K_{ϕ}, ϕ), R_{P}(G, (V_{0})) ⟶^{≅} R_{P}(N_{ϕ}/K_{ϕ}, ϕ),

and

Coker(rs_{G,(V_{0})}) ⟶^{≅} Coker(rs_{N_{ϕ}/K_{ϕ},ϕ}).

Proof. By Lemma 3.2, R_{P}(G) is the inverse limit of the representation rings R(H), taken over all H ⊆ G of finite index such that H/G_{0} has prime power order. Since each R(H) splits as a sum of finitely generated groups indexed by the G/G_{0}-orbits (V_{0}) ∈ Irr(G_{0}), we now see that R_{P}(G) also splits as such a sum. And hence rs_{G}^{U} also splits as a direct sum of homomorphisms rs_{G,(V_{0})}.

Now fix V_{0} ∈ Irr(G_{0}) and let ϕ be its maximal weight in C^{∗}. Write Φ = (ϕ) for short: the N/T -orbit of ϕ ∈ C_{T}^{∗}. By Proposition 1.4, the assignment [V] ↦ [V ⟨Φ⟩] defines a bijection from Irr(G, (V_{0})) to Irr(N, Φ), and hence an isomorphism R(G, (V_{0})) ⟶^{≅} R(N, Φ). Similarly, it induces isomorphisms R(H, (V_{0})) ⟶^{≅} R(H∩N, Φ) for each H ⊆ G of finite index, and upon taking the inverse limit over all such H for which H/G_{0} has prime power order we get an isomorphism R_{P}(G, (V_{0})) ⟶^{≅} R_{P}(N, Φ). And this in turn induces an isomorphism between the cokernels of rs_{G,(V_{0})} and rs_{N,Φ}.

The homomorphism R(N, Φ) ⟶ R(N_{ϕ}, ϕ) ≅ R(N_{ϕ}/K_{ϕ}, ϕ), defined by sending [V] to [V ⟨ϕ⟩], is an isomorphism: its inverse is the induction map [V] ↦ [Ind_{N_{ϕ}}^{N}(V)]. This

same assignment also defines an isomorphism R_{P}(N, Φ) ⟶^{≅} R_{P}(N_{ϕ}/K_{ϕ}, ϕ) (whose inverse is again the induction map); and hence defines an isomorphism between the cokernels of rs_{N,Φ} and rs_{N_{ϕ}/K_{ϕ},ϕ}. □

The above general description of Coker(rs_{G}^{U}) is rather complicated. In contrast, the conditions for the map rs_{G}^{U} to be onto can be formulated more simply.

Theorem 3.10. Let G be any compact Lie group. Fix a maximal torus T ⊆ G and a Weyl chamber C ⊆ 𝔱, and set N = N(T, C) ⊆ G. Let E′(N) denote the set of subgroups E ⊆ N of finite index such that E/T is elementary but not of prime power order. Then

expt(Coker(rs_{G}^{U})) = lcm{δ(E/[E, T]) | E ∈ E′(N)}.  (1)

In particular, rs_{G}^{U} is surjective if and only if rs_{N}^{U} is surjective, if and only if T ∩[E, E] = [E, T] for all E ∈ E′(N).

Proof. It is clear from part (c) that the exponent of Coker(rs_{G}^{U}) divides the number given in (1). To show that these are equal, fix any prime p, and choose E ⊆ N of finite index such that E/T is p-elementary but not a p-group. We must show that δ(E/[E, T])| expt(Coker(rs_{G}^{U})). Choose any ϕ′ ∈ (T/[E, T])^{∗} ⊆ T^{∗} such that δ(E/[E, T], ϕ′) = δ(E/[E, T]). Since N/T acts linearly on 𝔱^{∗} and leaves C^{∗} invariant, the fixed set (C^{∗})^{E} is a cone shaped subspace of (𝔱^{∗})^{E} with nonempty interior. Hence, we can choose ϕ ∈ C^{∗} ∩ (T/[E, T])^{∗} = (C_{T}^{∗})^{E} such that ϕ ≡ ϕ′ modulo the exponent of \frac{T ∩[E,E]}{[E,T]}. If q ≠ p is any other prime dividing |E/T |, then

δ(E/[E, T], qϕ) = δ(E/[E, T], ϕ) = δ(E/[E, T], ϕ′) = δ(E/[E, T])

by Lemma 3.7(d,e). And finally, if gT ∈ E/T is the element of order q, then gT ∈ S in the notation of Lemma 3.8: no two elements in gT/ Ker(qϕ) are conjugate. Thus,

δ(E/[E, T]) = δ(E/[E, T], qϕ)| expt(Coker(rs_{E,qϕ}))| expt(Coker(rs_{G}^{U}))

by Lemma 3.8; and this finishes the proof of formula (1). The necessary and sufficient conditions for rs_{G}^{U} to be surjective now follow from Lemma 3.7(a). □

Since the general condition for rs_{G}^{U} to be surjective is still somewhat complicated, we now list some special cases which are simpler to formulate.

Corollary 3.11. For any compact Lie group G, Coker(rs_{G}^{U}) has finite exponent, and

expt(Coker(rs_{G}^{U}))^{2}||π_{0}(G)|.  (1)

Furthermore, rs_{G}^{U} is surjective if G satisfies any of the following conditions:

(a) G is finite or connected.

(b) All elements of π_{0}(G) have prime power order.

(c) π_{0}(G) is a periodic group: all of its Sylow subgroups are cyclic or quaternion.

(d) Z(G_{0}) = 1.

(e) G is a semidirect product of the form G = G_{0} ⋊ Γ, where Γ ⊆ G normalizes some maximal torus T and leaves invariant some Weyl chamber in T.

Proof. Fix a maximal torus T ⊆ G_{0}, and a Weyl chamber C. Set N = N(T, C). As in Theorem 3.10, let E′(N) be the set of subgroups H ⊆ N of finite index such that H/T is elementary but not of prime power order.

By Lemma 3.7(b), δ(H/[T, H])^{2}||H/T |||π_{0}(G)| for each H ⊆ N of finite index. So (1) follows from Theorem 3.10.

(a) rs_{G}^{U} is onto by Lemma 3.4 if G is finite, and by (1) if G is connected.

(b) If all elements of π_{0}(G) = π_{0}(N) have prime power order, then E′(N) = ∅, and so rs_{G}^{U} is onto by Theorem 3.10.

(c) Note that H_{2}(Γ) = 0 for any finite periodic group Γ. Hence, if π_{0}(G) is periodic, then for any H ∈ E′(N), H/[H, T] ≅ T/[H, T] × H/T. So rs_{N}^{U} and rs_{G}^{U} are onto by Theorem 3.10.

(e) The conditions on Γ imply that N is a semidirect product of T with Γ, and hence that rs_{G}^{U} is onto by Theorem 3.10.

(d) By [Bo, §4.10, Corollaire], the surjection Aut(G_{0}) ↠ Out(G_{0}) is split by outer automorphisms which fix T and C. Let Γ ⊆ G be the subgroup of elements whose conjugation action lies in the image of any given splitting map. Then G = G_{0} ⋊ Γ (since G_{0} ∩ Γ = Z(G_{0}) = 1); and so rs_{G}^{U} is onto by (e). □

We remark here that G being a semidirect product G_{0} ⋊ Γ does not in itself imply that rs_{G}^{U} is onto. As an example, set

G = C_{3} × (SU(2)×_{C_{2}}Q(8)),

where C_{3} is cyclic of order 3, Q(8) is a quaternion group of order 8, and the product is taken by identifying the central subgroups of order 2 in SU(2) and Q(8). Then Theorem 3.10 applies to show that Coker(rs_{G}^{U}) has exponent 2. But SU(2)×_{C_{2}}Q(8) is also a semidirect product of SU(2) with C_{2}×C_{2}: the splitting comes from the diagonal subgroup

⟨(i, i)⟩ × ⟨(j, j)⟩ ⊆ Q(8)×_{C_{2}}Q(8) ⊆ SU(2)×_{C_{2}}Q(8).

So far, we have dealt mostly with the case of unitary representations. The general conditions for rs_{G}^{O} to be surjective seem to be much more complicated. For example, with a little more work, one can show that if G is a central extension of a torus by a finite group, then rs_{G}^{O} is onto if and only if rs_{G}^{U} is onto. In contrast, the following example provides a simple way of constructing groups G for which rs_{G}^{O} is not onto but rs_{G}^{U} is onto.

Example 3.12. Fix any pair (G′, V′), where G′ is a compact connected Lie group, and V′ an irreducible G′-representation of real type having the additional property that some central element z∈Z(G′) of order 2 acts on V′ by (− Id). Choose any odd prime power n > 1, and set G = G′×_{C_{2}}Q(4n): the central product of G′ with the quaternion group of order 4n, where z is identified with the central element of Q(4n). Then rs_{G}^{O} is not onto.

Proof. Let W be any effective irreducible representation of Q(4n), and set V = V′⊗_{ℂ}W. Then V is an irreducible G-representation of quaternion type, but its restriction to any p-toral subgroup of G (for any prime p) has real type. In particular,

[V] represents an element of RO_{P}(G); but since rs_{G}^{O} and rs_{G}^{U} are injective (all elements of π_{0}(G) ≅ D(2n) have prime power order), it does not lie in the image of rs_{G}^{O}. □

For example, we can take G′ = SO(2m) for any m ≥ 2, and let V′ be the standard G′-representation on ℂ^{2m}. Set G = G′×_{C_{2}}Q(4n), for some odd prime power n ≥ 3. Then rs_{G}^{O} : RO(G) ⟶ RO_{P}(G) fails to be onto, while rs_{G}^{U} is onto (in fact, an isomorphism) by Corollary 3.11(b) (all elements of π_{0}(G) have prime power order).

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