204 D.V. Millionschikov
where
αk+s=αs;1e^1 +αs;2e^2 +···+αs;kek,Pk+s(e^1 ,...,ek+s−^1 )=∑
1 ≤i<j≤k+s− 1Ps;i,jei∧ej. (11.27)It is convenient to defineαi=0,i=1,...,k. The set{α 1 ,...,αn}of closed
1-forms is in fact the set of the weights of completely reducible representation
associated to the adjoint representationX→ad(X).
For the proof we apply Lie’s theorem to the adjoint representation ad
restricted to the commutant [g,g]:
X∈g→ad(X):[g,g]→[g,g].Namely we can choose a basisek+1,...,enin [g,g] such that the subspaces
Vi,i=k+1,...,nspanned byei,...,en are invariant with respect to the
representation ad. Then we adde 1 ,...,ekin order to get a basis of the whole
g. For the forms of the dual basise^1 ,...,ening∗we have formulas (11.26).
Let us consider a new canonical basis ofg∗:
e ̃^1 =e^1 ,..., ̃ek=ek,
e ̃k+s=t2(s−1)ek+s,s=1,...,n−k,(11.28)
wheret>0 is a real parameter.
Then for the differential dωin the complexΛ∗( ̃e^1 ,..., ̃en)wehave:
dω=d 0 +ω∧+td 1 +t^2 d 2 +···, d 0 e ̃i=αi∧e ̃i.In particular
(d 0 +ω∧)( ̃ei^1 ∧···∧e ̃iq)=(αi 1 +···+αiq+ω)∧e ̃i^1 ∧···∧ ̃eiq.Now one can define the scalar product inΛq( ̃e^1 ,..., ̃en) declaring the set
{ei^1 ∧···∧eiq}of basicq-forms as an orthonormal basis ofΛq( ̃e^1 ,..., ̃en).
Then
d∗ωdω+dωd∗ω=R 0 +tR 1 +t^2 R 2 +···,
R 0 ( ̃ei^1 ∧···∧ ̃eiq)=‖αi 1 +···+αiq+ω‖^2 e ̃i^1 ∧···∧ ̃eiq.(11.29)
Ast→0 the minimal eigenvalue of d∗ωdω+dωd∗ωconverges to the minimal
eigenvalue ofR 0 .Thusif
αi 1 +···+αiq+ω=0, 1 ≤i 1 <i 2 <···<iq≤nthenHωq(g) = 0 (Fig. 11.2).
Recall thatα 1 =···=αk = 0 and let us introduce the finite subset
Ωg⊂H^1 (g) such that: