COHOMOLOGY OF LIE ALGEBRAS

3(b) Cohomology of Lie algebras

Let G be any connected semisimple Lie group, K ⊂ G( R ) a maximal compact subgroup, (ρ, V ) a finite-dimensional complex representation of G, Γ ⊂ G(R) a torsion-free discrete arithmetic subgroup. We let X = G(R)/K be the symmetric space attached to G and define the local system

V ˜ = Γ\(X × V )→X _Γ = Γ\X.

Here Γ acts diagonally on X × V . Note that Γ = π ₁ (X _Γ ) and ˜ V is a local system.

Define

C ^∞ ( ˜ V ) = {f ∈ C ^∞ (G(R), V ) | f (γg) = ρ(γ )f (g), γ ∈ Γ, g ∈ G(R)} = Ind ^G _Γ (V ).

(We write G for G(R) in much of this section.) This is a representation space for G and g, under the right regular representation R. There is an isomorphism of G-modules

C ^∞ ( ˜ V ) −→ ^∼ C ^∞ (Γ\G(R)) ⊗ V ; f 7→ F (g) = ρ(g) ⁻¹ f(g).

Indeed, if h ∈ G then R h (F )(g) = ρ(gh) ⁻¹ f (gh), hence under this isomorphism, R h (f ) maps to

[g 7→ ρ(g ⁻¹ )R h (f)(g) = ρ(h)·ρ((gh) ⁻¹ )R h (f )(g) = ρ(h)·R h F (g) = (R⊗ρ(h)F )(g)].

We want to compute the cohomology groups H ^∗ (Γ\G, V ˜ ) and especially H ^∗ (Γ\X, V ˜ ).

Now ˜ V is locally constant (in the euclidean topology) and so its cohomology is com- puted by the complex of global sections of the (twisted) de Rham complex:

0→ V ˜ →A ⁰ ( ˜ V ) →A ^{d 1} ( ˜ V ) → ^d . . . →A ^{d N} ( ˜ V )→0.

Here A ⁱ ( ˜ V ) is the locally constant sheaf that on a small ball U is just the C ^∞ differential i-forms on U with coefficients in V . Then

A ⁱ ( ˜ V ) := Γ(Γ\G, V ˜ ) = A ⁱ (G, V ) ^Γ .

Now A ⁱ (G, V ) is the space of smooth functions on G that at the point g takes V i

T G,g to V . But V i

T G,g = V i

g. So A ⁱ (G, V ) = C ^∞ (G, Hom(

^ i

g, V )) = Hom(

^ i

g, C ^∞ (G, V )).

We will compute exterior differentiation and then look at what this does to the Γ-invariant subspaces.

Typeset by AMS -TEX

1

Differential forms and differentials.

The exterior derivative of a differential form is calculated by linearity using the formula

d(f ω ₁ ∧ ω ₂ ∧ . . . ω q ) = df ∧ ω ₁ ∧ ω ₂ ∧ . . . ω q ) + X q

i=1

(−1) ^{i +1} f ω ₁ ∧ · · · ∧ dω i ∧ . . . ω q .

This simplifies if ω i = dx i for a coordinate system x 1 , . . . , x n , whose derivatives commute, but we have identified differentials with linear maps from the Lie alge- bra to functions, and elements of the Lie algebra certainly don’t commute. Let X ₁ , . . . , X N be a basis for g, ω ₁ , . . . , ω N the dual basis that parallelizes the cotan- gent space. Note in any case that if f ∈ C ^∞ (G) then df = P

X j (f )ω j . On the other hand, d ² f = 0, and this allows us to compute dω i for each i, as follows:

0 = d ² f = d( X

X j (f )ω j ) = X

j

d(X j (f ) ∧ ω j ) + X

k

X k (f )dω k

= X

i,j

X i ◦ X j (f )ω i ∧ ω j + X

k

X k (f)dω k

= X

i<j

(X i ◦ X j − X j ◦ X i )(f )ω i ∧ ω j + X

k

X k (f)dω k

= X

i<j

[X i , X j ](f )ω i ∧ ω j + X

k

X k (f)dω k .

Now we write [X i , X j ] = P

k c ^j _ij X k and thus 0 = X

k

X k (f)[ X

i<j

c ^k _ij ω i ∧ ω j + dω k ].

But this is true for any f , and by letting f vary, we see that the term in brackets vanishes, in other words

dω k = − X

i<j

c ^k _ij ω i ∧ ω j .

Continuing the manipulations, we eventually find that if

ω ∈ A ^q (G, V ) = Hom(

^ q

g, C ^∞ (G, V ))

dω(Y ₀ ∧ · · · ∧ Y q ) = X q

j =0

(−1) ^j Y j (ω(Y ₀ ∧ · · · ∧ Y ˆ j ∧ · · · ∧ Y q ))

+ X

r<s

ω([Y r , Y s ] ∧ Y ₀ ∧ · · · ∧ Y ˆ r ∧ · · · ∧ Y ˆ s ∧ · · · ∧ Y q )

The action Y j (ω(•)) is right-differentiation of functions. The complete calculation

can be found in Knapp, Lie Groups, Lie Algebras, and Cohomology, pp. 155-160.

The Lie algebra complex.

If (π, W ) is any (complex) U (g)-module, we can define a complex C ^• (g, W ) = Hom( V •

g, W ) with differential given by the same formula df (Y ₀ , . . . , Y q ) = X

(−1) ^j π(Y i )f (Y ₀ , . . . , Y ˆ j , . . . , Y q )+ X

r<s

f ([Y r , Y s ], Y ₀ , . . . , Y ˆ r , . . . , Y ˆ s , . . . , Y q ).

Let H ^q (g, W ) = ker(d q )/im(d q −1 ). Then

H ⁰ (g, W ) = {f ∈ Hom( C , W ) | df (X) = 0, ∀X ∈ g} = {f ∈ Hom( C , W ) | π(X ) = 0, ∀X ∈ g}.

In other words

H ⁰ (g, W ) = W ^g := Hom U ( g ) (C, W ).

Theorem. The functor W 7→ W ^g is left-exact and W 7→ H ^q (g, W ) are its right- derived functors.

Proof. Since W ^g := Hom _U(g) (C, W ), we know that the functor is left-exact and its right-derived functors are given by Ext ^q _U(g) (C, W ). So we need to identify C ^• (g, W ) with Hom U ( g ) (C ^• , W ) where C ^• is an acyclic resolution of C in the category of U (g)-modules. This will be sketched later in the setting of (g, K )-modules.

We certainly don’t want to compute the cohomology of C ^∞ (G, V ); but we have seen that

A ⁱ ( ˜ V ) = A ⁱ (G, V ) ^Γ = Hom(

^ i

g, C ^∞ (Γ\G) ⊗ V ) So

H ^q (Γ\G, V ˜ ) = H ^q (g, C ^∞ (Γ\G) ⊗ V ).

This is still not what we want to compute, which is H ^• (X _Γ , V ˜ ) = H ^• (Γ\G/K, V ˜ ).

We can compute this by a complex of differential forms A ^• (X _Γ , V ˜ ) = A ^• (X, V ) ^Γ and there is a functorial embedding

i K : A ^q (X, V ) ^Γ ֒ → A ^q (G, V ) ^Γ .

given by G 7→ X ; g 7→ gK (pullback of differentials). The image consists of f : V q

g→C ^∞ (Γ\G) ⊗ V such that

(1) f (Y 0 , . . . , Y q−1 ) = 0 if one of the Y i ∈ k = Lie(K );

(2) f ∈ (A ^q (G, V ) ^Γ ) ^K (right-invariant under K).

Here (1) says that the pullback of differentials from X to G vanish on vectors tangent to K , and (2) says that the coefficients of the differential are functions on X = G/K. More precisely – ignore Γ for the moment, since the right and left actions don’t interfere: Say f (g) ∈ Hom( V q

T G,g , V );

f (gk) ∈ Hom(

^ q

T G,gk , V ) −→ ^∼

R(k

) Hom(

^ q

T G,g , V )

and the condition that f be K -invariant is that R(k ⁻¹ )f (gk) = f(g). But R(k ⁻¹ )

acting on left-invariant vector fields is just Ad(k) acting on g. So to conclude

Lemma. The image of i K (A ^q (X, V ) ^Γ ) is Hom K (

^ q

(g/k), C ^∞ (Γ\G, V )) = Hom K (

^ q

(g/k), C ^∞ (Γ\G) ⊗ V ).

Define C ^∞ (Γ\G) ₀ ⊂ C ^∞ (Γ\G) to be the subspace of K -finite vectors: a vector whose translates under K generate a finite-dimensional subspace. Then it is clear that for any q,

Hom K (

^ q

(g/k), C ^∞ (Γ\G) ⊗ V ) = Hom K (

^ q

(g/k), C ^∞ (Γ\G) 0 ⊗ V ).

because both V and V q

(g/k) are finite-dimensional.

Thus for any representation (π, W ) of U (g) on which the action of k integrates to a consistent K -finite action of K, we define

C ^q (g, K; W ) = Hom K (

^ q

(g/k), W ).

This is a subspace of C ^q (g, W ) and it is easy to see that it is preserved by the differential, so that we have a complex and can define the relative Lie algebra cohomology H ^q (g, K; W ).

Proposition. H ^• (X _Γ , V ˜ ) −→ ^∼ H ^• (g, K; C ^∞ (Γ\G) ₀ ⊗ V ).

Again H ^q (g, K ; •) is the right-derived functor of a left-exact functor on the cat- egory of (g, K)-modules (see below). In what follows, G is a connected reductive Lie group, K ⊂ G is a maximal compact subgroup (modulo the center of G) and g = Lie(G), k = Lie(K ).

Definition. 1. A (g, k)-module is a U (g)-module whose restriction to U (k) is semi- simple and a sum of finite-dimensional k-modules.

2. A (g, K )-module is a (g, k)-module (π, V ) whose k action integrates to an action of K (note: K may be disconnected) in such a way that, for k ∈ K and X ∈ g, we have π(k)π(X)π(k ⁻¹ )v = π(ad(k)X)v for all v ∈ V .

3. A (g, K)-module V is admissible if for any irreducible representation τ of K, Hom K (τ, V ) is finite-dimensional.

One also calls (g, K )-modules Harish-Chandra modules. They form an abelian category (a full subcategory of the category of U (g)-modules) with enough projec- tives and injectives. If V, W are (g, K)-modules, one can compute Ext ^• (V, W ) – extensions in the category of (g, K)-modules – by using a projective resolution of V :

. . . →P i →P i −1 → . . . →P ₀ →V ; C ⁱ = Hom(P i , W ); Ext ^q (V, W ) = H ^q (C ^• ).

Let P i = U (g) ⊗ U (k) V i

(g/k) and define ∂ q : P q →P q−1 by the expected formula

∂ q (r ⊗ x ₁ ∧ · · · ∧ x q ) = X

(−1) ⁱ⁻¹ x i · r ⊗ x ₁ ∧ · · · ∧ x ˆ i ∧ · · · ∧ x q )

+ X

i<i

(−1) ^i+j r ⊗ ([x i , Y j ] ∧ x ₁ ∧ · · · ∧ x ˆ i ∧ · · · ∧ x ˆ j ∧ · · · ∧ x q )

Let ε : P ₀ = U (g) ⊗ U ( k ) C→C be the augmentation.

Theorem. The P j are projective, the maps ∂ q and ε are well-defined, and the sequence . . . →P q →P q −1 → . . . →→P ₀ →C is exact.

Proof. The detailed proofs are contained in §VII.8 of Knapp’s yellow book. There are two points:

(1) tensor product U (g) ⊗ U (k) • takes projectives to projectives and everything in the category is U (k)-projective;

(2) The sequence is exact.

Exactness is a long calculation that reduces ultimately using filtrations to the exactness of the Koszul complex. The first point can be easily explained. Let V be any (g, K )-module and let U be a K-module. Let I(U ) = U (g) ⊗ U (k) U . We check that this is a (g, k)-module (that it is a (g, K )-module follows easily). Let W ⊂ U (g) be a finite-dimensional subspace invariant under ad(k), X ∈ W , Y ∈ U (k); then

Y X ⊗ u = [Y, X ] ⊗ u + X ⊗ Y u ⊂ W ⊗ U.

So the action is semisimple and I (U ) is a (g, k)-module.

Suppose f : B→A is a surjective morphism of (g, K )-modules. Now Frobenius reciprocity is a canonical isomorphism

Hom _{(g,K )} (I (U ), •) = Hom K (U, •).

So the map Hom _(g,K) (I (U ), B)→Hom _(g,K) (I(U ), A) is surjective if and only if Hom K (U, B)→Hom K (U, A) is surjective; but this is clear because B and A are semisimple as K-modules. Thus I(U ) is projective for any U , and in particular all the P j are projective.

Corollary. The functors H ^• (g, K ; •) are the right-derived functors of W 7→ Hom g,K (C, W ).

In particular short exact sequences give rise to long exact sequences in the usual way.

Definition. Let U be a (g, K )-module. The contragredient of U is the subspace U ˜ ⊂ Hom(U, C ) of vectors on which K acts finitely.

Suppose U is admissible, so that as K -module, U = ⊕ _σ∈ K ˆ U (σ) with U (σ) = Hom K (σ, U ) ⊗ σ finite-dimensional. Then ˜ U = ⊕ _σ∈ K ˆ U (σ) ^∗ where ∗ is the usual contragredient for finite-dimensional representations. In particular, ˜ U is again ad- missible.

Definition. Let χ : Z (g)→C be an algebra homomorphism. The (g, K )-module U has infinitesimal character χ if zu = χ(z)u for all z ∈ Z(g), u ∈ U .

Corollary. Suppose U and V are (g, K )-modules with infinitesimal characters χ U and χ V , respectively. Suppose V is finite-dimensional and χ U 6= χ _{V ˜} . Then H ^q (g, K ; U ⊗ V ) = 0 for all q.

Proof. The natural equivalence of (bi)-functors:

Hom ₍ g,K ) (C, U ⊗ V ) −→ ^∼ Hom ₍ g,K ) ( ˜ V , U )

gives rise to isomorphisms of derived functors

H ^q (g, K ; U ⊗ V ) −→ ^∼ Ext ^q ( ˜ V , U ).

So it suffices to show that Ext ^q ( ˜ V , U ) = 0 for all q. Now by hypothesis, there is z ∈ Z(g) that acts as 1 on U and as 0 on ˜ V . If q = 0 and h : ˜ V →U is a (g, K )- morphism then h(v) = zh(v) = h(zv) = 0. If q > 0, let S ∈ Ext ^q ( ˜ V , U ) correspond to a Yoneda extension, i.e. a long exact sequence of (g, K)-modules

0→U →E q −1 → . . . →E ₀ → V ˜ →0.

Then z acts consistently on this sequence and as the identity on U and as 0 on ˜ V , and defines an equivalence with the trivial Yoneda extension.

Alternatively, multiplication by z defines two natural transformations of bifunc- tors (A, B) 7→ Ext ^q _(g,K) (A, B), say z ₁ and z ₂ , by acting in the first and second variable, respectively. For q = 0 we have z 1 = z 2 . Suppose they are equal up to q−1. Let B→C→B ^′ be an exact sequence of (g, K )-modules with C injective. Then the exact cohomology sequence breaks up as Ext ^j−1 (U, V ^′ )→Ext ^j (U, V ) which is surjective for j = 1 and an isomorphism for j ≥ 2, and which commutes with z ₁ and z 2 . This reduces the case of q to that of q − 1. In our setting, z 1 = 0 and z ₂ = 1, so we are done.

Compact quotients.

Suppose Γ\G is compact. Then C ^∞ (Γ\G) 0 ⊂ L 2 (Γ\G) where the measure defin- ing L ₂ is any right-invariant Haar measure dg on G = G(R). It is not hard to show that if G is reductive, then any right-invariant Haar measure is also left-invariant.

Now L ₂ (Γ\G) is a Hilbert space on which G acts by a unitary representation – unitary because

< r(h)f, r(h)f ^′ > L

= Z

Γ\ G

f(gh)f ^′ (gh)dg = Z

Γ\ G

f (g)f ^′ (g)dg =< f, f ^′ > L

, where the second equality follows from invariance of dg.

The following theorem is due to Gelfand and Piatetski-Shapiro:

Theorem. Assume Γ\G is compact. Then L 2 (Γ\G) decomposes as G-module as a countable Hilbert space direct sum:

L ₂ (Γ\G) = M d

π

m(π, Γ)π

where π runs over irreducible unitary representations of G and m(π, Γ) ∈ N. In particular, the multiplicity of π is always finite, and is zero except for a countable set of π.

The classification of irreducible unitary representations is incomplete, but the

following theorem is fundamental. Choose a maximal compact (or compact con-

nected) subgroup K of G. We define a K -finite vector in a Hilbert space repre-

sentation (π, V) as above; a smooth vector v ∈ V is one for which, for any element

X ∈ Lie(G) and any w ∈ V the function t 7→< π(exp(tX )v, w > from R to V is

infinitely differentiable.

Theorem(Harish-Chandra). Let π be an irreducible unitary Hilbert space rep- resentation of the reductive Lie group G and let π ₀ ⊂ π denote the subspace of K- finite smooth vectors. Then U (g) acts naturally on π 0 by π(X)v = _dt ^d π(exp(tX)v and makes it an irreducible (g, K) module.

This is proved in several stages, the most important being that any irreducible representation of K occurs with finite multiplicity in π 0 – in other words, that π 0 is an admissible (g, K) module. This implies that every K -finite vector is automati- cally smooth. The proofs can be found in the beginning of Chapter VIII of Knapp’s book Representation Theory of Semisimple Groups.

We see that L ² (Γ\G) ₀ = C ^∞ (Γ\G) ₀ and that this in turn is L d

π m(π, Γ)π ₀ . Thus Theorem. Assume Γ\G is compact. Then for any finite-dimensional representa- tion V of G,

H ^• (X Γ , V ˜ ) −→ ^∼ M d

π

m(π, Γ)H ^• (g, K; π 0 ⊗ V ).

Thus the calculation of the cohomology of X _Γ divides into two parts: a global part, which is the determination of m(π, Γ), and a purely Lie-theoretic part, which is the calculation of H ^• (g, K ; π ₀ ⊗ V ). The first part is very hard, the second part was solved some time ago. First I switch to the adelic setting.

Theorem (Borel-Harish-Chandra). Let G be a reductive group over Q with

center Z , and for any open compact subgroup K ⊂ G(A f ), let K S(G) = G(Q)\G(A)/K _∞ Z(R)×

K . Then K S(G) is compact for one K if and only it it is compact for all K if and only if G/Z has Q-rank 0; in other words, if G/Z does not contain a subgroup isomorphic to GL(1).

One says that G/Z is anisotropic if it is of Q-rank 0. The adelic version of the Gelfand-Piatetski-Shapiro theorem is the following:

Theorem. Let G be a (connected) reductive group over Q with center Z , and assume G/Z is anisotropic. Then L ₂ (G(Q)\G(A)) decomposes as G(A)-module as a countable Hilbert space direct sum:

L ₂ (G(Q)\G(A)) = M d

π

m(π)π

where π runs over irreducible unitary representations of G(A).

We fix a maximal compact subgroup K _∞ ⊂ G(R). With π as in the theorem, a vector v ∈ V π is smooth if it is C ^∞ with regard to the action of G(R) and if there is an open compact subgroup K f ⊂ G(A f ) that fixes v. Write π ₀ ⊂ π for the space of K _∞ -finite smooth vectors. Then

(1) π 0 determines π (and vice versa); in particular, π 0 is irreducible as a repre- sentation of (U (g), K _∞ ) × G(A f );

(2) π ₀ admits a unique factorization (up to scalar multiples) π ₀ −→ ^∼ π _∞ ⊗ ⊗ ^′ _p π p

where the product is taken over all prime numbers, π _∞ is an irreducible

(g, K _∞ )-module, and each π p is an irreducible (smooth) representation of

G(Q p ).

If V is a representation of G, we can then define H ^• (g, K ∞ ; π⊗V ) := H ^• (g, K ∞ ; π ∞ ⊗ V ) ⊗ π f where π f = ⊗ ^′ _p π p is an irreducible smooth representation of G(A f ). If K f ⊂ G(A f ) is open compact, we can similarly define

H ^• (g, K _∞ ; π ^K

⊗ V ) := H ^• (g, K _∞ ; π _∞ ⊗ V ) ⊗ π _f ^K

.

Working through the comparison of K S(G) with a union of spaces of the form Γ i \G(R)/K ∞ Z(R), we find

Proposition. For any representation V of G, there is a canonical isomorphism

H ^• ( K S(G), V ˜ ) −→ ^∼ M

π

H ^• (g, K ; π _∞ ⊗ V ) ⊗ π _f ^K

;

H ^• (S(G), V ˜ ) −→ ^∼ M

π

H ^• (g, K ; π _∞ ⊗ V ) ⊗ π f ,

where the latter isomorphism commutes with the action of G(A f ) on both sides.

Automorphic vector bundles. Now assume (G, X ) a Shimura datum, so that X is of Hermitian type. Fix h ∈ X and let [W ] be the automorphic vector bundle on S(G, X ) attached to a finite-dimensional representation of K h . We can replace the de Rham complex by the Dolbeault complex in the discussion above. Instead of

A ⁱ (G, V ) = C ^∞ (G, Hom(

^ i

g, V )) = Hom(

^ i

g, C ^∞ (G, V )).

we have

A ^{0 ,q} ([W ]) = [C ^∞ (G(Q)\G(A), Hom(

^ i

(p ⁻ ), W ))] ^K

= Hom K

(

^ i

(p ⁻ ), C ^∞ (G(Q)\G(A)) ⊗ W )

= [

^ i

(p ⁻ ) ^∗ ⊗ C ^∞ (G(Q)\G(A)) ⊗ W ] ^K

.

Now k h ⊕ p ⁻ is the Lie algebra of a subgroup P h ⊂ G ^C (not defined over R and the definition of relative Lie algebra cohomology in this case identifies A ^{0 ,q} ([W ]) with C ^q (Lie(P h ), K h ; C ^∞ (G(Q)\G(A)) ⊗ W ). Thus Dolbeault’s theorem gives us an isomorphism

H ^q (S(G, X ), [W ] −→ ^∼ H ^q (Lie(P h ), K h ; C ^∞ (G( Q )\G(A)) ⊗ W ).

Again, we can replace C ^∞ (G(Q)\G(A)) by L

π m(π)π ₀ and then we find an iso- morphism of G(A f )-spaces

H ^q (S(G, X ), [W ]) −→ ^∼ M

π

m(π)H ^q (Lie(P h ), K h ; π _∞ ⊗ W ) ⊗ π f .

Hodge theory for (g, K)-modules.

The analytic arguments used to prove the Hodge theorem in complex geometry becomes completely algebraic in the setting of relative Lie algebra cohomology.

Change notation: let π be a (g, K )-module that comes from a unitary representation of G. We want to compute H ^q (g, K ; π ⊗ V ) for any irreducible finite-dimensional representation (ρ, V ). We already know that this vanishes unless π and V have opposite infinitesimal characters. To apply Hodge theory, we endow W with an admissible scalar product: one that is invariant under K and with respect to which the action of p is self-adjoint. The existence of such a scalar product is easy to show:

V is a representation of G, therefore of the compact real form G u , and therefore possesses a G u -invariant (hermitian) scalar product (unique up to positive scalar multiples), say (, ) V . This means that Lie(G u ) = k ⊕ ip acts by skew-adjoint operators: if X ∈ Lie(G u ) then (Xv, v ^′ ) + (v, Xv ^′ ) = 0. For X = iY with Y ∈ p this means

0 = i(Y v, v ^′ ) + ¯ i(v, Y v ^′ ) = i[(Y v, v ^′ ) − (v, Y v ^′ )]

which means that p is self-adjoint.

Let D ^q (π ⊗ V ) = V q

p ^∗ ⊗ π ⊗ V with scalar product given by the tensor product of the hermitian Killing form on p C :

(x, y) = B(x, y) ¯

(dualized and taken to the q-th power) with the scalar products already defined on the other two factors. All of these scalar products are invariant under k. Write τ = π ⊗ ρ, so that τ(x) = π(x) ⊗ 1 + 1 ⊗ ρ(x) for x ∈ g. The adjoint with respect to the scalar product on τ is given by

τ ^∗ (x) = −τ (x), x ∈ k; τ ^∗ (x) = −π(x) + ρ(x), x ∈ p.

In what follows, we choose a (real) basis x ₁ , . . . , x D of p (and don’t confuse the dimension D with the differential d. If η ∈ D ^q (π ⊗ V ) = Hom( V q

p, π ⊗ V ) and J ⊂ {1, . . . , D}, |J| = q, we write

η J = η(x j

, . . . , x j

).

Proposition. Define d ^∗ : D ^q (π ⊗ V )→D ^{q −1} (π ⊗ V ) by the formula (d ^∗ η) J =

X D

j=1

τ (x j ) ^∗ η _{j}∪J .

Then d ^∗ commutes with K, maps the K -invariant subspace C ^q (π ⊗ V ) into the K -invariant subspace C ^{q −1} (π ⊗ V ), and is formally adjoint to d:

(d ^∗ η, f ) = (η, df ) for η ∈ D ^q , f ∈ D ^q−1 .

Proof. We recall the formula for df (with a shift of indices):

df (Y j

, . . . , Y q+1 )

= X

j

(−1) ^j−1 τ(Y j )f (Y ₁ , . . . , Y ˆ j , . . . , Y q +1 ) + X

r<s

f ([Y r , Y s ], Y ₁ , . . . , Y ˆ r , . . . , Y ˆ s , . . . , Y q +1 )

= X

j

(−1) ^{j −1} τ(Y j )f (Y ₁ , . . . , Y ˆ j , . . . , Y q +1 )

because [p, p] ⊂ k so the bracket terms vanish. Thus (replacing the Y u ’s by x j

(η, df ) = X

|I|=q

(η I , (df ) I ) τ = X

|I|=q

(η I , X

u

(−1) ^{u −1} τ (x j

)f _I(u) ) τ

where I = {j ₁ , . . . , j q } and I(u) is I with the j u term removed. And thus by adjunction

(η, df) = X

I,u

((−1) ^u−1 τ ^∗ (x j

)η I , f I ( u ) . Using the anticommutation formula

η I = (−1) ^{u −1} η _{j

_}∪I(u)

this shows after reviewing the notation that (η, df) = (d ^∗ η, f ).

Now if x ∈ k we find

−(τ (x)d ^∗ η, f ) = (τ (x) ^∗ d ^∗ η, f ) = (d ^∗ η, τ (x)f) = (η, dτ (x)f )

= (η, τ (x)df ) = (−τ(x)η, df ) = −(d ^∗ τ (x)η, f ).

Thus d ^∗ commutes with K and in particular takes the K-invariants to K-invariants.

Let ∆ = dd ^∗ + d ^∗ d. For each q, ∆ is an endomorphism of D ^q (π ⊗ V ) and of C ^q (π ⊗ V ). Moreover, for η ∈ D ^q (π ⊗ V )

(∆η, η) = (dη, dη) + (d ^∗ η, d ^∗ η) and since the scalar product is positive non-degenerate,

∆η = 0 ⇔ dη = 0 and d ^∗ η = 0 ⇔ (∆η, η) = 0.

In this case we say η is harmonic; and we let H ^q (π ⊗ V ) ⊂ C ^q (π ⊗ V ) denote the subspace of harmonic forms.

Proposition. For every q, the map H ^q (π ⊗ V )→H ^q (g, K ; π ⊗ V ) is an isomor- phism.

Proof. This is equivalent to the orthogonal decomposition C ^q (π ⊗ V ) = H ^q (π ⊗ V ) ⊕ Im(d) ⊕ Im(d ^∗ ).

Note for example that for η ∈ C ^q ,

dη = 0 ⇔ (dη, f) = 0 ∀f ∈ C ^q+1 ⇔ η ⊥ Im(d ^∗ ).

So C ^q = ker(d) ⊕ Im(d ^∗ ) is an orthogonal decomposition. Similarly, Z ^q := ker d = H ^q ⊕ Im(d) is an orthogonal decomposition because H ^q = Z ^q ∩ ker d ^∗ .

Kuga’s formula calculates the action of ∆ in terms of a specific element, the

Casimir element, in Z (g).