1. Preliminaries
The Heisenberg group (of order
), 
is a noncommutative nilpotent Lie group whose underlying manifold is 
with coordinates 
and group law given by
Setting
, then 
forms a real coordinate system for
. In this coordinate system, we define the following vector fields:
It is clear from [1] that 
is a basis for the left invariant vector fields on 
These vector fields span the Lie algebra 
of 
and the following commutation relations hold:
Similarly, we obtain the complex vector fields by setting
(1)
In the complex coordinate, we also have the commutation relations
The Haar measure on 
is the Lebesgue measure 
on 
[2]. In particular, for
, we obtain the 3-dimensional Heisenberg group 
(since
). Hence 
may also be referred to as (2n + 1)-dimensional Heisenberg group.
One significant structure that accompanies the Heisenberg group is the family of dilations
This family is an automorphism of
. Now, if 
is an automorphism, there exists an induced automorphism, 
such that
For simplicity, assume that 
and 
coincide. Thus we may simply assume that if 
we have 
2. Heisenberg Laplacian
An operator that occurs as an analogue (for the Heisenberg group) of the Laplacian
on 
is denoted by 
where 
is a parameter and defined by
where 
are as defined in (1) so that 
can be written as
(2)
is called the sublaplacian. 
satisfies symmetry properties analogous to those of 
on
. Indeed, we have that 
1) is left-invariant on
;
2) has degree 2 with respect to the dilation automorphism of 
and 3) is invariant under unitary rotations.
Several methods for the determination of solutions, fundamental solutions of (2) and conditions for local solvability are well known [3-5].
The Heisenberg-Laplacian is a subelliptic differential operator defined for 
as 
on 
and denoted by
. It is obtained from the usual vector fields as
(3)
By a technique in [6], the operator 
is factorized into two quasi-linear first order operators on 
as:
and
so that
Introducing the Lie algebra structure, we have
indicating that the Heisenberg algebra is noncommutative and 
is hypoelliptic [4]. We thus obtain an operator (which is a homogeneous element of
, the universal enveloping algebra of the Heisenberg group when 
is the Heisenberg algebra) [5] consistent with that of Hans Lewy [7]. In [2], it has been shown that none of the factors of
, 
or 
is solvable and as such, 
is not solvable.
In this paper, we shall prove that 
only possesses a trivial group-invariant solution and for 
a compact subgroup of 
we have that
the K-invariant universal enveloping algebra of the Heisenberg group is generated by 
and
.
Now, by a solution of a factor 
say, we shall mean that if 
are independent real variables, and 
such that 
has a solution 
in the neighbourhood 
of the point
, with 
then 
is analytic at
.
Definition 2.0. Let 
be any open subset of
, and 
a number such that 
A function 
on 
satisfying
is said to be uniformly Holder continuous with Holder exponent 
if 
when 
they are called uniformly Lipschitz continuous. When 
they are simply continuous and bounded. A function is said to be in 
-space if its first partial derivatives satisfy a Holder condition with positive exponent, provided the distance of the points involved does not exceed 1.
Theorem 2.1. Let 
be a periodic real 
-function which is analytic in no t-interval. Then there exists a 
-function 
determined by the derivative 
of 
such that
has no 
-solution,(no matter what open 
-set taken as domain of existence).
For Proof, see [8].
Theorem 2.2. The Heisenberg Laplacian, 
defined in (3) has no non-trivial group invariant solution.
Proof. Let 
be a group-invariant solution of (3). We wish to show that 
To do this, let 
be a map generated by the group of automorphisms, dilations 
where 
determines the growth or decay rate. If 
is defined by
then obtaining the first and second order derivatives of 
with respect to the independent variables we have
Substituting these into (3), we obtain a trivial equation. But by Group-invariant method, we should obtain a system of ordinary differential equations of lower order (see [9] p. 185). Thus, there exists no non-trivial groupinvariant solution for
. □
Theorem 2.3. Let 
be a compact subgroup of
, then 
the 
-invariant universal enveloping algebra of the Heisenberg group is generated by 
and
.
Proof. Let 
be the algebra of 
-invariant differential operators on 
and let 
be the symmetric algebra generated by the set
We note that the derived action of 
on 
is given by
and 
acts on 
via
and on 
the 
-valued polynimial functions on 
-vector space 
via
Now, if we identify 
with the complexified symmetric algebra 
then the symmetric product 
of 
becomes the polynomial 
given by
Now, define a symmetrization map by
with
Now since 
acts on 
and 
by automorphism and 
defined by
induces an algebra map on the associated graded algebras and by induction [10, p. 282] the eigenfunctions of 
and 
are eigenfunctions of any element in 
we have that the following diagram is commutative.
for 
Since 
is a linear isomorphismit maps 
onto 
Since the action of
preserves degree on
, and by [11], if
generates 
then,
generates 
If 
then
where the sum is finite and each 
is a polynomial which is 
-invariant. Thus, the result follows by the fact that the eigenfunctions of 
and 
are the eigenfunctions of 
[12]. □