Space of Linear Differential Operators on the Real Line as a Module over the Lie Algebra of Vector Fields

Abstract

Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To define this family, one considers arguments of differential operators as tensor-densities of degree $\lambda$.) In this paper we solve the problem of isomorphism between $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to different values of $\lambda$. The result is as follows: for $k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each other for every values of $\lambda\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm \frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant) intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case $(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to two different values of the degree: $\lambda$ and $\lambda^{\prime}$, are isomorphic if and only if $\lambda+\lambda^{\prime}=1$.

Full text

arXiv:dg-ga/9602004v1 4 Feb 1996
Space of linear differential operators on the
real line as a module over the Lie algebra of
vector fields
H. Gargoubi, V.Yu. Ovsienko
CNRS, Centre de Physique Th´eorique ∗
Abstract
Let Dkbe the space of k-th order linear differential operators on R:A=
ak(x)dk
dxk+···+a0(x). We study a natural 1-parameter family of Diff(R)- (and
Vect(R))-modules on Dk. (To define this family, one considers arguments of differ-
ential operators as tensor-densities of degree λ.) In this paper we solve the problem
of isomorphism between Diff(R)-module structures on Dkcorresponding to differ-
ent values of λ. The result is as follows: for k= 3 Diff(R)-module structures on D3
are isomorphic to each other for every values of λ6= 0,1,1
2,1
2±√21
6, in this case
there exists a unique (up to a constant) intertwining operator T:D3→ D3. In the
higher order case (k≥4) Diff(R)-module structures on Dkcorresponding to two
different values of the degree: λand λ′, are isomorphic if and only if λ+λ′= 1.
∗CPT-CNRS, Luminy Case 907, F–13288 Marseille, Cedex 9 FRANCE

1 Introduction
Space of linear differential operators on a manifold Mhas various algebraic
structures: the structure of associative algebra and of Lie algebra, in the
1-dimensional case it can be considered as an infinite-dimensional Poisson
space (with respect to so-called Adler-Gelfand-Dickey bracket).
1.1 Diff(M)-module structures. One of the basic structures on the
space of linear differential operators is a natural family of module structures
over the group of diffeomorphisms Diff(M) (and of the Lie algebra of vec-
tor fields Vect(M)). These Diff(M)- (and Vect(M))-module structures are
defined if one considers the arguments of differential operators as tensor-
densities of degree λon M.
In this paper we consider the space of differential operators on R.1Denote
Dkthe space of k-th order linear differential operators:
A(φ) = ak(x)dkφ
dxk+···+a0(x)φ(1)
where ai(x), φ(x)∈C∞(R).
Define a 1-parameter family of Diff(R)-module structures on C∞(R) by:
g∗
λφ:= φ◦g−1· dg−1
dx !−λ
where λ∈R(or λ∈C) is a parameter. Geometrically speaking, φis a
tensor-density of degree −λ:
φ=φ(x)(dx)−λ.
A 1-parameter family of actions of Diff(R) on the space of differential
operators (1) is defined by:
g(A) = g∗
λA(g∗
λ)−1.
1Particular cases of actions of Diff(R) and Vect(R) on this space were considered by
classics (see [1], [14]). The well-known example is the Sturm-Liouville operator d2
dx2+
a(x) acting on −1
2-densities (see e.g. [1], [14], [13]). Already this simplest case leads to
interesting geometric structures and is related to so-called Bott-Virasoro group (cf. [9],
[12]).
2

Denote Dk
λthe space of operators (1) endowed with the defined Diff(R)-
module structure.
Infinitesimal version of this action defines a 1-parameter family of Vect(R)-
module structures on Dk(see Sec. 3 for details).
1.2 The problem of isomorphism. Let Mbe a manifold, dim M≥2.
The problem of isomorphism of Diff(M)- (and Vect(M))-module structures
for different values of λwas stated in [4] and saved in the case of second order
differential operators. In this case, different Diff(M)-module structures are
isomorphic to each other for every λexcept 3 critical values: λ= 0,−1
2,−1
(corresponding to differential operators on: functions, 1
2-densities and volume
forms respectively).
Geometric quantization gives an example of such a special Diff(M)-module:
differential operators are considered as acting on 1
2-densities (see [7]).
Recently P.B.A. Lecomte, P. Mathonet, and E. Tousset [8] showed that in
the case of differential operators of order ≥3, Diff(M)-modules corresponding
to λand λ′-densities are isomorphic if and only if λ+λ′= 1. The unique
isomorphism in this case is given by conjugation of differential operators.
These results solve the problem of isomorphism in the multi-dimensional
case.
It was shown in [4], [8], that the case dim M= 1 (M=Ror S1) is
particular. It is reacher in algebraic structures and therefore is of a special
interest.
In this paper we solve the problem of isomorphism of Diff(R)-modules
Dk
λfor any k. The result is as follows.
1) The modules D3
λof third order differential operators (1) are isomorphic
to each other for all values of λexcept 5 critical values:
{0,−1,−1
2,−1
2+√21
6,−1
2−√21
6}.
(this result was announced in [4]).
2) The Diff(R)-modules Dk
λand Dk
λ′on the space of differential operators
(1) of order k≥4 are isomorphic if and only if λ+λ′=−1.
1.3 Intertwining operator. The most important result of the paper is
a construction of the unique (up to a constant) equivariant linear operator
3

on the space of third order differential operators:
T:D3
λ→ D3
µ(2)
for λ, µ 6= 0,−1,−1
2,−1
2±√21
6, see the explicit formulæ (3), (7) and (8)
below. It has nice geometric and algebraic properties and seems to be an
interesting object to study.
Operator Tis an analogue of the second order Lie derivative from [4]
intertwining different Diff(M)-actions on the space of second order differential
operators on a multi-dimensional manifold M.
1.3 Normal symbols. The main tool of this paper is the notion of a
normal symbol, which we define in the case of 4-th order differential operators.
We define a sl2-equivariant way to associate a polynomial function of degree
4 on T∗Rto a differential operator A∈ D4
λ. In the case of second order
operators the notion of normal symbol was defined in [4]. This construction
is related with the results of [3]. We discuss the geometric properties of the
normal symbol and its relations to the intertwining operator (2).
2 Main results
We formulate here the main results of this paper, all the proofs will be given
in Sec. 3-7.
2.1 Classification of Diff(R)-modules. First, remark that for each
value of k, there exists an isomorphism of Diff(R)-modules:
Dk
λ∼
=Dk
−1−λ.
It is given by conjugation A7→ A∗:
A∗=
k
X
i=1
(−1)idi
dxi◦ai(x)
The following two theorems give a solution of the problem of isomorphism
of Diff(R)-modules Dk
λon space Dk.
The first result was announced on [4]:
4

Theorem 1. (i) All the Diff(R)-modules D3
λwith λ6= 0,−1,−1
2,−1
2+
√21
6,−1
2−√21
6are isomorphic to each other.
(ii) The modules D3
0,D3
−1
2
,D3
−1
2+√21
6
are not isomorphic to D3
λfor general λ.
It follows from the general isomorphism ∗:Dk
λ∼
=Dk
−1−λ, that
D3
0∼
=D3
−1and D3
−1
2+√21
6∼
=D3
−1
2−√21
6
.
Therefore, there exist 4 non-isomorphic Diff(R)-module structures on the
space D3.
Theorem 2. For k≥4, the Diff(R)-modules Dk
λand Dk
λ′are isomorphic if
and only if λ+λ′=−1.
This result shows that operators of order 3 play a special role in the
1-dimensional case (as operators of order 2 in the case of a manifold of di-
mension ≥2, cf. [4], [8]).
2.2 Intertwining operator T.
Theorem 3. For λ, µ 6= 0,−1,−1
2,−1
2±√21
6there exists a unique (up to a
constant) isomorphism of Diff(R)-modules D3
λand D3
µ.
Let us give an explicit formula for the operator (2).
Every differential operator of order 3 can be written (not in a canonical
way) as a linear combination of four operators:
1) a zero order operator of multiplication by a function: φ(x)7→ φ(x)f(x),
2) a first order operator of Lie derivative:
Lλ
X=X(x)d
dx −λX′(x),
where X′=dX
dx ,
3) symmetric “anti-commutator” of Lie derivatives:
[Lλ
X, Lλ
Y]+:= Lλ
X◦Lλ
Y+Lλ
Y◦Lλ
X
4) symmetric third order expression:
[Lλ
X, Lλ
Y, Lλ
Z]+:= SymX,Y,Z(Lλ
X◦Lλ
Y◦Lλ
Z)
for some vector fields X(x)d
dx , Y (x)d
dx , Z(x)d
dx .
5

Theorem 4. The following formula:
T(f) = µ(µ+ 1)(2µ+ 1)
λ(λ+ 1)(2λ+ 1) f
T(Lλ
X) = 3µ2+ 3µ−1
3λ2+ 3λ−1Lµ
X
T([Lλ
X, Lλ
Y]+) = 2µ+ 1
2λ+ 1[Lµ
X, Lµ
Y]+
T([Lλ
X, Lλ
Y, Lλ
Z]+) = [Lµ
X, Lµ
Y, Lµ
Z]+
(3)
defines an intertwining operator (2).
A remarkable fact is that the formula (3) does not depend on the choice
of X, Y, Z and frepresenting the third order operator. (Indeed, the formulæ
(7) and (8) below give the expression of Tdirectly in terms of coefficients of
differential operators.) Moreover, this property fixes the coefficients in (3) in
a unique way (up to a constant).
Remarks. 1) In the case of multi-dimensional manifold M, almost all
Diff(M)-module structures on the space of second order differential oper-
ators are isomorphic to each other and the corresponding isomorphism is
unique (up to a constant) [4]; there is no isomorphism between different
Diff(M)-module structures on the space of third order operators, except the
conjugation [8].
2) The formula (3) gives an idea that it would be interesting to study the
commutative algebra structure (defined by the anti-commutator) on the Lie
algebra of all differential operators.
3 Action of Vect(R)on space D4
To prove Theorems 1-4, it is sufficient to consider only the Vect(R)-action on
Dk. Indeed, since the Diff(R)-action on the space of differential operators is
local, therefore, the two properties: of Vect(R)- and of Diff(R)-equivariance
are equivalent.
6

3.1 Definition of the family of Vect(R)-actions. Let Vect(R) be the
Lie algebra of smooth vector fields on R:
X=X(x)d
dx
with the commutator
[X(x)d
dx, Y (x)d
dx] = (X(x)Y′(x)−X′(x)Y(x)) d
dx,
where X′=dX/dx.
The action of Vect(R) on space Dkis defined by:
adLλ
X(A) := Lλ
X◦A−A◦Lλ
X
where
Lλ
Xφ=X(x)φ′(x)−λX′(x)φ(x)
The last formula defines a 1-parameter family of Vect(R)-actions on C∞(R).
One obtains a 1-parameter family of Vect(R)-modules on Dk.
Notation. 1. The operator Lλ
Xis called the operator of Lie derivative of
tensor-densities of degree −λ. Denote Fλthe corresponding Vect(R)-module
structure on C∞(R).
2. As in the case of Diff(R)-module structures, we denote Dk
λspace Dk
as a Vect(R)-module.
3.2 Explicit formula. Let us calculate explicitly the action of Lie alge-
bra Vect(R) on space D4. Given a differential operator A∈ D4, let us use
the following notation for the Vect(R)-action adLX:
adLX(A) = aX
4(x)d4
dx4+aX
3(x)d3
dx3+aX
2(x)d2
dx2+aX
1(x)d
dx +aX
0(x).
Lemma 3.1. The action adLλ
Xof Vect(R)on space D4is given by :
aX
4=L4
X(a4)
aX
3=L3
X(a3) + 2(2λ−3)a4X′′
aX
2=L2
X(a2) + 3(λ−1)a3X′′ + 2(3λ−2)a4X′′′
aX
1=L1
X(a1) + (2λ−1)a2X′′ + (3λ−1)a3X′′′ + (4λ−1)a4XIV
aX
0=L0
X(a0) + λ(a1X′′ +a2X′′′ +a1XIV +a0XV)
(4)
7

Proof. One gets easily the formula (4) from the definition:
adLλ
X(A) = [Lλ
X, A] = (Xd
dx −λX′)(a4
d4
dx4+···+a0)
−(a4
d4
dx4+···+a0)(Xd
dx −λX′)
3.3 Remarks. It is convenient to interpret the action (4) as a deformated
standard action of Vect(R) on the direct sum:
F4⊕ F3⊕ F2⊕ F1⊕ F0.
(given by the first term of the right hand side of each equality in the formula
(4)). This interpretation is the motivation of the main construction of Sec.
4, it will be discussed in Sec. 7.2.
The main idea of proof of Theorems 1 and 2 is to find some normal form
(cf. [4]) of the coefficients a4(x),...,a0(x) for 4-order differential operators
on Rwhich reduce the action (4) to a canonical form.
4 Normal form of a symbol
It is convenient to represent differential operators as polynomials on the
cotangent bundle. The standard way to define a (total) symbol of an an
operator (1) is to associate to Athe polynomial
PA(x, ξ) =
k
X
i=0
ξiai(x),
on T∗R∼
=R2(where ξis a coordinate on the fiber). However, this
formula depends on coordinates, only the higher term ξkak(x) of PA(the
principal symbol) has a geometric sense.
4.1 The main idea. Lie algebra Vect(R) naturally acts on C∞(T∗R)
(it acts on the cotangent bundle).
Consider a linear differential operator A∈ D4. Let us look for a natural
definition of a symbol of Ain the following form:
PA(x, ξ) = ξ4¯a4(x) + ξ3¯a3(x) + ξ2¯a2(x) + ξ¯a1(x) + ¯a0(x),
8

where the functions ¯ai(x) are linear expressions of the coefficients ai(x) and
their derivatives.
Any symbol P(x, ξ) can be considered as a linear mapping
D4→ F4⊕ F3⊕ F2⊕ F1⊕ F0.
Indeed, the Lie algebra Vect(R) acts on each coefficient ¯ai(x) of the polyno-
mial PA(x, ξ) as on a tensor-density of degree −i:
LX(PA) =
4
X
i=0
ξiLi
X(¯ai).
However, there is no such a mapping which is Vect(R)-equivariant.
4.2 Definition. The normal symbol of A∈ D4
λas a polynomial PA(x, ξ)
such that the linear mapping A7→ PAis equivariant with respect to the
subalgebra sl2⊂Vect(R)generated by the vector fields
{d
dx, x d
dx, x2d
dx}.
Proposition. 4.1. (i) The following formula defines a normal symbol of a
differential operator A∈ D4
λ:
¯a4=a4
¯a3=a3+1
2(2λ−3)a′
4
¯a2=a2+ (λ−1)a′
3+2
7(λ−1)(2λ−3)a′′
4
¯a1=a1+1
2(2λ−1)a′
2+3
10 (λ−1)(2λ−1)a′′
3
+1
15 (λ−1)(2λ−1)(2λ−3)a′′′
4
¯a0=a0+λa′
1+1
3λ(2λ−1)a′′
2+1
6λ(λ−1)(2λ−1)a′′′
3
+1
30 λ(λ−1)(2λ−1)(2λ−3)a(IV )
4
(5)
(ii) The normal symbol is defined uniquely (up to multiplication of each
function ¯ai(x)by a constant).
Proof. Direct calculation shows that the Vect(R)-action adLλon D4
9

given by the formula (4) reads in terms of ¯aias:
¯aX
4=L4
X(¯a4)
¯aX
3=L3
X(¯a3)
¯aX
2=L2
X(¯a2) + 2
7(6λ2+ 6λ−5)J3(X, ¯a4)
¯aX
1=L1
X(¯a1) + 2
5(3λ2+ 3λ−1)J3(X, ¯a3)
+1
6λ(λ+ 1)(2λ+ 1)J4(X, ¯a4)
¯aX
0=L0
X(¯a0) + 2
3λ(λ+ 1)J3(X, ¯a2)
+1
6λ(λ+ 1)(2λ+ 1)J4(X, ¯a3)
+1
420 λ(λ+ 1)(12λ2+ 12λ+ 11)J5(X, ¯a4)
(6)
where ¯aX
iare coefficients of the normal symbol of the operator adLλ
X(A) and
the expressions Jmare:
J3(X, ¯as) = X′′′¯as
J4(X, ¯as) = sX (I V )¯as+ 2X′′′¯a′
s
J5(X, ¯as) = s(2s−1)X(V)¯as+ 5(2s−1)X(IV )¯a′
s+ 10X′′′¯a′′
s
It follows that the mapping D4→ F4⊕ ··· ⊕ F0defined by (5) is sl2-
equivariant. Indeed, for a vector field X∈sl2(which is a polynomial in xof
degree ≤2) all the terms Jm(X, ¯as) in (6) vanish.
Proposition 4.1 (i) is proven.
Let us prove the uniqueness. By definition, the functions ¯ai(x) are linear
expressions in as(x) and their derivatives:
¯ai(x) = X
s,t
αs
t(x)a(t)
s(x),
where a(t)
s=dtas/dxt,αj
k(x) are some functions. The fact that the normal
symbol PAis sl2-equivariant means that for a vector field X∈sl2, ¯aX
i=
Li
X(ai).
a) Substitute X=d/dx to obtain that the coefficients αj
kdoes not depend
on x;
b) substitute X=xd/dx to obtain the condition j−k=i:
¯ai(x) =
i
X
j=4
αja(j−i)
j(x),
10

and αi6= 0;
c) put αi= 1 and, finally, substitute X=x2d/dx to obtain the coefficients
from (5).
Proposition 4.1 (ii) is proven.
The notion of normal symbol of a 4-th order differential operator plays a
central role in this paper.
4.3 Remark: the transvectants. Operations J3(X, as), J4(X, as), J5(X, as)
are particular cases of the following remarkable bilinear operations on tensor-
densities. Consider the expressions:
jn(φ, ψ) = X
i+j=n
(−1)i n
i!(2λ−i)!(2µ−j)!
(2λ−n)!(2µ−n)!φ(i)ψ(j)
where φ=φ(x), ψ =ψ(x) are smooth functions.
This operations defines unique (up to a constant) sl2-equivariant map-
ping:
Fλ⊗ Fµ→ Fλ+µ−n
Operations jn(φ, ψ) were discovered by Gordan [6], they are also known as
Rankin-Cohen brackets (see [11], [2]).
Note, that the operations Jmfrom the formula (6) are proportional to jn
for X∈ F1, as∈ F−s.
5 Diagonalization of operator T
We will obtain here an important property of the intertwining operator (2):
in terms of normal symbol it has a diagonal form. We will also prove the
part (i) of Theorem 1 and Theorem 4.
5.1 Proof of Theorem 1, part (i). Let us define an isomorphism of
modules D3
λand D3
µfor λ6= 0,−1,−1
2,−1
2±√21
6. Associate to A∈ D3
λthe
operator T(A)∈ D3
µ:
T:a3
d3
dx3+a2
d2
dx2+a1
d
dx +a07−→ aT
3
d3
dx3+aT
2
d2
dx2+aT
1
d
dx +aT
0
such that its standard symbol
PT(A)=ξ3aT3(x) + ξ2aT2(x) + ξaT1(x) + aT0(x)
11

is given by:
aT3(x) = ¯a3(x)
aT2(x) = 2µ+ 1
2λ+ 1¯a2(x)
aT1(x) = 3µ2+ 3µ−1
3λ2+ 3λ−1¯a1(x)
aT0(x) = µ(µ+ 1)(2µ+ 1)
λ(λ+ 1)(2λ+ 1) ¯a0(x)
(7)
It follows immediately from the formula (6), that this formula defines an
isomorphism of Vect(R)-modules: T:D3
λ∼
=D3
µ.
Theorem 1 (i) is proven.
5.2 Proof of Theorem 4. Let us show that the operator (7) in terms
of symmetric expressions of Lie derivatives is given by (3).
The first equality in (3) coincides with the last equality in (7).
1) Consider a first order operator of a Lie derivative
Lλ
X=X(x)d
dx −λX′(x).
Its normal symbol defined by (5) is
PLλ
X=ξX (x).
One obtains the second equality of the formula (3).
2) The anti-commutator
[Lλ
X, Lλ
Y]+= 2XY d2
dx2+ (1 −2λ)(XY )′d
dx −λ(XY ′′ +X′′Y) + 2λ2X′Y′
has the following normal symbol:
P[Lλ
X,Lλ
Y]+= 2ξ2XY −2
3λ(λ+ 1)(XY ′′ +X′′Y−X′Y′)
which also following from (5). The third equality of (3) follows now from the
second and the fourth ones of (7).
12

3) The normal symbol of a third order expression [Lλ
X, Lλ
Y, Lλ
Z]+:=
SymX,Y,Z (Lλ
XLλ
YLλ
Z) can be also easily calculated from (5). The result is:
P[Lλ
X,Lλ
Y,Lλ
Z]+= 6ξ3XY Z
−(3λ2+ 3λ−1)ξ(XY Z′′ +XY ′′Z+X′′Y Z −1
5(XY Z)′′)
−λ(λ+ 1)(2λ+ 1)(XY Z′′′ +XY ′′′Z+X′′′Y Z)
This formula implies the last equality of (3).
6.3 Remarks. a) The normal symbols of [Lλ
X, Lλ
Y]+and [Lλ
X, Lλ
Y, Lλ
Z]+
are given by very simple and harmonic expressions (which implies the diag-
onal form (3) of operator T). It would be interesting to understand better
the geometric reason of this fact.
b) Comparing the formulæ (3) and (7), one finds a coincidence between
coefficients. This fact shows that, in some sense, the symmetric expressions
of Lie derivatives and the normal symbol represent the same thing in terms of
differential operators and in terms of polynomial functions on T∗R, respec-
tively. We do not see any reason a-priori for this remarkable coincidence.
6 Uniqueness of operator T
In this section we prove that the isomorphism Tdefined by the formula (7) is
unique (up to a constant). We also show that in the higher order case k≥4
there is no analogues of this operator.
6.1 Proof of Theorem 3. The normal symbol of an operator A∈
D3
λis at the same time a normal symbol of T(A)∈ D3
µ, since operator
Tis equivariant. The normal symbol is unique up to normalization (cf.
Proposition 4.1, part (ii)), therefore the polynomial PT(A)(x, ξ) defined by
the formula (5), is of the form:
PT(A)(x, ξ) = α3ξ3¯a3(x) + α2ξ2¯a2(x) + α1ξ¯a1(x) + α0¯a0(x),
where αi∈Rare some constants depending on λand µ. Choose α3= 1.
It follows immediately from the formula (6) (after substitution a4≡0) that
the formula (7) gives the unique choice of the constants α2, α1, α0such that
operator Tis equivariant.
Theorem 3 is proven.
13

6.2 Proof of Theorem 2. Suppose now that Φ : D4
λ→ D4
µis an
isomorphism. In the same way, it follows that in terms of normal symbols,
operator Φ is diagonal. More precisely, if A∈ D4
λ, then the normal symbol
of the operator Φ(A)∈ D4
µis:
PΦ(A)(x, ξ) = α4ξ4¯a4(x) + α3ξ3¯a3(x) + α2ξ2¯a2(x) + α1ξ¯a1(x) + α0¯a0(x),
where ¯aiare the components of the normal symbol of A,αi∈R. The
condition of equivariance implies :
α2
α0
=λ(λ+ 1)
µ(µ+ 1),α3
α0
=λ(λ+ 1)(2λ+ 1)
µ(µ+ 1)(2µ+ 1)
α4
α0
=λ(λ+ 1)(12λ2+ 12λ+ 11)
µ(µ+ 1)(12µ2+ 12µ+ 11),α4
α2
=6λ2+ 6λ−5
6µ2+ 6µ−5
α3
α1
=3λ2+ 3λ−1
3µ2+ 3µ−1,α4
α1
=λ(λ+ 1)(2λ+ 1)
µ(µ+ 1)(2µ+ 1)
This system of equation has solutions if and only if λ=µor λ+µ=−1.
For λ=µone has: α0=α1=α2=α3=α4and for λ+µ=−1 one has:
α0=−α1=α2=−α3=α4.
Theorem 2 is proven for k= 4.
Theorem 2 follows now from one of the results of [8]: given an isomor-
phism Φ : Dk
λ→ Dk
µ, then the restriction of Φ to D4
λis an isomorphism
of Vect(R)-modules: D4
λ→ D4
µ. (To prove this, it is sufficient to suppose
equivariance of Φ with respect to the affine algebra with generators d
dx , x d
dx ,
see [8]).
This implies that λ=µor λ+µ=−1.
Theorem 2 is proven.
7 Relation with the cohomology group
H1(Vect(R); Hom(Fλ,Fµ))
The problem of isomorphism of Vect(R)-modules Dk
λfor different values of
λis related to the first cohomology group H1(Vect(R); Hom(Fλ,Fµ)). This
cohomology group has already been calculated by B.L. Feigin and D.B. Fuchs
(in the case of formal series) [5].
14

7.1 Nontrivial cocycles.The relation of Vect(R)-action on the space of
differential operators and the cohomology groups H1(Vect(R); Hom(Fλ,Fµ))
is given by the following construction.
Let us associate to the bilinear mappings Jmdefined by the formula (6),
a linear mapping cm: Vect(R)→Hom(Fs,Fs+1−m):
cm(X)(a) := Jm(X, a),
where a∈ Fs.
A remarkable property of transvectants J3and J4is:
Lemma 7.1. For each value of s, the mappings c3and c4are non-trivial
cocycles on Vect(R):
(i) c3∈Z1(Vect(R); Hom(Fs,Fs−2)),
(ii) c4∈Z1(Vect(R); Hom(Fs,Fs−3)).
Proof. From the fact that the formula (6) defines a Vect(R) action one
checks that for any X, Y ∈Vect(R) :
[LX, cm(Y)] −[LY, cm(X)] = cm([X, Y ])
with m= 3,4. This means, that c3and c4are cocycles.
The cohomology classes [c3],[c4]6= 0. Indeed, verify that c3and c4are
cohomological to the non-trivial cocycles:
e
c3(X)(a) = X′′′a+ 2X′′a′,
e
c4(X)(a) = X′′′a′+X′′a′′
from [5].
Lemma 7.1 is proven.
7.2 Proof of Theorem 1, part (ii). First, remark that for every
α1, α2, α3∈R, the formula
ρX(a3) = L3
X(a3)
ρX(a2) = L2
X(a2)
ρX(a1) = L1
X(a1) + α1J3(X, a3)
ρX(a0) = L0
X(a0) + α2J3(X, a2) + α3J4(X, ¯a3)
15

defines a Vect(R)-action. Indeed, this formula coincides with (6) in the
case a4≡0 and for the special values of α1, α2, α3, however, the constants
α1, α2, α3are independent.
The Vect(R)-action ρis a non-trivial 3-parameter deformation of the
standard action on the direct sum F3⊕ F2⊕ F1⊕ F0.
The fact, that the cocycles c3and c4are non-trivial, is equivalent to the
fact that the defined Vect(R)-modules with:
1) α1, α2, α36= 0,
2) α1= 0, α26= 0, α36= 0,
3) α16= 0, α2= 0, α36= 0,
4) α16= 0, α26= 0, α3= 0,
are not isomorphic to each other.
The Vect(R)-modules D3
λ(given by the formula (6) with a4≡0) corre-
sponds to the case 1) for general values of λ, to the case 2) for λ=−1
2±√21
6,
to the case 3) for λ=−1
2and to the case 4) for λ= 0,−1. Therefore, one
obtains 5 critical values of the degree for which Vect(R)-module structure
on the space of third order operators is special.
Theorem 1 (ii) is proven.
Remark. For each value of λat least one of constants α1, α2, α36=
0. This implies that the module D3
λis not isomorphic to the direct sum
F3⊕ F2⊕ F1⊕ F0.
8 Explicit formula for the intertwining oper-
ator
We give here the explicit formula for the operator (2) intertwining Vect(R)-
actions on D3which follows from the expression for the operator Tin terms
of the normal symbol (7).
For every A∈ D3
λ, the operator T(A)∈ D3
µ
T(A) = aT
3
d3
dx3+aT
2
d2
dx2+aT
1
d
dx +aT
0
is given by the following formula:
16

aT
3=a3
aT
2=2µ+ 1
2λ+ 1a2+3(µ−λ)
2λ+ 1 a′
3
aT
1=3µ2+ 3µ−1
3λ2+ 3λ−1a1
+(λ−µ)(µ(12λ−1) −λ+ 3)
2(2λ+ 1)(3λ2+ 3λ−1) a′
2
+3
2
µ2(5λ−1) −µ(6λ2+λ−1) + λ3+ 2λ2−λ
(2λ+ 1)(3λ2+ 3λ−1) a′′
3
aT
0=µ(µ+ 1)(2µ+ 1)
λ(λ+ 1)(2λ+ 1) a0
−µ3(3λ+ 5) −µ2(3λ2−6) −µ(5λ2+ 6λ)
(λ+ 1)(2λ+ 1)(3λ2+ 3λ−1) a′
1
+µ3(3 −λ)−µ2(6λ2+ 7λ−5) + µ(7λ3+ 4λ2−5λ)
2(λ+ 1)(2λ+ 1)(3λ2+ 3λ−1) a′′
2
−µ3(3λ2+ 1) −3µ2(λ2+ 2λ−1) −µ(3λ4−3λ3−5λ2+ 3λ)
2(λ+ 1)(2λ+ 1)(3λ2+ 3λ−1) a′′′
3
(8)
We de not prove this formula since we do not use it in this paper.
Remarks. 1) If λ=µ, then the operator Tdefined by this formula is
identity, if λ+µ=−1, then Tis the operator of conjugation.
2) The fact that operator Tis equivariant implies that the formula (8)
does not depend on the choice of the coordinate x.
9 Discussion and final remarks
Let us give here few examples and applications of the normal symbols (5).
9.1 Examples. The notion of normal symbol was introduced in [4] in the
case of second order differential operators. In this case, for λ=1
2(operators
on −1
2-densities), the normal symbol (5) is just the standard total symbol:
17

¯a2=a2,¯a1=a1,¯a0=a0. This corresponds to the classical example of second
order operators on −1
2-densities (cf. the footnote at the introduction).
In the same way, the semi-integer values λ= 1,3
2,2,... corresponds to
particularly simple expressions of the normal symbol for operators of order
k= 3,4,5,....
9.2 Modules of second order differential operators on R. a) The
module of second order operators D2
λwith λ= 0,−1, is decomposed to a
direct sum:
D2
0∼
=D2
−1∼
=F2⊕ F1⊕ F0.
Indeed, the coefficients ¯a2,¯a1,¯a0transform as tensor-densities (cf formula
(6)). This module is special: D2
λis not isomorphic to D2
0for λ6= 0,−1 (see
[4]).
b) For every λ, µ 6= 0,−1, D2
λ∼
=D2
µ.
9.3 Operators on 1
2-densities. For every k≥3, the module Dk
−1
2
(corresponding to 1
2-densities) is special. It is decomposed into a sum of
submodules: of symmetric operators and of skew-symmetric operators.
9.4 Normal symbol and Weil symbol. The Weil quantization defines
a 1-parameter family of mappings from the space of polynomials C[ξ, x] to
the space of differential operators on Rwith polynomial coefficients. One
associate to a polynomial the symmetric expression in ¯hd
dx and x:F(ξ, x)7→
SymF(¯hd
dx , x). This (1,1)-correspondence between differential operators and
polynomials is sl2-equivariant. However, in the Weil quantization the action
of the Lie algebra sl2on differential operators is generated by x2, x d
dx +d
dx x, d2
dx2
and therefore, is completely different from the normal symbol.
9.5 Automorphic (pseudo)differential operators. The notion of
canonical symbol is related (and in some sense inverse) to the construction of
the recent work P. Cohen, Yu. Manin and D. Zagier [3] of a P S L2-equivariant
(pseudo)differential operator associated to a holomorphic tensor-density on
the upper half-plane.
9.6 Exotic ⋆-product. Another way to understand this sl2-equivariant
correspondence between linear differential operators and polynomials in ξ, x
leads to a ⋆-product on the algebra of Laurent polynomials on T∗Rwhich is
not equivalent to the standard Moyal-Weil quantization (see [10]).
18

Acknowledgments. It is a pleasure to acknowledge enlightening discussions
with C. Duval, P.B.A. Lecomte, Yu.I. Manin and E. Mourre.
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19