The Real K-Theory of Compact Lie Groups

Let $G$ be a compact, connected, and simply-connected Lie group, equipped with a Lie group involution $\sigma_G$ and viewed as a $G$-space with the conjugation action. In this paper, we present a description of the ring structure of the (equivariant) $KR$-theory of $(G, \sigma_G)$ by drawing on previous results on the module structure of the $KR$-theory and the ring structure of the equivariant $K$-theory.

The main results of [13] are stated in the following Theorem 2.4. Let G be a compact connected Lie group with torsion-free fundamental group. Then 1. K * (G) is torsion-free.
1. Although the definition of δ G (ρ V ) given by Brylinski-Zhang in [9] is incorrect, their proof of Theorem 2.7 can be easily corrected by using the correct definition as in Definition 2.5, which does not affect the validity of the rest of their arguments, and Theorem 2.7 still stands.
Proof . If f ∈ Hom G (V, σ * G V ), then it is anti-linear on V and f (gv) = σ G (g)f (v) for g ∈ G and v ∈ V . The assumption that f 2 = Id V just says that f is an involution. So V together with σ V = f is a Real representation of G. Proposition 2.15. Let V be an irreducible complex representation of G and suppose that V ∼ = σ * G V . Let f ∈ Hom G (V, σ * G V ). Then 1. f 2 = k Id V for some k ∈ R.
2. There exists g ∈ Hom G (V, σ * G V ) such that g 2 = Id V or g 2 = − Id V . Proof . Note that f 2 ∈ Hom G (V, V ). By Schur's lemma, f 2 = k Id V for some k ∈ C. On the other hand, But f is an anti-linear map on V . It follows that k = k and hence k ∈ R. For part (2), we may first simply pick an isomorphism f ∈ Hom G (V, σ * G V ). Then f 2 = k Id V for some k ∈ R × . Schur's lemma implies that any g ∈ Hom G (V, σ * G V ) must be of the form g = cf for some c ∈ C. Then g 2 = cf • cf = ccf 2 = |c| 2 k Id V . Consequently, if k is positive, we choose c = 1 √ k so that g 2 = Id V ; if k is negative, we choose c = 1 √ −k so that g 2 = − Id V . Proposition 2.16. Let V be an irreducible Real representation of G.

The commuting field of
3. The commuting field of V is isomorphic to H iff V ∼ = W ⊕ σ * G W as complex G-representations, where W is an irreducible complex G-representation and there exists f ∈ Hom G (V, σ * G V ) such that f 2 = − Id V , and σ V (w 1 , w 2 ) = (w 2 , w 1 ). Proof . One can easily establish the above proposition by modifying the proof of Proposition 3 in Appendix 2 of [6], which is a special case of the above proposition where σ G is trivial. Proposition 2.17.
1. The map i : RR(G) → R(G) which forgets the Real structure is injective.

Any complex G-representation V which is a Real representation can only possess a unique
Real structure up to isomorphisms of Real G-representations.
, which is easily seen to be true because of the Real G-representation isomorphism It follows that ρ • i amounts to multiplication by 2 on RR(G), and is therefore injective because RR(G) is a free abelian group generated by irreducible Real representations. Hence i is injective.
(2) is simply a restatement of (1). Proposition 2.17 makes it legitimate to regard RR(G) as a subring of R(G). From now on we view [V ] ∈ R(G) as an element in RR(G) if V possesses a compatible Real structure.
Proposition 2.18. Let G be a compact Real Lie group. Let V be an irreducible complex representation of G. Then Proof . By Proposition 2.16, V is a Real representation of real type iff there exists f ∈ where , is a G-invariant Hermitian inner product on V . It can be easily seen that B is G-invariant, symmetric and non-degenerate. Conversely, given a G-invariant symmetric nondegenerate bilinear form on V ×σ * G V and using equation (2.3), we can define f ∈ Hom(V, σ * G V ) G , which squares to identity. Part (2) and (3) The abelian group generated by classes of irreducible complex representation of type F is denoted by R(G, F).
Definition 2.20. If V is a complex G-representation equipped with an anti-linear endomorphism J V such that J V (gv) = σ G (g)J(v) and J 2 = − Id V , then we say V is a Quaternionic representation of G. Let Rep H (G) be the category of Quaternionic representations of G. A morphism between V and W ∈ Rep H (G) is a linear transformation from V to W which commutes with G and respect J V and J W . We denote Mor(V, W ) by Hom G (V, W ) (J V ,J W ) . An irreducible Quaternionic representation is an irreducible object in Rep H (G, σ G ). The Quaternionic representation group of G, denoted by RH(G), is the Grothendieck group of Rep H (G). Let RH(G, F) be the abelian group generated by the isomorphism classes of irreducible Quaternionic representations with F as the commuting field.
Remark 2.21. The tensor product of two Quaternionic representations V and W is a Real representation as J V ⊗ J W is an anti-linear involution which is compatible with σ G . Similarly the tensor product of a Real representation and a Quaternionic representation is a Quaternionic representation. To put it succinctly, RR(G)⊕RH(G) is a Z 2 -graded ring, with RR(G) being the degree 0 piece and RH(G) the degree −1 piece. Later on we will assign RH(G) with a different degree so as to be compatible with the description of the coefficient ring KR * G (pt).
Proposition 2.22. RH(G), as an abelian group, is generated by the following where V is an irreducible complex representation of real type with J(u, w) = (−w, u). Its commuting field is H.
where V is an irreducible complex representation of complex type with J(u, w) = (−w, u). Its commuting field is C.

[V ]
, where V is an irreducible complex representation of quaternionic type. Its commuting field is R.
Proof . The proof proceeds in the same fashion as does the proof for Proposition 2.16. 1. The map j : RH(G) → R(G) which forgets the Quaternionic structure is injective.
2. Any complex G-representation which is a Quaternionic representation can only possess a unique Quaternionic structure up to isomorphisms of Quaternionic G-representations.
Proof . The proof proceeds in the same fashion as does the proof for Proposition 2.17. It suffices to show that, if η : . This is true because of the Quaternionic G-representation isomorphism Example 2.25. We shall illustrate the similarities and differences of the various aforementioned representation groups with an example. Let G = Q 8 × C 3 , the direct product of the quaternion group and the cyclic group of order 3, equipped with the trivial involution. There are 5 irreducible complex representations of Q 8 , namely, the 4 1-dimensional representations which become trivial on restriction to the center Z of Q 8 and descend to the 4 1-dimensional representations of Q 8 /Z ∼ = Z 2 ⊕ Z 2 , and the 2-dimensional faithful representation. We denote these representations by 1 Q 8 , ρ (0,1) , ρ (1,0) , ρ (1,1) and ρ Q respectively. Similarly, we let 1 C 3 , ρ ζ and ρ ζ 2 be the three 1-dimensional complex irreducible representations of C 3 . It can be easily seen that It follows that Some representations above should be equipped with suitable Real or Quaternionic structures given in Propositions 2.16 and 2.22. For example, the Real structure of ρ Q ⊗1 C 3 ⊕ ρ Q ⊗1 C 3 in RR(G, H) is given by swapping the two coordinates.

KR-theory
KR-theory was first introduced by Atiyah in [2] and used to derive the 8-periodicity of KOtheory from the 2-periodicity of complex K-theory. KR-theory was motivated by the index theory of real elliptic operators.

1.
A Real space is a pair (X, σ X ) where X is a topological space equipped with an involutive homeomorphism σ X , i.e. σ 2 X = Id X . We will sometimes suppress the notation σ X and simply use X to denote the Real space, if there is no danger of confusion about the involutive homeomorphism. A Real pair is a pair (X, Y ) where Y is a closed subspace of X invariant under σ X .
2. Let R p,q be the Euclidean space R p+q equipped with the involution which is identity on the first q coordinates and negation on the last p-coordinates. Let B p,q and S p,q be the unit ball and sphere in R p,q with the inherited involution.

3.
A Real vector bundle (to be distinguished from the usual real vector bundle) over X is a complex vector bundle E over X which itself is also a Real space with involutive homeomorphism σ E satisfying A Quaternionic vector bundle (to be distinguished from the usual quaternionic vector bundle) over X is a complex vector bundle E over X equipped with an anti-linear lift σ E of σ X such that σ 2 E = − Id E . 4. Let X be a Real space. The ring KR(X) is the Grothendieck group of the isomorphism classes of Real vector bundles over X, equipped with the usual product structure induced by tensor product of vector bundles over C. The relative KR-theory for a Real pair KR(X, Y ) can be similarly defined. In general, the graded KR-theory ring of the Real pair (X, Y ) is given by The ring structure of KR * is extended from that of KR, in a way analogous to the case of complex K-theory. The number of graded pieces, which is 8, is a result of Bott periodicity for KR-theory (cf. [2]).
Note that when σ X = Id X , then KR(X) ∼ = KO(X). On the other hand, if X × Z 2 is given the involution which swaps the two copies of X, then KR(X × Z 2 ) ∼ = K(X). Also, if X is equipped with the trivial involution, then KR(X × S 2,0 ) ∼ = KSC(X), the Grothendieck group of homotopy classes of self-conjugate bundles over X (cf. [2]). In this way, it is natural to view KR-theory as a unifying thread of KO-theory, K-theory and KSC-theory.
On top of the Real structure, we may further add compatible group actions and define equivariant KR-theory.

A
Real G-vector bundle E over a Real G-space X is a Real vector bundle and a G-bundle over X, and it is also a Real G-space.
3. In a similar spirit, one can define equivariant KR-theory KR * G (X, Y ). Notice that the Gactions on B 0,q and S 0,q in the definition of KR −q G (X, Y ) are trivial.
1. Let K * (+) be the complex K-theory of a point extended to a Z 8 -graded algebra over Here β ∈ K −2 (+) is the class of the reduced canonical bundle on CP 1 ∼ = S 2 .
2. Let σ * X be the map defined on (equivariant) vector bundles on X by σ * In the following proposition, we collect, for reader's convenience, some basic results of KRtheory (cf. [15,Section 2]), some of which are stated in the more general context of equivariant KR-theory. Proposition 2.29.
be the homomorphism which forgets the Real structure of Real vector bundles, and r : (1) is given in [15,Section 2]. The proof of (2) is the same as in the nonequivariant case, which is given in [2].
be the corresponding K-theory constructed using Quaternionic G-bundles over X.
By generalizing the discussion preceding Lemma 5.2 in [15] to the equivariant and graded setting, we define a natural transformation One can easily show by generalizing the discussion in the last section of [5] that The module structure of KR-theory of compact simply-connected Lie groups The following structure theorem for KR-theory, due to Seymour, is crucial in his computation of KR * (pt)-module structure of KR * (G).
Theorem 2.32 ( [15,Theorem 4.2]). Suppose that K * (X) is a free abelian group and decomposed by the involution σ * X into the following summands Then, as KR * (pt)-modules, where F is the free KR * (pt)-module generated by h 1 , . . . , h n .
Remark 2.33. If T = 0, then the conditions in Theorem 2.32 are equivalent to K * (X) being free abelian and c : KR * (X) → K * (X) being surjective. In this special case the theorem implies that This smacks of the definition of weakly equivariant formality (cf. Remark 2.8) and inspires us to define a similar notion for equivariant KR-theory (cf. Definition 4.2). We say a real space is real formal if it satisfies the conditions of Theorem 2.32.
. For any Real space X, KR −1 (X) is isomorphic to the abelian group of equivariant homotopy classes of maps from X to U (∞) which respect σ X and σ R on U (∞). Similarly, KR −5 (X), which is isomorphic to KH −1 (X) by Proposition 2.31, is isomorphic to the abelian group of equivariant homotopy classes of maps from X to U (2∞) which respect σ X and σ H on U (2∞) (cf. remarks in the last two paragraphs of Appendix of [15]). We can define maps analogous to those in Definition 2.1 in the context of KR-theory.
If ρ ∈ RH(G), then δ H (ρ) can be similarly represented, with the Real structure replaced by the Quaternionic structure.
In fact Theorems 2.4 and 2.32 also yield the following description of module structure of KR-theory of a compact connected Real Lie group with torsion-free fundamental group with a restriction on the types of the Real representations.
As we see from Theorem 2.37 and Corollary 2.39, to get a full description of the ring structure of KR * (G), it remains to figure out δ R (ϕ i ) 2 and δ H (θ j ) 2 . We will, in the end, obtain formulae for the squares by way of computing the ring structure of KR * G (G) and applying the forgetful map. In particular, we will show that δ R (ϕ i ) 2 and δ H (θ j ) 2 in general are non-zero. So, unlike the complex K-theory, KR * (G) is not an exterior algebra in general. Nevertheless, KR * (G) is not far from being an exterior algebra, in the sense of the following Corollary 2.41.
1. KR * (pt) 2 , which is the ring obtained by inverting the prime 2 in KR * (pt), is isomorphic to 2 , which is the ring obtained by inverting the prime 2 in KR * (G), is isomorphic to, as KR * (pt) 2 -algebra 3 The coef f icient ring KR * G (pt) In this section, we assume that G is a compact Real Lie group, and will prove a result on the coefficient ring KR * G (pt). In [5], all graded pieces of KR * G (pt) were worked out using Real Clifford G-modules. We record them in the following Proposition 3.1. KR −q G (pt), as abelian groups, for 0 ≤ q ≤ 7, are isomorphic to RR(G), RR(G)/ρ(R(G)), R(G)/j(RH(G)), 0, RH(G), RH(G)/η(R(G)), R(G)/i(RR(G)) and 0 respectively, where the maps i, j, ρ, η are as in Propositions 2.17 and 2.24.
is an isomorphism of graded rings.
Proof . The proposition follows by verifying the isomorphism in different degree pieces against the description in Proposition 3.1. For example, in degree 0, (3) follows from Proposition 2.29.
Remark 3.4. In [5], KR * G (X), where the G-action is trivial, is given as the following direct sum of abelian groups where KC * (X) and KH * (X) are Grothendieck groups of the so-called 'Complex vector bundles' and 'Quaternionic vector bundles' of X. We find Proposition 3.3, which is motivated by this description, better because the ring structure of the coefficient ring is more apparent when cast in this light. The proposition is, as we will see in the next section, a consequence of a structure theorem of equivariant KR-theory (Theorem 4.5), and therefore still holds true if the point is replaced by any general space X with trivial G-action.

Equivariant KR-theory rings of compact simply-connected Lie groups
Throughout this section we assume that G is a compact, connected and simply-connected Real Lie group unless otherwise specified. We will prove the main result of this paper, Theorem 4.33, which gives the ring structure of KR * G (G). Our strategy is outlined as follows.
1. We obtain a result on the structure of KR * G (G) (Corollary 4.10) which is analogous to Theorem 2.37 and Proposition 3.3. We define δ G R (ϕ i ), δ G H (θ j ), λ G k and r G ρ,i,ε 1 ,...,εt,ν 1 ,...,νt (cf. Definition 4.8 and Corollary 4.11), which generate KR * G (G) as a KR * G (pt)-algebra, as a result of Corollary 4.10. We show that (λ G k ) 2 = 0 (cf. Proposition 4.13). 2. We compute the module structure of KR * (U (n),σ F ) (U (n), σ F ) for F = R and H. 3. Let T be the maximal torus of diagonal matrices in U (n) and, by abuse of notation, σ R be the inversion map on T , σ H be the involution on U (n)/T (where n = 2m is even) defined by gT → J m gT . We show that the restriction map and the map induced by the Weyl covering map q G : U (2m)/T × T → U (2m), (gT, t) → gtg −1 , are injective.
1. Seymour first suggested the analogues of Steps 3, 4 and 5 in the ordinary KR-theory case in [15] in an attempt to compute δ R (ϕ i ) 2 and δ H (θ j ) 2 , but failed to establish Step 3, which he assumed to be true to make conjectures about δ R (ϕ i ) 2 .
2. In equivariant complex K-theory, K * G (G/T × T ) ∼ = K * T (T ) for any compact Lie group G, and the two maps p * G (the restriction map induced by the inclusion T → G) and q * G which is induced by the Weyl covering map are the same. If π 1 (G) is torsion-free, then these two maps are shown to be injective (cf. [9]. In fact it is even shown there that the maps inject onto the Weyl invariants of K * T (T )). In the case of equivariant KR-theory, things are more complicated. First of all, while in the case where (G, σ G ) = (U (n), σ R ), it is true that KR * G (G/T × T ) ∼ = KR * T (T ), and p * G and q * G are the same, it is no longer true in the case where (G, σ G ) = (U (2m), σ H ). In Step 3, we use q * G for the quaternionic type involution case because we find that it admits an easier description than p * G does. Second, we do not know whether p * G and q * G are injective for general compact Real Lie groups (equipped with any Lie group involution). For our purpose it is sufficient to show the injectivity results in Step 3. 2) X is a weakly equivariantly formal G-space, and 3) the forgetful map KR * G (X) → KR * (X) admits a section s R : KR * (X) → KR * G (X) which is a KR * (pt)-module homomorphism. Remark 4.3. If X is a weakly equivariantly formal G-space, then the forgetful map K * G (X) → K * (X) admits a (not necessarily unique) section s : K * (X) → K * G (X) which is a group homomorphism.
We first prove a structure theorem of equivariant KR-theory of Real equivariantly formal spaces.
Theorem 4.5. Let X be a Real equivariantly formal space. For any element a ∈ K * (X) (resp. a ∈ KR * (X)), let a G ∈ K * G (X) (resp. a G ∈ KR * G (X)) be a (Real) equivariant lift of a with respect to a group homomorphic section s (resp. s R which is a KR * (pt)-module homomorphism). Then the map is a group isomorphism. In particular, if R(G, C) = 0, then f is a KR * G (pt)-module isomorphism.
Proof . Consider the following H(p, q)-systems For the last H(p, q)-system, G acts on S q−p,0 trivially. The spectral sequences induced by these H(p, q)-systems converge to KR * (X), K * (X) and KR * G (X) respectively (for the assertion for the first two H(p, q)-systems, see the proofs of Theorem 3.1 and Lemma 4.1 of [15]. That the third H(p, q)-system converges to KR * G (X) follows from a straightforward generalization of the aforementioned proofs by adding equivariant structure throughout). Consider the two long exact sequences for the pair (X × B q−p,0 , X × S q−p,0 ), with the top exact sequence involving equivariant KR-theory and the bottom one ordinary KR-theory, and the vertical maps being forgetful maps. By applying the five-lemma, we have that each element in the first two H(p, q)systems has a (Real) equivariant lift. Define a group homomorphism As RR(G, R), RH(G, R) and R(G, C) are free abelian groups, and tensoring free abelian groups and taking cohomology commute, f is the abutment of f (p, q). On the E p,q 1 -page, f (p, q) becomes With the above identification, , which is an isomorphism by weak equivariant formality of X. It follows that f is also an isomorphism. If R(G, C) = 0, then by (1)    C.-K. Fok and KR * (U (n)×U (n), σ C ) ∼ = F ⊕r(K * (+)⊗T ), where F is the KR * (pt)-module freely generated by monomials in h 1 , . . . , h n . By Corollary 4.10, (3.4) in [2] and its equivariant analogue). Since η · r(·) = 0, we may assume that k i is from the component RR(U (n) × U (n), σ C , R) ⊗ F . But the degree −7 piece of the later is 0. So (h G i ) 2 = 0. Consider the map .
It can be easily seen that (γ Definition 4.14. Let σ n be (the class of) the standard representation of U (n).
Proof . For the involution σ R and ∧ i σ n , define the bilinear form Obviously the form is U (n)-invariant, symmetric and non-degenerate. By Proposition 2.18, each of ∧ i σ n , 1 ≤ i ≤ n is a Real representation of real type. Similarly, define, for the involution σ H and ∧ i σ 2m , a bilinear form So by Propositions 2.16, 2.18 and 2.22, ∧ i σ 2m is a Real representation of real type when i is even and a Quaternionic representation of real type when i is odd. There are no complex representations of complex type because ∧ i σ n ∼ = σ * F ∧ i σ n for F = R and H. and Putting all these together and applying Theorem 3.3, we get the desired conclusion. Remark 4.17. As ungraded KR * (pt)-modules, both

Injectivity results
This step involves proving that the restriction map p * G to the equivariant KR-theory of the maximal torus and the map q * G induced by the Weyl covering map are injective.
Lemma 4.18. Let G be a compact Lie group and X a G-space. Let i * 1 : K * G (X) → K * T (X) be the map which restricts the G-action to T -action. Then is injective for any ring R.
Proof . By [4,Proposition 4.9], i * 1 is split injective. So is i * 1 ⊗ Id R for any ring R.
where τ j are the weights of ρ.
Proof . Let V be the vector space underlying the representation ρ. δ T (ρ) is represented by the complex of T -equivariant vector bundles which, on restricting to is decomposed into a direct sum of complexes of 1-dimensional T -equivariant vector bundles, each of which corresponds to a weight of ρ.
Lemma 4.20. Let G be a simply-connected, connected compact Lie group and ρ 1 , . . . , ρ l be its fundamental representations. Then Proof . It suffices to show that k f is injective for 1 ≤ k ≤ l. Suppose I ⊆ {1, . . . , l}, |I| = k, m I := i∈I m i and f (m I ) := i∈I f (m i ). If |I|=k r I m I ∈ ker( k f ), then for any J with |J| = k, Hence |I|=k r I m I = 0 and the conclusion follows.
is injective for any ring R.
Proof . Note that is an R(T )-algebra isomorphism (cf. [12,Theorem 4.4]). Using Theorem 2.7, we have that K * T (G) is isomorphic, as an R(T )-algebra, to * R(T ) M , where M is the R(T )-module freely generated by δ T (ρ 1 ), . . . , δ T (ρ l ). We also observe that K * T (T ) is isomorphic, as an R(T )-algebra, to * R(T ) N , where N is the R(T )-module freely generated by δ( 1 ), . . . , δ( l ). Note that the hypotheses of Lemma 4.21 are satisfied by f = i * 2 ⊗ Id Zm for any m ≥ 2, as r δ( i ) (by Lemma 4.20) is indeed nonzero for any nonzero r in Z m (the coefficients of d G are either 1 or −1, so after reduction mod m rd G is still nonzero). Now that i * 2 ⊗ Z m is injective, so is i * 2 ⊗ Id R for any ring R.
On the other hand, From the above equations we obtain By Proposition 4.23, Theorem 4.30. Let G be a Real compact Lie group. Then The result now follows from Proposition 4.29.
To further express δ G R (∧ 2 ϕ i ) and δ G R (∧ 2 θ j ) in terms of the module generators associated with the fundamental representations, we may use the following derivation property of δ G R and δ G H . Proof . We refer the reader to the proof of Proposition 3.1 of [9] with the definition of δ G (ρ) given there (which is incorrect) replaced by the one in Definition 2.5. One just need to simply check that the homotopy ρ s in the proof for t ≥ 0 intertwines with both σ R and σ H .
By Proposition 4.31 Now the first equation is immediate. The second and third equations can be derived similarly.
The sign can be determined using formulae in (2) of Proposition 2.29.
Proof . Only (4.6) and (4.7) need explanation, but they are just equivariant analogues of Corollary 2.39 and follow from (2) of Proposition 2.29 and Remark 4.9.