REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 1, Pages 31–36 (November 4, 1996) S 1088-4165(97)00004-6

COHOMOLOGY OF CLASSIFYING SPACES AND HERMITIAN REPRESENTATIONS

GEORGE LUSZTIG

Abstract. It is shown that a large part of the cohomology of the classifying space of a Lie group satisfying certain hypotheses can be obtained by a difference construction from hermitian representations of that Lie group. This result is relevant to the study of Novikov’s higher signatures.

1. Statement of the Theorem

1.1. Let G be a connected Lie group. We set H_{G}^{k} = H^{k}(B_{G}, C) (cohomology with complex coefficients) where B_{G} is a classifying space of G. We set H_{G}^{∗} = H_{G}^{0} × H_{G}^{1} × H_{G}^{2} × …; we regard this as a topological C-algebra in which a fundamental system of neighbourhoods of 0 is provided by the subspaces 0×0×…×0×H_{G}^{l} ×H_{G}^{l+1} ×… for various integers l ≥ 0.

For any continuous finite dimensional complex representation V′ of G, we can form the associated complex vector bundle on B_{G} and consider its Chern character ch_{V′} ∈ H_{G}^{∗}. It is well known that, in the case where G is compact, the elements ch_{V′} (for various V′ as above) span over C a dense subspace of H_{G}^{∗}.

1.2. How to extend this result to not necessarily compact groups?

Let V be a finite dimensional C-vector space with a given non-degenerate hermitian form on which G acts linearly and continuously, preserving the hermitian form. We associate to V an element c̃h_{V} ∈ H_{G}^{∗} as follows.

We choose a maximal compact subgroup K of G. We can find a direct sum decomposition V = V^{+} ⊕ V^{−} with V^{+}, V^{−} orthogonal to each other for the hermitian form such that V^{+}, V^{−} are K-invariant subspaces and the hermitian form is positive definite on V^{+} and negative definite on V^{−}. The Chern characters ch_{V^{+}} ∈ H_{K}^{∗}, ch_{V^{−}} ∈ H_{K}^{∗} are then well defined since V^{+}, V^{−} are representations of K. We may identify H_{K}^{∗} = H_{G}^{∗} since the inclusion K → G induces a homotopy equivalence B_{K} ⟶^{∼} B_{G}; we define

c̃h_{V} = ch_{V^{+}} − ch_{V^{−}} ∈ H_{G}^{∗}.

Note that the element c̃h_{V} is independent of the choices of K and of the decomposition V = V^{+} ⊕ V^{−}, since the set of these choices is a contractible space.

Our main result is the following:

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors August 13, 1996 and, in revised form, August 21, 1996.

1991 Mathematics Subject Classification. Primary 20G99.

Supported in part by the National Science Foundation.

Theorem 1.3. Assume that G is the group of R-rational points of a connected, simply connected semisimple algebraic group G over C with a given R-structure. Assume also that G possesses some compact Cartan subgroup. Then the elements c̃h_{V} for various V as above span over C a dense subspace of H_{G}^{∗}.

In the case where G is a real symplectic group, this was proved in [L] where an application to the study of Novikov’s higher signatures (for discrete cocompact subgroups of G) was given. Gromov [G, p.139-140] realized that the result of [L] has also interesting differential-geometric applications, and asked the author (in January 1995) whether the more general statement of the Theorem above might be true. This provided the impetus for the present work.

2. Proof of the Theorem

2.1. Let G be as in 1.3 and let 𝔤 be the Lie algebra of G (over C). Let Z_{G} be the centre of G. Let T be a maximal torus of G. We choose standard Chevalley generators e_{i}, f_{i}, h_{i}(i ∈ I) for 𝔤 such that {h_{i}|i ∈ I} is a C-basis of the Lie algebra of T.

Let π: G → G be the involutive automorphism of G such that the tangent map dπ: 𝔤 → 𝔤 is given by

dπ(e_{i}) = −f_{i}, dπ(f_{i}) = −e_{i}, dπ(h_{i}) = −h_{i}

for all i ∈ I. Let ^{−}: G → G be the antiholomorphic involution of G whose tangent map is the conjugate-linear map ^{−}: 𝔤 → 𝔤 given by ē_{i} = e_{i}, f̄_{i} = f_{i}, h̄_{i} = h_{i} for all i ∈ I. Clearly, π: G → G commutes with ^{−}: G → G. It is well known that

K = {g ∈ G|π(ḡ) = g}

is a maximal compact subgroup of G.

2.2. For any λ = (λ_{i})_{i∈I} ∈ N^{I}, let V_{λ} be a finite dimensional C-vector space with a non-zero vector η on which G acts linearly as an algebraic group such that the corresponding representation of 𝔤 on V_{λ} is irreducible and satisfies e_{i}η = 0, h_{i}η = λ_{i}η for all i. Note that (V_{λ}, η) is uniquely determined by λ up to unique isomorphism. It follows easily that there is a unique conjugate-linear involution x ↦ x̄ of V_{λ} such that \overline{ax} = āx̄ for all a ∈ 𝔤, x ∈ V_{λ} and η̄ = η. It is well known and easy to prove that there is a unique hermitian form ⟨, ⟩: V × V → C (linear in the first variable, conjugate-linear in the second variable) such that

⟨η, η⟩ = 1 and ⟨ax, y⟩ = ⟨x, −dπ(ā)y⟩ for all x, y ∈ V_{λ} and all a ∈ 𝔤. It is well known that this hermitian form is positive definite.

From the definition we deduce the identity:

⟨gx, y⟩ = ⟨x, π(ḡ^{−1})y⟩

for all x, y ∈ V_{λ} and all g ∈ G.

2.3. Let σ ∈ T be such that σ^{2} ∈ Z_{G}. We have (a) σ̄ = σ^{−1},

(b) π(σ) = σ^{−1}. Indeed, it is easy to check that, if t ∈ T is of finite order, then t̄= t^{−1}. Since Z_{G} is a finite group, σ is of finite order. Hence (a) holds. Clearly, for any t ∈ T, we have π(t) = t^{−1}. Hence (b) holds.

COHOMOLOGY OF CLASSIFYING SPACES AND HERMITIAN REPRESENTATIONS 33

2.4. We define an antiholomorphic map τ: G → G by

g ↦ τ(g) = σπ(ḡ)σ^{−1}.

It is easy to see that τ is an involution. Let

G = {g ∈ G|τ(g) = g}. (a)

Then G is the group of real points for an R-rational structure on G. Note that

G ∩ T = {t ∈ T |τ(t) = t} = {t ∈ T |tt̄= 1}

is a compact Cartan subgroup of G. Thus, G satisfies the assumptions of Theorem 1.3. Conversely, it is known that any G as in the assumptions of Theorem 1.3 can be obtained by the previous construction. (For example, σ = 1 gives rise to a compact G.) Hence it suffices to prove Theorem 1.3 for G given by (a).

2.5. Let G^{σ} be the centralizer of σ in G. Let

K = G^{σ} ∩ G = G^{σ} ∩ K = G ∩ K.

We have

σ ∈ K.

Indeed, from 2.3(a),(b) we see that σ ∈ K. Clearly, σ ∈ G^{σ}; our assertion follows. The following result is well known.

Lemma 2.6. (a) K is a maximal compact subgroup of G^{σ}.

(b) K is a maximal compact subgroup of G.

2.7. Since η is an eigenvector for the T -action on V_{λ}, we have

ση = δη

for some δ ∈ C^{∗}. Applying ^{−} to the last equality we obtain

δ̄η = σ̄η = σ^{−1}η = δ^{−1}η;

hence

δ̄ = δ^{−1}.

By Schur’s lemma, σ^{2} acts as a scalar on V_{λ} and this scalar is necessarily δ^{2}. Thus,

σ^{2} = δ^{2}

on V_{λ}.

For x, y ∈ V_{λ} we set

⟨⟨x, y⟩⟩ = δ^{−1}⟨σx, y⟩.

Lemma 2.8. (a) ⟨⟨, ⟩⟩ is a non-degenerate hermitian form on V_{λ}.

(b) For any x, y ∈ V_{λ} and any g ∈ G, we have ⟨⟨gx, gy⟩⟩ = ⟨⟨x, y⟩⟩.

Let x, y ∈ V_{λ}. We must show that

⟨⟨x, y⟩⟩ = \overline{⟨⟨y, x⟩⟩}. (c)

The right hand side is

\overline{δ^{−1}⟨σy, x⟩} = δ̄^{−1}⟨x, σy⟩ = δ⟨x, σy⟩

while the left hand side is

δ^{−1}⟨σx, y⟩ = δ^{−1}⟨x, π(σ̄^{−1})y⟩ = δ^{−1}⟨x, π(σ̄^{−1})y⟩
= δ^{−1}⟨x, σ^{−1}y⟩ = ⟨x, δσ^{−1}y⟩ = ⟨x, δ^{−1}σy⟩ = δ⟨x, σy⟩.

Thus (c) is proved. The fact that ⟨⟨, ⟩⟩ is non-degenerate follows from the analogous property of ⟨, ⟩. Next we show that, for g in G, we have

⟨⟨gx, y⟩⟩ = ⟨⟨x, τ(g^{−1})y⟩⟩. (d)

The right hand side is

⟨⟨gx, y⟩⟩ = δ^{−1}⟨σgx, y⟩ = δ^{−1}⟨x, π(ḡ^{−1})π(σ̄^{−1})y⟩ = δ^{−1}⟨x, π(ḡ^{−1})σ^{−1}y⟩
= δ^{−1}⟨x, σ^{−1}τ(g^{−1})y⟩,

while the left hand side is

⟨⟨x, τ(g^{−1})y⟩⟩ = δ^{−1}⟨σx, τ(g^{−1})y⟩ = δ^{−1}⟨x, π(σ̄^{−1})τ(g^{−1})y⟩ = δ^{−1}⟨x, σ^{−1}τ(g^{−1})y⟩.

Thus (d) is proved. In the case where g ∈ G, identity (d) becomes ⟨⟨gx, y⟩⟩ = ⟨⟨x, g^{−1}y⟩⟩. Replacing here y by gy we obtain (b). The lemma is proved.

Lemma 2.9. Let V_{λ}^{+} = {x ∈ V_{λ}|σx = δx}, V_{λ}^{−} = {x ∈ V_{λ}|σx = −δx}. Then (a) V_{λ} = V_{λ}^{+} ⊕ V_{λ}^{−} and ⟨⟨V_{λ}^{+}, V_{λ}^{−}⟩⟩ = 0;

(b) ⟨⟨, ⟩⟩|_{V_{λ}^{+}} is positive definite and ⟨⟨, ⟩⟩|_{V_{λ}^{−}} is negative definite.

The first statement of (a) follows from the fact that σ^{2} = δ^{2} on V_{λ}. We have σ ∈ G (see 2.5) hence, by 2.8(b), σ acts as an isometry of ⟨⟨, ⟩⟩. Hence the ζeigenspace of σ is orthogonal under ⟨⟨, ⟩⟩ to the ζ′-eigenspace of σ provided that ζ′ζ̄ ≠ 1. The last condition is satisfied by ζ = δ, ζ′ = −δ since −δδ̄ = −1. This proves (a).

We prove (b). If x ∈ V_{λ}^{+} and x ≠ 0, we have ⟨⟨x, x⟩⟩ = δ^{−1}⟨σx, x⟩ = ⟨x, x⟩ > 0; if x ∈ V_{λ}^{−} and x ≠ 0, we have ⟨⟨x, x⟩⟩ = δ^{−1}⟨σx, x⟩ = −⟨x, x⟩ < 0. The lemma is proved.

2.10. Let W be the Weyl group of G with respect to T and let W′ be the Weyl group of G^{σ} with respect to T. We regard W′ naturally as a subgroup of W. Now W hence, by restriction, W′ acts on T by conjugation. This induces an action of W, hence of W′, through algebra automorphisms on O, the algebra of regular functions T → C. We will denote the action of w ∈ W on O by f ↦ w^{∗}f. Let O^{W} be the algebra of W -invariant elements of O; let O^{W′} be the algebra of W′-invariant elements of O. We have O^{W} ⊂ O^{W′} ⊂ O.

For any t ∈ T, the set of elements of O that vanish at t is a maximal ideal I_{t} of O; we denote the completion of O with respect to the maximal ideal I_{t} by Ô_{t}. If W_{t} is the stabilizer of t in W, then the W_{t}-action on O preserves I_{t}; hence it induces a W_{t}-action on Ô_{t}. Let (Ô_{t})^{W_{t}} be the space of invariants of this W_{t}-action. In particular, for t = σ we have W_{σ} = W′; hence (Ô_{σ})^{W′} is well defined. For t = 1, we have W_{1} = W; the W -action on Ô_{1} may be restricted to W′ and we denote by (Ô_{1})^{W′} the space of W′-invariants for this action.

For any continuous finite dimensional complex representation V′ of K, the function t → tr(t, V) on T ∩ K = T ∩ K = {t ∈ T |tt̄= 1} extends uniquely to a regular function χ_{V′}: T → C which belongs to O^{W′}.

By classical results on cohomology of classifying spaces [B], we may identify

H_{K}^{∗} = (Ô_{1})^{W′}

as topological algebras so that the following holds: For any continuous finite dimensional complex representation V′ of K, the element ch_{V′} ∈ H_{K}^{∗} corresponds to the image of χ_{V′} under the obvious imbedding j: O^{W′} → (Ô_{1})^{W′}.

If V_{λ} is as in 2.2, we have for any t ∈ T ∩ K

tr(t, V_{λ}^{+}) − tr(t, V_{λ}^{−}) = δ^{−1}tr(tσ, V_{λ}).

Let χ̃_{V_{λ}} ∈ O^{W′} be the function on T given by t ↦ δ^{−1}tr(tσ, V_{λ}). (This function is W′-invariant since σ is W′-invariant; note also that δ depends on λ.) We see that the element c̃h_{V_{λ}} ∈ H_{G}^{∗} = H_{K}^{∗} = (Ô_{1})^{W′} associated in 1.2 to the G-module V_{λ} (restriction from G to G) with its G-invariant hermitian form ⟨⟨, ⟩⟩ is precisely the image of χ̃_{V_{λ}} under the obvious imbedding j: O^{W′} → (Ô_{1})^{W′}. Now O^{W} is spanned as a vector space by the functions t ↦ tr(t, V_{λ}) for various λ as in 2.2. It follows that the subspace of H_{G}^{∗} = H_{K}^{∗} = (Ô_{1})^{W′} spanned by the elements c̃h_{V_{λ}} for various λ as in 2.2 is precisely the image of the composition

O^{W} ⟶^{p_{σ}} O^{W′} ⟶^{j} (Ô_{1})^{W′} (a)

where p_{σ} attaches to f ∈ O^{W} the function t ↦ f(tσ) in O^{W′}.

Hence Theorem 1.3 is a consequence of the following result, in which σ may be taken to be an arbitrary element of T.

Proposition 2.11. The image of the composition 2.10(a) is dense in (Ô_{1})^{W′}.

Consider the diagram

O^{W} ⟶^{α} (Ô_{σ})^{W′} ⟶^{β} (Ô_{1})^{W′}

where α is induced by the obvious imbedding O → Ô_{σ} and β is the isomorphism induced by the isomorphism Ô_{σ} → Ô_{1} which comes from the translation by σ on T (an isomorphism of varieties which takes 1 to σ). It is clear that β ∘ α is equal to the composition 2.10(a). Since β is an isomorphism of topological algebras, it is therefore enough to show that the image of α is dense in (Ô_{σ})^{W′}. Let f ∈ (Ô_{σ})^{W′}. Let X be the W -orbit of σ in T. For each t ∈ X, we define an element f_{t} ∈ Ô_{t} as follows. We choose w ∈ W such that w(t) = σ. Now w^{∗}: O → O induces an isomorphism w^{∗}: Ô_{σ} ⟶^{∼} Ô_{t} and we set f_{t} = w^{∗}(f). This element is independent of the choice of w since f is W′-invariant. Note that f_{σ} = f.

Let Î_{t} be the maximal ideal of Ô_{t}. Let n ≥ 1 be an integer. By the Chinese remainder theorem, we can find ϕ ∈ O such that ϕ = f_{t} mod Î_{t}^{n} for all t ∈ X. For any w ∈ W, the function w^{∗}ϕ ∈ O satisfies again w^{∗}ϕ = f_{t} mod Î_{t}^{n} for all t ∈ X since, by definition, the family (f_{t})_{t∈X} is W -invariant in an obvious sense. Setting ϕ′ = ♯(W)^{−1} ∑_{w∈W} w^{∗}ϕ ∈ O, we deduce that ϕ′ = f_{t} mod Î_{t}^{n} for all t ∈ X. In particular, taking t = σ, we see that ϕ′ = f mod Î_{σ}^{n}. Since ϕ′ ∈ O^{W}, we see that O^{W} is dense in (Ô_{σ})^{W′}.

This completes the proof of the Proposition hence, that of Theorem 1.3.

References

[B] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. Math. 57 (1953), 115-207. MR 14:490e

[G] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21-st century, in honor of I. M. Gelfand, vol. II, Progr. in Math. 132, Birkhäuser, Boston, 1996. CMP 96:12

[L] G. Lusztig, Novikov’s higher signature and families of elliptic operators, J. Diff. Geom. 7 (1971), 225-256.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

   E-mail address: gyrui@math.mit.edu