we have

(LY —¦ β s)(0), v = ’ ( β s)(0), LY v

so that only the terms with |γ| = q or |γ| = q + 1 can survive in the sum (notice

k

Y ∈ gp , p ≥ 1, implies LY v = 0). Since = j0 Y ∈ g0 acts on (the jet of constant

section) v by .v = L’Y v(0), we get the result.

34.7. Example. In order to demonstrate the computations with this formula,

let us now discuss the linear operations in dimension one.

We say that V is a gk -module homogeneous in the order if there is k0 such

n

that gk0 .v = 0 implies v = 0 and gl .v = 0 for all v and l > k0 . Each gk -module

n

includes a submodule homogeneous in order. Indeed, the isotropy algebra of

each vector v contains some kernel bl , l ¤ k, denote lv the minimal one. Let p

be the minimum of these l™s. Then the set of vectors with lv = p is a submodule

homogeneous in order. In particular, every irreducible module is homogeneous

in order.

Consider a g1 module V homogeneous in order. For every non zero vector

1

a = p — v ∈ S p (g’1 ) — V — ‚ (T1 V )— and ∈ g1 we get

k

1

p’1 p’2

p

.a = ’p — [‚1 , ]v + —¦ [‚1 , [‚1 , ]] — v.

1 1

2

d d

= x2 dx so that [‚1 , ] = 2x dx =: 2h and [‚1 , [‚1 , ]] = 2‚1 .

Take

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

34. Multilinear natural operators 285

Assume now b1 .a = 0. Then

p’1

— (’2ph.v + p(p ’ 1)v)

0 = .a = 1

so that 2h.v = (p ’ 1)v or p = 0.

d

Further, set = x3 dx . We get

p p’2

— [‚1 , [‚1 , ]].v ’ p p’3 .[‚1 , [‚1 , [‚1 , ]]] — v

0 = .a = 21 31

p’2

— (3p(p ’ 1)h.v ’ p(p ’ 1)(p ’ 2)v).

= 1

Hence either p = 0 or p = 1 or 3h.v = 3 (p ’ 1)v = (p ’ 2)v. The latter is not

2

possible, for it says p = ’1. The case p = 0 is not interesting since the action

of b1 on all vectors in V — = S 0 (g’1 ) — V — is trivial. But if p = 1 we get h.v = 0

and so the homogeneity in order implies the action of g1 on V is trivial. Hence

1

V = R if irreducible. Moreover, the submodule generated by a in (T1 R)— is 1

1

1 — R with the action h.t 1 = 0 + t 1 . Hence if • : T1 V ’ W is a g-module

homomorphism and if both V and W are irreducible, then either • factorizes

through • : V ’ W which means V = W , • = k.idV , or V = R, W = R— with

the minus identical action of g1 . In this way we have proved theorem 34.2 in the

1

dimension one.

34.8. The situation in higher dimensions is much more di¬cult. Let us mention

some concepts and results from representation theory. Our source is [Zhelobenko,

Shtern, 83] and [Naymark, 76].

Consider a Lie algebra g and its representation ρ in a vector space V . An

element » ∈ g— is called a weight if there is a non zero vector v ∈ V such that

ρ(x)v = »(x)v for all x ∈ g. Then v is called a weight vector (corresponding

to »). If h ‚ g is a subalgebra, then the weights of the adjoint representation

of h in g are called roots of the algebra g with respect to h. The corresponding

weight vectors are called the root vectors (with respect to h).

A maximal solvable subalgebra b in a Lie algebra g is called a Borel subalgebra.

A maximal commutative subalgebra h ‚ g with the property that all operators

adx, x ∈ h, are diagonal in g is called a Cartan subalgebra.

In our case g = gl(n), the upper triangular matrices form the Borel subalgebra

b+ while the diagonal matrices form the Cartan subalgebra h. Let us denote

n+ the derived algebra [b+ , b+ ], i.e. the subalgebra of triangular matrices with

zeros on the diagonals. Consider a gl(n)-module V . A vector v ∈ V is called

the highest weight vector (with respect to b+ ) if there is a root » ∈ h— such that

x.v ’ »(x)v = 0 for all x ∈ h and x.v = 0 for all x ∈ n+ . The root » is called

the highest weight. In our case we identify h— with Rn .

The highest weight vectors always exist for complex representations of com-

plex algebras and are uniquely determined for the irreducible ones. The pro-

cedure of complexi¬cation allows to use this for the real case as well. So each

¬nite dimensional irreducible representation of gl(n) is determined by a high-

est weight (»1 , . . . , »n ) ∈ C such that all »i ’ »i+1 are non negative integers,

i = 1, . . . , n ’ 1.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

286 Chapter VII. Further applications

34.9. Examples. Let us start with the weight of the canonical representation

on Rn corresponding to the tangent bundle T . The action of a = (ak ), ak = δj δl ki

l l

‚

for some j < i, (corresponding to the action of X = xi ‚xj given by the negative

of the Lie derivative) on a highest weight vector v must be zero, so that only its

¬rst coordinate can be nonzero. Hence the weight is (1, 0, . . . , 0).

Now we compute the weights of the irreducible modules Λp Rn— . The action

‚

of X = xi ‚xj on a (constant) form ω is L’X ω. Since LX dxl = δj dxi we l

get (cf. 7.6) that if X.ω = 0 for all j < i then ω is a constant multiple of

dxn’p+1 § · · · § dxn . Further, the action of L’xi /‚xi on dxi1 § · · · § dxip is

minus identity if i appears among the indices ij and zero if not. Hence the

highest weight is (0, . . . , 0, ’1, . . . , ’1) with n ’ p zeros. Similarly we compute

the highest weight of the dual Λp Rn , (1, . . . , 1, 0, . . . , 0) with n ’ p zeros.

Analogously one ¬nds that the highest weight vector of S p Rm— is the sym-

metric tensor product of p copies of dxn and the weight is (0, . . . , 0, ’p).

34.10. Let us come back to our discussion on singular vectors in (Tn V )— for an

k

irreducible gl(n)-module V . In our preceding considerations we can take suitable

subalgebras with grading instead of the whole algebra g of formal vector ¬elds.

It turns out that one can describe in detail the singular vectors in dimension two

and for the subalgebra of divergence free formal vector ¬elds. We shall denote

this algebra by s(2) and we shall write sr for the Lie algebras of the corresponding

2

jet groups. We shall not go into details here, they can be found in [Rudakov, 74,

pp. 853“859]. But let us indicate why this description is useful. A subalgebra

a ‚ g is called a testing subalgebra if there is an isomorphism s(2) ’ a ‚ g

of algebras with gradings and a distinguished subspace w(a) ‚ g’1 such that

g’1 = a’1 • w(a), [a, w(a)] = 0.

k

Lemma. Let V be a g1 -module, (Tn V )— = i=0 S i (g’1 ) — V — and a ‚ g be a

k

n

k

¯

testing subalgebra. Then V = i=0 S i (w(a)) — V — ‚ (Tn V )— is an a0 -module

k

k ¯

and there is an a-module isomorphism (Tn V )— ’ i=0 S i (a’1 ) — V onto the

k

image.

∞

¯ S i (w(a)) — V — is an a0 -module, for [a, w(a)] = 0. We have

Proof. V = i=0

∞ ∞ ∞

¯ S j (w(a)) — V —

S i (a’1 ) — V = S i (a’1 ) —

i=0 i=0 j=0

∞

S i (a’1 • w(a)) — V — .

= i=0

34.11. It turns out that there are enough testing subalgebras in the algebra of

formal vector ¬elds. Using the results on s(2), Rudakov proves that for every g1 - n

— 1

module V the homogeneous singular vectors can appear only in V • S (g’1 ) —

V — ‚ (Tn V )— . This is equivalent to the assertion that all linear natural operators

k

are of order one.

Let us remark that this was also proved by [Terng, 78] in a very interesting

way. She proved that every tensor ¬eld is locally a sum of ¬elds with polynomial

coe¬cients of degree one in suitable coordinates (di¬erent for each summand)

and so the naturality and linearity imply that the orders must be one.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

34. Multilinear natural operators 287

34.12. Now, we know that if there is a homogeneous singular vector x which

does not lie in V — ‚ (Tn V )— then there must be a highest weight singular vector

1

x ∈ g— — V — , for all linear representations in question are ¬nite dimensional.

’1

k

i=1 li — ui , where k ¤ n and all i are assumed linearly

Let us write x =

independent.

k

Proposition. Let x = i=1 li — ui be a singular vector of highest weight with

respect to the Borel algebra b+ ‚ g1 . If ui = 0, i = 1, . . . , p, and up+1 = 0,

n

then ui = 0, i = p + 1, . . . , k, and up is a highest weight vector with weight

» = (1, . . . , 1, 0, . . . , 0) with n ’ p + 1 zeros. Then the weight of x is µ =

(1, . . . , 1, 0, . . . , 0) with n ’ p zeros.

Proof. Since x is singular, we have for all k, j, l (we do not use summation

conventions now)

(1)

0 = ’xk xl ‚xj . p p — up = p 1 — [ ‚xp , xk xl ‚xj ].up = xl ‚xj .uk + xk ‚xj .ul .

‚ ‚ ‚ ‚ ‚

In particular, for all k, j

xk ‚xj .uk = 0

‚

(2)

xj ‚xj .uk = ’xk ‚xj .uj .

‚ ‚

(3)

Further, x is a highest weight vector with weight µ = (µ1 , . . . , µn ) and for all i,

j we have

xi ‚xj .x =

‚ ‚ ‚ i‚

— xi ‚xj .up + — up

(4) p [’ ‚xp , x ‚xj ]

pp

‚

— xi ‚xj .up + — ui .

= pp j

‚

If i > j, we have xi ‚xj .x = 0 and so

xi ‚xj .up = 0

‚

(5) p=j

xi ‚xj .uj = ’ui .

‚

(6)

‚

Further, xi ‚xi .x = µi x and so (4) implies for all p, i

p

xi ‚xi .up = (µi ’ δi )up .

‚

(7)

The latter formula shows that the vectors up are either zero or root vectors

of V — with respect to the Cartan algebra h with weights »(p) = (»1 , . . . , »n ),

p

»i = µi ’ δi . Formula (2) implies that either up = 0 or µp = 1. If up = 0,

then all ul = 0, l ≥ p, by (6). Assume up = 0 and up+1 = 0, i.e. µi = 1, i ¤ p.

‚

Then (5) and (6) show that up is a highest weight vector. By (3), xj ‚xj .uk =

‚

»(k)j uk = ’xk ‚xj .uj , so that for k = p, j > p, (7) implies »(p)j up = µj .up =

‚

’xp ‚xj .uj = 0. Hence µi = 1, i = 1, . . . , p, and µi = 0, i = p + 1, . . . , n.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

288 Chapter VII. Further applications

34.13. Now it is easy to prove theorem 34.2. If D : E1 E is a linear natural

1

operator between bundles corresponding to irreducible Gn -modules V , W , then

either V — = Λp Rn , p = 0, . . . , n ’ 1, and W — = Λp+1 Rn , or D is a zero order

operator. The dual of the inclusion W — ’ (Tn V )— corresponds to the exterior

1

di¬erential up to a constant multiple.

Let us remark, that the only part of the proof we have not presented in detail

is the estimate of the order, but we mentioned a purely geometric way how to

prove this, cf. 34.11. It might be useful in concrete situations to combine some

general methods with ¬nal computations in the above style.

34.14. The method of testing subalgebras is heavily used in [Rudakov, 75] deal-

ing with subalgebras of divergence free formal vector ¬elds and Hamiltonian vec-

tor ¬elds. The aim of all the mentioned papers by Rudakov is the study of in¬nite

dimensional representations of in¬nite dimensional Lie algebras of formal vector

∞

¬elds. His considerations are based on the study of the space i=0 S i (g’1 ) — V —

and so the results are relevant for our purposes as well. We should remark that in

‚

the cited papers the action slightly di¬ers in notation and the vector ¬elds xi ‚xj

are identi¬ed with the transposed matrix (ai ) to our (aj ) and so the weights cor-

j i

respond to the Borel subalgebra of lower triangular matrices. Due to Rudakov™s

results, a description of all linear operations natural with respect to unimodular

or symplectic di¬eomorphisms is also available. In the unimodular case we get

the following result. We write S n for the category of n-dimensional manifolds

with ¬xed volume forms and local di¬eomorphisms preserving the distinguished

forms.

Theorem. All non zero linear natural operators D : E1 E between two ¬rst

order natural bundles on category S n corresponding to irreducible representa-

tions of the ¬rst order jet group are

(1) E1 = E, D = k.id, k ∈ R

(2) E1 = Λp T — , E = Λp+1 T — , D = k.d, k ∈ R, n > p ≥ 0

i

(3) E1 = Λn’1 T — , E = Λ1 T — , D = k.(d —¦ i —¦ d) : Λn’1 T — ’ Λn T — ’ Λ0 T — ’

’

∼ =

Λ1 T — , k ∈ R.

Let us point out that this theorem describes all linear natural operations not

only up to decompositions into irreducible components but also up to natural

equivalences. For example, to ¬nd linear natural operations with vector ¬elds

we have to notice Rn ∼ Λn’1 Rn— , ‚xp ’ i( ‚xp )(dx1 § · · · § dxn ). Hence the

‚ ‚

=

Lie di¬erentiation of the distinguished volume forms corresponds to the exte-

rior di¬erential on (n ’ 1)-forms, the identi¬cation of n-forms with functions

yields the divergence of vector ¬elds and the exterior di¬erential of the diver-

gence represents the ˜composition™ of exterior derivatives from point (3). Beside

the constant multiples of identity, there are no other linear operations (with

irreducible target).

We shall not describe the Hamiltonian case. We remark only that then not

even the di¬erential forms correspond to irreducible representations and that

the interesting operations live on irreducible components of them.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

34. Multilinear natural operators 289

34.15. Next we shall shortly comment some results concerning m-linear oper-

ations. We follow mainly [Kirillov, 80]. So ρ— will denote a representation dual

to a representation ρ of G1 + and we write ∆ for the one-dimensional represen-

n

tation given by a ’ deta’1 . Further “ρ (M ) is the space of all smooth sections

of the vector bundle Eρ over M corresponding to ρ. In particular “∆ (M ) co-

incides with „¦n M . To every representation ρ we associate the representation

ρ := ρ— — ∆. The pointwise pairing on “ρ (M ) — “ρ— (M ) gives rise to a bilinear

˜

mapping “ρ (M ) — “ρ (M ) ’ „¦n (M ), a natural bilinear operation of order zero.

˜

Given two sections s ∈ “ρ (M ), s ∈ “ρ (M ) with compact supports we can inte-

˜ ˜

grate the resulting n-form, let us write s, s for the outcome. We have got a

˜

bilinear functional invariant with respect to the di¬eomorphism group Di¬M .

For every m-linear natural operator D of type (ρ1 , . . . , ρm ; ρ) we de¬ne an

(m + 1)-linear functional

FD (s1 , . . . , sm , sm+1 ) = D(s1 , . . . , sm ), sm+1 ,

de¬ned on sections si ∈ “ρi (M ), i = 1, . . . , m, sm+1 ∈ “ρ (M ) with compact

˜

supports. The functional FD satis¬es

(1) FD is continuous with respect to the C ∞ -topology on “ρi and “ρ ˜

(2) FD is invariant with respect to Di¬M

(3) FD = 0 whenever ©m+1 suppsi = ….

i=1

We shall call the functionals with properties (1)“(3) the invariant local func-

tionals of the type (ρ1 , . . . , ρm ; ρ).

˜

Theorem. The correspondence D ’ FD is a bijection between the m-linear

natural operators of type (ρ1 , . . . , ρm ; ρ) and local linear functionals of type

(ρ1 , . . . , ρm ; ρ).

˜

The proof is sketched in [Kirillov, 80] and consists in showing that each such

functional is given by an integral operator the kernel of which recovers the natural

m-linear operator.

34.16. The above theorem simpli¬es essentially the discussion on m-linear nat-

ural operations. Namely, there is the action of the permutation group Σm+1

on these operations de¬ned by (σFD )(s1 , . . . , sm+1 ) = FD (sσ1 , . . . , sσ(m+1) ),

σ ∈ Σm+1 . Hence a functional of type (ρ1 , . . . , ρm ; ρ) is transformed into a

functional of type (ρσ’1 (1) , . . . , ρσ’1 (m+1) ) and so for every operation D of the

type (ρ1 , . . . , ρm ; ρ) there is another operation σD. If σ(m + 1) = m + 1, then

this new operation di¬ers only by a permutation of the entries but, for example,

if σ transposes only m and m + 1, then σD is of type (ρ1 , . . . , ρm’1 , ρ; ρm ).

˜˜

In the simplest case m = 1, the exterior derivative d : „¦ M ’ „¦p+1 M is

p

transformed by the only non trivial element in Σ2 into d : „¦n’p’1 M ’ „¦n’p M .

If m = 2, the action of Σ3 becomes signi¬cant. We shall now describe all

operations in this case. Those of order zero are determined by projections of

ρ1 — ρ2 onto irreducible components.

34.17. First order bilinear natural operators. We shall divide these op-

erations into ¬ve classes, each corresponding to some intrinsic construction and

the action of Σ3 .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

290 Chapter VII. Further applications

1. Write „ for the canonical representation of G1 + on Rn , i.e. “„ (M ) are the

n

smooth vector ¬elds on M . For every representation ρ we have the Lie derivative

L : “„ (M ) — “ρ (M ) ’ “ρ (M ), a natural operation of type („, ρ; ρ). The action

of Σ3 yields an operation of type (ρ, ρ; „ ) allowing to construct invariantly a

˜˜

covector density from any two tensor ¬elds which admit a pointwise pairing into

a volume form. This operation appears often in the lagrangian formalism and

Nijenhuis called it the lagrangian Schouten concomitant.

2. This class contains the operations of the types (Λk „ —∆κ , Λl „ —∆» ; Λm „ —

∆µ ), where k, l, m are certain integers between zero and n while κ, », µ are

certain complex numbers.

Assume ¬rst k + l > n + 1. Then an operation exists if m = k + l ’ n ’ 1,

µ = κ + » ’ 1. Let us choose an auxiliary volume form v ∈ “∆ (M ) and use the

identi¬cation Λk „ —∆κ ∼ Λn’k „ — —∆κ’1 , i.e. we shall construct an operation of

=

the type (Λ „ —∆ , Λ „ —∆» ; Λm „ — —∆µ ) with k +l ¤ n’1, m = k +l +1

k— l—

κ

and µ = κ + » . Then we can write a ¬eld of type Λk „ — ∆κ in the form ω.v κ’1 ,

ω ∈ Λn’k T — M . We de¬ne

D(ω1 .v κ’1 , ω2 .v »’1 ) = (c1 dω1 § ω2 + c2 ω1 § dω2 ).v µ’1

(1)

where ω1 is a (n’k)-form, ω2 is a (n’l)-form, and c1 , c2 are constants. The right

hand side in (1) should not depend on the choice of v. So let us write v = •.˜ v

κ’1 κ’1 »’1 »’1

where • is a positive function. Then ω1 .v = ω1 .˜

˜v , ω2 .v = ω2 .˜