4(a) Discrete series
We again assume (G, X) is a Shimura datum, with Gder anisotropic, so the associated Shimura varieties are compact. Thus all the automorphic representations π ofGare unitary, modulo the action of the center (which I will attempt to ignore).
Recall the isomorphisms
H•(S(G),V˜) −→∼ M
π
H•(g, K;π∞⊗V)⊗πf, Hq(S(G, X),[W]) −→∼ M
π
m(π)Hq(Lie(Ph), Kh;π∞⊗W)⊗πf, and the isomorphism from Hodge theory
H•(g, K;π∞⊗V) −→ H∼ •(π∞⊗V).
The next step is to explain how harmonic forms are computed in the most inter- esting cases, but first I add a fourth isomorphism expressing Hodge theory locally for coherent cohomology:
H•(Lie(Ph), Kh;π∞⊗W) −→ H∼ •(π∞⊗W).
This is if anything even easier than the previous case, because every representation of Kh is a representation of Ph and carries a Ph-invariant hermitian metric. The Laplacian for ∂ is denoted =∂∂∗+∂∗∂ rather than ∆ but it also satisfies
(η, η) = (∂η, ∂η) + (∂∗η, ∂∗η).
There is also a relation between H•(g, Kh;∗) and H•(Lie(Ph), Kh;∗). Indeed, we have
Ci(g, Kh;π∞⊗V) =HomK(
^i
p, π∞⊗V) =HomK(
^i
(p+⊕p−), π∞⊗V)
=⊕p+q=iHomK(
^p
p+⊗
^q
p−, π∞⊗V) =⊕p+q=iCp,q(g, Kh;π∞⊗V) where
Cp,q(g, Kh;π∞⊗V) =HomK(
^q
p−, Hom(
^p
p+, π∞⊗V)).
We write d = ∂ +∂ as in K¨ahler geometry, with ∂ of degree (1,0) and ∂ of degree (0,1). Each of them has an adjoint with respect to the hermitian form, and we construct
+ =∂∂∗+∂∗∂;− =∂∂∗+∂∗∂
Typeset byAMS-TEX
1
Proposition. ∆ =++−.
This is equivalent to the equations
∂∂∗+∂∂∗ =∂∗∂+∂∗∂ = 0
which is a computation we omit. (This is in the 1963 Annals paper of Matsushima- Murakami. The point is that∂is given by derivatives with respect top+whereas∂∗ is given by the adjoints of derivatives with respect top−, but these can be expressed as derivatives with respect top+again, because of the nature of the hermitian form, so the derivatives in ∂ and ∂∗ commute, as do those of ∂ and ∂∗, and these give the relation when the signs are taken into account.
Corollary. There is an isomorphismHi(g, Kh;π∞⊗V) −→ ⊕∼ p+q=iHp,q(π∞⊗V) where Hp,q is given by the forms harmonic with respect to both + and −.
This is a formal consequence, as in K¨ahler geometry. The next statement is less formal.
Theorem. There are canonical isomorphisms
Hp,q(π∞ ⊗V) −→ H∼ q(π∞⊗Hp(p+, V)) =Hq(Lie(Ph), Kh;π∞⊗Hp(p+, V)).
The proof is omitted because it is long and here is only intended to motivate the study of coherent cohomology as a way to understand the Hodge theory of cohomology of local systems.
Kuga’s formula.
We continue to assume π unitary. Kuga’s formula calculates the action of ∆ in terms of a specific element, the Casimir element C ∈Z(g). The existence of C in the image of g⊗g in U(g) is equivalent to the existence of the Killing form. ViaB we can identify g∗ −→∼ g as Ad(g)-modules, and thus we have an isomorphism
B: g⊗g∗ −→∼ g⊗g.
Let X1, . . . , Xm be a basis for g, Yi the dual basis of g∗. Then the element T r = P
iXi⊗Yi ∈g⊗g∗ is invariant under the representation ofg(orG) and corresponds to the trace in the given basis; it does not depend on the choice of basis. Then
B(T r) =X
Xi⊗Xi∗
where B(Xi, Xj∗) =δij. Let C denote the image of B(T r) under the natural map g⊗g→U(g). Then C is invariant under the adjoint action ofg, which means that C ∈Z(g). This can also be checked by hand using the definition ofB.
Now π and ρ have been assumed irreducible; thus by Schur’s lemma they have infinitesimal characters (this needs proof).
Kuga’s Formula. Assume π(C) = s and ρ(C) = r. Then ∆ acts as r −s on Cq(π⊗V) for any q.
The proof is a long computation (see pp. 49-51) based on the Cartan decompo- sition; one writes C in terms of a basis ofk plus a basis of p.
Corollary. If r 6= s then Hq(π⊗V) = 0. If r =s then every element of Cq(π⊗ V) =HomK(Vq
p, π⊗V) is harmonic, so that
Hq(g, K;π⊗V) =HomK(
^q
p, π⊗V).
The first statement follows becauseC ∈Z(g) and we have already seen that if the infinitesimal characters of π and V∗ do not coincide then there is no cohomology, hence no harmonic forms. But the Laplacian is self-adjoint, so the eigenvalues on V and V∗ coincide. (Alternatively: ifr 6=s thenC just multiplies by r−s 6= 0, so there are no harmonic forms.) The second statement is then obvious.
A similar calculation shows that, ifπ is unitary, thenHq(Lie(Ph), Kh;π⊗W) = Cq(Lie(Ph), Kh;π⊗W) =HomK(Vq
p−, π⊗W) (all cochains are harmonic). Using the bigrading, the previous corollary already gives
Hp,q(g, K;π⊗V) =HomK(
^q
p−, Hom(
^q
p+, π⊗V)) =HomK(
^q
p−, π⊗Hom(
^p
p+, V)) and the latter can be identified by a similar calculation with
HomK(
^q
p−, π⊗Hp(p+, V))) =Cq(Lie(Ph), Kh;π⊗Hp(p+, V) as promised above.
To motivate the next section, I state Kostant’s theorem.
Kostant’s theorem. Let G be any reductive Lie group, V an irreducible finite- dimensional representation of G, P = L·U a parabolic subgroup. Let W and WL be the Weyl groups of G and L, respectively, relative to a maximal torus contained in L. Choose a positive root system ∆+ so that µ is the highest weight of V and let ρ+ be the half sum of positive roots. There is a unique set of (shortest) coset representatives WP for W/WL so that w ⋆ µ:=w(µ+ρ+)−ρ+ is a highest weight of a representation Vw⋆µ for L relative to this positive root system. Then for any p,
Hp(Lie(U), V) −→ ⊕∼ ℓ(w)=p,w∈WP Vw⋆µ as representation of L.
Kostant’s original proof was by Hodge theory for the finite-dimensional Lie(U)- cohomology complex, and a calculation. Apply this to G as above and P = Ph+ with Lie algebrakh⊕p+h. It follows that whenπ is unitary andV has highest weight µ,
Hi(g, Kh;π⊗V) =⊕p+q=i⊕ℓ(w)=pHomK(
^q
p−, π⊗Vw⋆µ) =⊕p+q=i⊕ℓ(w)=pHq(Ph, Kh;π⊗Vw⋆µ).
In the next section, I describe a collection of|WG/WKh|unitary representations πw of G(R), with the property that dimHq(Ph, Kh;πw⊗Vw⋆µ) = 1 whenq+ℓ(w) = dimp and is 0 otherwise.
Discrete series.
Suppose G is a semisimple Lie group with maximal compact subgroup K. We consider L2(G) with respect to a Haar measure dg. Since dg is both right and left invariant under the action of G, there is an action of G×G on L2(G). If G is compact, L2(G) is the Hilbert space completion of C[G] (functions on G, or the group algebra of G) and then the Peter-Weyl theorem (generalizing the theory for finite groups) states that
L2(G) = ˆ⊕τ∈Gˆτ∗⊗τ.
In general,L2(G) includes a Hilbert sum of subrepresentations – this is the discrete spectrum – and a continuous spectrum. Note that ifGis non-compact, the constant functions are not in L2. If the discrete spectrum is non-trivial it is called the discrete series. Harish-Chandra proved that G has a discrete spectrum if and only if a maximal torusT ofK is a maximal torus ofG. We have seen that this is always true ifG/K has a hermitian structure (because then the center ofK contains a non- trivial torus, and it is contained in a maximal torus which is necessarily contained in its centralizer).
Harish-Chandra also classified the discrete series. Fix a maximal torusT ⊂K = Kh with Lie algebra t and sets of positive roots ∆+K ⊂ ∆+ for T in K and G, respectively; let
ρ= 1 2
X
α∈∆+
α; ρk = 1 2
X
α∈∆+K
α.
Write W = W(G, T), WK = W(K, T). The most natural parametrization of the discrete series starts with a finite-dimensional representation (σ, V) of G with highest weight Λ. The Harish-Chandra isomorphism identifies
Z(g) −→∼ S(t)W
where w∈W acts on characters of t by the twisted action w ⋆(χ) =w(χ+ρ)−ρ.
Thus the character of Z(g) on V corresponds to the W-orbit of λ := Λ +ρ ∈ t∗. Since Λ is dominant integral, λ is regular: < λ, α >6= 0 for all roots α. Indeed
< λ, α >>0 for allα ∈∆+. We want to considerπ such that H∗(g, K;π⊗V∗)6= 0 where V∗ is the contragredient of V, and such that π is in the discrete series.
(Every member of the discrete series arises in this way.) Letting χπ and χV be the infinitesimal characters of π and V respectively, we know that χπ = χV =χλ. The set ΠV of discrete series π with infinitesimal character χλ is then indexed by w ∈ WPh+ ≡ W/WK; in Harish-Chandra’s notation these can be denoted πw−1λ; then for λ′, λ′′ in the W-orbit of λ, πλ′ ≃ πλ′′ if and only if λ′ = w(λ′′) for some w∈WK.
ByK-typewe mean an irreducible (finite-dimensional) representation ofK. The following theorem is valid for reductive as well as semi-simple groups; a discrete series for a reductive group G is a finite sum of square-integrable representations of Gder.
Theorem. Let d = 12dimRG/K, so that d = dimX if G/K is of hermitian type.
Let λ′ ∈ W(λ) and let ∆+λ′ be the set of roots α such that < λ′, α >> 0 (it is a W-translate of the original ∆+ = ∆+λ, and let
ρλ′ = 1 2
X
α∈∆+λ′
α.
Then
(i) (Weak Blattner formula) TheK-type with highest weightτλ′ :=λ′+ρλ′−2ρk occurs in πλ′ with multiplicity one, and any K-type that occurs in πλ′ has highest weight of the form λ′+ρλ′ −2ρk+Q′ where Q′ is a sum of elements of ∆+λ′.
(ii) If V′ 6= V is a finite-dimensional irreducible representation of G, then Ext∗g,K(V′, π) = H∗(g, K;π⊗(V′)∗) = 0, whereas Extig,K(V, π) = 0 unless i = d;
dimExtdg,K(V, π) = 1.
Proof. Part (i) is due to Schmid (after work of many others). I derive (ii) from (i). The first statement follows from the fact that two distinct finite-dimensional irreducible representations of G have distinct infinitesimal characters. Let ∆+n =
∆+λ′\∆K, ∆n = ∆+n `
−∆+n, the set of roots of T acting onp. The weights of T on Vi
p are thus of the form α1+· · ·+αi for α• ∈ ∆n distinct. Write ρn =ρλ′−ρk. The weights on Vi
p are then of the form
α1+· · ·+αj −(αj+1+· · ·+αi) where all the α• are now in ∆+n. We can write
α1+· · ·+αj = 2ρn−(γ1+· · ·+γt)
where ∆+n ={α1, . . . , αj;γ1, . . . , γt}. It follows that every weight of T on Vi
pis of the form 2ρn−Q whereW is a sum of elements of ∆+n. Moreover, if 2ρn is itself a weight of Vi
p then i=d=|∆+n| and 2ρn occurs in Vd
pwith multiplicity 1.
On the other hand, the weights of (σ, V) are all of the form λ′−ρλ′−Q withQ a sum of positive roots and Λ′ =λ′ −ρλ′ has mutliplicity 1 by the theorem of the highest weight (for the positive root system ∆+λ′). Thus
(i) The weights ofT on Vi
p⊗V are of the form 2ρn+λ′−ρλ′ −Q whereQ is a sum of elements of ∆+λ′. If 2ρn+λ′−ρλ′ +Q′ is a weight on Vi
p⊗V (with Q′ a sum of positive roots) then Q′ = 0, i=d, and the multiplicity equals 1.
(ii) Now suppose τ is a dominant integral weight for ∆+K (still relative to λ′) such that HomK(Wτ,Vi
p⊗V)6= 0. Then τ = 2ρn+λ′−ρλ′ −Q as before, and if τ = 2ρn+λ′ −ρλ′ +Q′ then Q′ = 0, i =d, and dimHomK(Wτ,Vq
p⊗V) = 1.
But now we are done by the weak Blattner formula.
Consequences for the cohomology and coherent cohomology of Shimura varieties.
Note that we have fixedK above in order to viewπ as a (g, K)-module. In what follows, K =Kh for some point h∈X.
We continue to assume that π is in the discrete series and that V is the repre- sentation in the theorem above. The bigrading shows that
Hd(g, K;π⊗V∗) =⊕p+q=dHp,q(g, K;π⊗V∗)
= ⊕p+q=dHq(Lie(Ph);K;π⊗Hp(p+, V∗)).
=⊕p+q=d⊕ℓ(w)=pHq(Lie(Ph);K;π⊗(V∗)w)).
Here we are using Kostant’s theorem: if µ is the highest weight of V∗ (relative to any positive root system compatible with Ph+) then Vw∗ is the representation of K with highest weight w(µ+ρ)−ρ. But since the left-hand side is of dimension 1,
only one term is non-trivial. Thus for eachπ in the discrete series, there is a unique representation σ of Kh such that
Hd(g, K;π⊗V∗) −→∼ Hq(Lie(Ph);K;π⊗σ))
where σ has highest weight w(µ+ρ)−ρ as above for some (unique) w, and then q =d−ℓ(w). Moreover, this is an isomorphism of 1-dimensional spaces.
Returning to the calculation of cohomology of automorphic vector bundles, we find
Hq(S(G, X),[W])−→∼ M
π
m(π)Hq(Lie(Ph), Kh;π∞⊗W)⊗πf.
−→∼ M
π∞∈/Gˆd
m(π)Hq(Lie(Ph), Kh;π∞⊗W)⊗πf ⊕ M
π∞∈Gˆd
m(π)πf.
Here we write ˆGd for the collection of discrete series representations of G(R).
The second isomorphism is not canonical, because it depends on a choice of basis of the 1-dimensional space Hq(Lie(Ph), Kh;π∞ ⊗W). However, it is not hard to see that this space is defined over a number field (and I’ve even written a paper about refining this definition).
Now we can use the existence of canonical models to draw conclusions about rationality. (Here mention that the Ωiand their tensor products are all automorphic vector bundles.)
