HOF, HANS-CHRISTOPH IM

RUH, ERNST A.

Hof, Hans-Christoph Im

Ruh, Ernst A.

Osaka University Knowledge Archive

The vanishing of cohomology associated to discrete subgroups of complex simple Lie groups

Osaka City University Repository

Im Hof, H.-C. and Ruh, E. A.
Osaka J. Math.
19 (1982), 669-675
THE VANISHING OF COHOMOLOGY ASSOCIATED
TO DISCRETE SUBGROUPS OF COMPLEX
SIMPLE LIE GROUPS
HANS-CHRISTOPH IM HOF AND ERNST A. RUH
(Received September 1, 1980)
1. Introduction
Let G denote a connected complex simple Lie group and K a maximal
compact subgroup of G. The quotient M—G/K is a riemannian symmetric
space of non-compact type. Let Γ denote a discrete subgroup of G with compact
quotient Γ\G, and let p denote an irreducible non-trivial complex representation
of G in a finite dimensional complex vector space F. In this paper we prove that
for such representations a certain quadratic form defined by Matsushima and
Murakami [3] is positive definite, and hence H*(T, M, p) vanishes.
The motivation for this paper is a result of Min-Oo and Ruh [4] on com-
parison theorems for non-compact symmetric spaces, where an estimate from
below for the first eigenvalue of the Laplace operator on 2-forms with values in
a bundle associated to the adjoint representation is essential. This estimate is an
immediate consequence of the positivity of the above quadratic form. The
vanishing of ίf*(Γ, M, p), without the information on the first eigenvalue, is a
special case of [1, Ch. VII, Th. 6. 7].
2. The result
Let c; denote the Lie algebra of left-invariant vector fields of the simple Lie
group G, p: Q-+&1 (F) the representation induced by p: G-*GL(F), and 9=i0p
a Cartan decomposition of g with ϊ the Lie algebra of a maximal compact sub-
group K. We identify the Lie algebra Q with the corresponding vector fields on
Γ\G.
Let A(T, M, p) (^40(Γ, M, p) in the notation of Matsushima and Murakami
[3]) denote the vector space of F-valued differential forms on Γ\G which are
horizontal and ad^-equivariant, i.e., -η^A(Γ, M, p) satisfies i
x
η=Q and θ
x
η=
— ρ(X)η for all X^l, where i
x
 is interior multiplication and θ
x
 is the Lie
* This work was done under the program Sonderforschungsbereich "Theoretische Mathematik'
(SFB 40) at the University of Bonn.
670 H.C. IM HOP AND E. A. RUH
derivative. A <7-form η^A(T, M, p) is determined by its values 97 .^..^  =
η(Yi^ ••-, Yiq) on (/-tuples of basis vectors of p. According to [3, (6.7)], the
Laplace operator
Δ: A(Γ,M,p)-»A(Γ,M,p)
is a sum of a differential operator Δ^ and an operator Δp associated to the re-
presentation p. Restricted to </-forms these operators have the following co-
ordinate expressions.
ΣΣ
,
v
 ..-, y(f) = - Σ Yl^,.. ,iq+ (-i)" [Ylm,
lt •-, yίf) =
where {y, ; ί=l, •••, w— dim M} is an orthonormal basis of £ with respect to the
Killing form φ of g restricted to p. As in [3], the definition of Δ requires a choice
of an admissible hermitean inner product on F. The inner product < , >F is
called admissible if for all u, v^F the following conditions hold:
<p(X)u,vyF=-<u,p(X)vyF for
(P<Y)u,vyF= <u,p(Y)vyF for
Matsushima and Murakami [3] prove that admissible hermitean inner products
always exist.
The following result is well known.
Proposition 1. The vector space H*(T,My p) is canonically isomorphίc to
the vector space {η^A(T, M, p); Δτ7=0} of harmonic forms.
The restriction of the Killing form φ to p together with the scalar product
< , yF on F induce a hermitean scalar product ( , ) on A(Γ, M, p), obtained by
integrating the pointwise defined scalar product
Here 17^,...,^ and ω/^...^ are the coordinates of g-forms with respect to an or-
thonormal basis in p, and ζη, ω)> is defined to be zero if η and ω are of different
degrees.
The following result is proved in [3].
Proposition 2. The quadratic forms ηϊ-*(ΔDη, -η) and 971— »(ΔPT7, 77) are positive
semi-definite.
A differential form ^^^(Γ, M, p) is a section of the trivial vector bundle on
VANISHING OF COHOMOLOGY 671
T\G with fibre Hom(Λp, ί1), the homomorphisms from the exterior algebra over
p to F. The operator Δp does not involve derivatives and thus can be viewed
as a linear map
Δp: Horn (Λp, F) -> Horn (Λp, F) .
Our main result concerns the positivity of the quadratic form ηt-*(Δpη, ^> on
Hom(Λp, F), which by Proposition 2 implies the vanishing of the cohomology
vector space #*(Γ, M, p).
Theorem. Let p denote an irreducible non-trivial complex representation of
a complex simple Lie algebra Qona finite dimensional complex vector space F. Then
the quadratic form 97i-><Δp?7, 97) on Hom(Λp, F) is positive definite, and therefore
The basic ideas of the proof are similar to those of Raghunathan [6]. Our
restriction to complex Lie groups allows us to prove the optimal result. In
addition, Assertions III and IV of [6], which lead to difficulties, can be avoided.
3. The proof
The restriction to complex Lie algebras g allows us to identify HomR(Λp, F)
with Home (Λg, F). In the following we suppress the subscripts R and C. Since
g — Iφp and p = ίl, multiplication with i is a R-vector space isomorphism
/: ϊ->p. Let Λ/: Λf->Λp denote the induced isomorphism and define
Horn (Λp, F) -=» Horn (Λϊ, F) -=; Horn (Λg, F) ,
where the first isomorphism is composition with ΛJ, and the image ξ of
£'eHom(ΛΪ, F) under the second isomorphism is defined by ξ(X®\)=\ξ'(X),
From now on we identify Hom(Λg, F) with jF®Λg* and view Δp as an
element in the endomorphism ring of F®ΛQ*. Let c denote the Casimir
element with respect to the Killing form φ in the universal enveloping algebra
U(Q) of g. The representation p extends to C/(g). In the following lemma σ
denotes the dual of the representation Λad induced by the adjoint representation
of g.
Lemma 1. 2ΔP = 3(p®l)(c)+(l®σ)(c)-(p®σ)(c)
This lemma proves in particular that Δp is a selfadjoint endomorphism with
respect to the scalar product < , > introduced earlier.
Proof. Let {Xk\ k=l, " ,n} be an prthonormal basis of ϊ with respect to
—φ restricted to ϊ, and {Yk\ = {iXk} the corresponding basis in Jx The image
672 H.C. IM HOP AND E. A. Run
£E:Hom(Λg, F) of ^G:Hom(Λp, F) under the isomorphism defined above evalu-
ated on (Xi^ •••, Xiq) is
ξ(X,
ί
, »., Xiq] = ,(1^ , -, ίAΓ,f) =
With this identification of Hom
c
(Λg, F) and HomR(Λt>, ί1), Δp operates on ξ as
follows:
, »., X,,) = Σ P
ι
, .-, Xlt)+(Tξ)(X,lt ..-, JΓ<f) .
In view of the identification Hom(Λg, F)=F®AQ*y the first summand is given
in terms of the Casimir element c as
S=(p®l)(c),
since {Xk} and {— -3ΓJ are dual bases with respect to φ and therefore c== —*Σ XI
To deal with the second summand, we abbreviate E"=Λg* and specialize to
— ith/EΞF and e^E. The immediate goal is to prove that in this case@e wit
Let cΊj denote the structure constants of ϊ (and g) with respect to the basis {Xk}
thus Σ t?ί}Xt=[Xh X,}, andA = l
P([iXia, ίΛΓJ) = -P([Xia, Xk}) = - ±clkP(Xs) = - ±c»sίap(Xs) ,
where the last equality holds because ckijy in terms of an orthonormal basis with
respect to —φ, is skew symmetric in each pair of indices. We have
(Tξ)(Xtl, -, x,,) = ΣΣ
Abbreviating X=XlίΛ—X,ι we obtain (Tξ)(X)=Σ p(X^(-Aad(Xk)X), and
for ζ=f®e and σ the dual representation of Λad we obtain
T(f®e) = Σ p(Xk)f®σ(Xk)e .
To conclude the proof we compute as in [6]
2p(Xk)®σ(Xk) = (p®σ)(Xk)2~p(Xkγ®idE-idF®σ(Xk) ,
VANISHING OF COHOMOLOGY 673
and obtain
2T = (p®l)(c)+(l®σ)(c)-(p®σ)(c) .
To prove the Theorem we will show that all the eigenvalues of Δp are positive.
The basic observation (see Lemma 2 below) is that for any irreducible represen-
tation p, the endomorphism p(c) is a scalar operator whose eigenvalue is given in
terms of the highest weight of p. This fact will be applied individually to the
irreducible components of p®l, l®σ, and p®σ.
First we introduce some notation. As above we fix a Cartan decomposition
g=ϊ©t>, where ϊ is a compact real form of g and {>=/!. Let ^ be a Cartan sub-
algebra of g compatible with the given Cartan decomposition. Then §=1^0 ,^
where 1^ =1) Π ϊ and f) =^)f]\>=iί)
Γ
 Let Δ denote the root system of the pair
(g, t)). To each #eΔ we associate ί/α>et) such that a(H)=(H
a
, Hy for all
/f ef), where the Killing form is denoted by < , > from now on. Then IL
coincides with the real vector space spanned by {H
a
\ αeΔ}, so Δ may be
viewed as a subset of §*, the real dual of §p. The Killing form < , > is real and
positive definite on ΪL, hence it induces a scalar product X , )> on §*. By fixing
a basis of Δ we once and for all determine a set Δ+ of positive roots. We define
δ=
β
Σ,α.
Lemma 2. Let p: g—>Ql(F) be any irreducible representation of Q with highest
weight λ, then p(c)=(\, λ+δ> *WF.
For a proof see Raghunathan [5, Lemma 4], or Bourbaki [2, Ch. 8, §6, n° 4].
Lemma 2 immediately applies to our given representation p and thus enables us
to compute the contribution of 3(p®l)(c) to the eigenvalues of 2ΔP. The
second term (l®σ)(c) involves the representation σ-=Λad* of g on E=ΛQ*.
This representation is no longer irreducible, so Lemma 2 applies to each compo-
nent of σ separately. Thus the knowledge of the highest weights of the ir-
reducible components of σ is required.
Lemma 3. Let μ be the highest weight of an irreducible component of σ.
Then μ L of the form μ= Σ m<*a, with /w
Λ
e {0,1}.
Proof. The weight space decomposition of ad: g-^gl(g) with respect to the
Cartan subalgebra § equals the root space decomposition of the pair (g. r)),, i.e.,
g = §θ Σ 9*.
The dual representation ad*: g->gl(g*) leads to the analogous decomposition
g* = ^*0 Σ (9*)« , with (8*)Λ = (g_*)* .
674 H.C. IM HOP AND E. A. RUH
Now let w=dimg*, r=dimξ)*, and observe dim(8*)
β
=l. Then E=λQ* =
Λ(ϊj*θ Σ (9*)*) is isomorphic to a sum of subspaces
 o
f the form
where 0<λ<r, 0<&<ra— 1, α,eΔ, and α, Φαy for / Φj.
Such a subspace is invariant under the action of f), it has weight αH ----- \-ak
This implies in particular that the highest weights of the irreducible components
occuring in σ are of the form a
λ
-\ ----- \-ak, α,eΔ, α,φαy for /Φj.
Let now E=^E^ be the decomposition of E into its irreducible com-
ponents Eμ indexed by their respective highest weights. Lemma 2 enables us to
compute the eigenvalues of (l®σ)(c) on F®Eμ.. The third term, (ρ®σ)(c), in
Lemma 1 involves the representation ρ®σ of g on F®E. This space certainly
decomposes into the sum ^F®E^ but each of the components F®Eμ may
further decompose into a sum F®E^=^ FJ, where the subspaces Vjί, irreduci-
ble under p®σ, are indexed bei their respective heighest weights v.
For each μ there is exactly one component F ,^+μ of F®Eμ with highest
weight \-\-μ. All other components F£ have highest weights v<\+μ. The
following lemma allows us to restrict our attention to the spaces V^ μ.
Lemma 4. Let ρlt ρ2 be two irreducible representations of g zcith respective
highest weights \ι, λ2. Then λι>X2 implies
<λι, λ!+δ»<λ2, X2+δ>
Proof. Let β=\ι— X2 and assume /3>0. Then
<λι,λ1+δ>-<λ2,X2+δ> =
, δ»0,
since X2 and δ are dominant.
According to Lemma 4, the maximal eigenvalue of (ρ®σ)(c) restricted to
F®Eμ is attained on the space V^
+μ
. Since (ρ®l)(c) and (l®σ)(c) are positive
scalar operators on the whole space F®Eμ.y and (ρ®σ)(c) occurs with a minus
sign in 2ΔP, the minimal eigenvalue of 2ΔP restricted to F®Eμ is attained on the
space V$,+μ. This minimal eigenvalue involves only X and μ, according to
Lemma 2. Our claim is now reduced to the
Assertion. Let μ be any of the highest weights occuring in the decomposition
Then the eigenvalue of 2ΔP is positive on V^.+μ.
Proof. On F£+μ we have
2ΔP = {3<X, X+δ>+<μ, μ+δ>-<χ+μ, χ+μ+δ» -id .
By a straightforward computation this reduces to
VANISHING OF COMOHOLOGY 675
Δp = {<J\,, λ^+<(λ, δ—μ)>} -id .
The term <(λ, \y is obviously positive, since λ is the highest weight of a non-
trivial representation. Now 8= 2 #> and according to Lemma 3, μ= 2 m^oi
with w
Λ
e {0,1}, hence δ—μ= 2 w*# with w
Λ
e {0,1,2}. Therefore <λ, δ—μ>
= Σ ^«<λ, α»0, since X is dominant.
References
[1] A. Borel and N. Wallach: Continuous cohomology, discrete subgroups, and re-
presentations of reductive groups, Ann. of Math. Study 94, Princeton University
Press 1980.
[2] N. Bourbaki: Groupes et algebres de Lie, Ch. 7, 8, Act. Sci. Ind. 1364, Hermann,
Paris, 1975.
[3] Y. Matsushima and S. Murakami: On vector bundle valued harmonic forms and
automorphic forms on symmetric riemannian manifolds, Ann. of Math. 78 (1963),
365-416.
[4] Min-Oo and E.A. Ruh: Vanishing theorems and almost symmetric spaces of non-
compact type, Math. Ann. 257 (1981), 419-433.
[5] M.S. Raghunathan: On the first cohomology of discrete subgroups of semisimple Lie
groups, Amer. J. Math. 87 (1965), 103-139.
[6] M.S. Raghunathan: Vanishing theorems for cohomology groups associated to discrete
subgroups of semisimple Lie groups, Osaka J. Math. 3 (1966), 243-256; Corrections
ibid, 16 (1979), 295-299.
Hans-Christoph Im Hof and Ernst A. Ruh
Mathematisches Institut der Universitat Bonn
Wegelerstrasse 10
D-5300 Bonn, Germany



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The vanishing of cohomology associated to discrete subgroups of complex simple Lie groups

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