SUB-LAPLACIANS WITH DRIFT
7
by an abelian Lie group A (for the semisimple splitting and other splittings of S
and their applications to the structure of solvmanifolds see [Al, A2, Go]).
It turns out that SN is isomorphic to the nil-shadow constructed by Auslander
and Green. This is trivial to verify when S = N \Rk. In the general case, one has
to go carefully through the proof of the semisimple splitting (the proof given in[V3
pp. 404-405] is more appropriate for this purpose). The construction that we give
here though of the nil-shadow SN is more convenient for our purposes. It is also
elementary, in the sense that it does not use the theory of algebraic groups.
Since S has polynomial volume growth, the eigenvalues of Ad:r,ir G S are of
the type
eia
with a G l . Thus the semisimple part of the action of S/N =
Rk
on
T V gives rise to rotations and hence to phenomena of homogenisation.
More precisely, let *s and *sN denote the group products of S and SN respec-
tively. Let us also denote by 5 and SN the Lie algebras of S and and SN and identify
them, respectively, with the *s and *5^-left invariant vector fields. Finally let us
fix a base {Xi, ...Xq} of SN- The interest of SN is due to the following fact:
Any *5-left invariant vector field E can be written as
(1.4.1) E(x) = a1(x)X1{x) + .... 4- aq(x)Xq(x),
where the coefficients ai(rr),..., aq(x) are quasi-periodic functions in the following
sense:
Let us denote by TT the quotient map 7r : S Rk = S/N. Then for all i = 1,..., q
there are constants ci, c'l5..., Cj, d- G R and vectors &i, &]_..., bj, b'- G Rfc such that
di(x) = 2Z Q s m (6^,7r(x)) + QCOS (^,7r(x)).
l£j
So, any left invariant sub-Laplacian on S can be written as a differential oper-
ator with quasi-periodic coefficients on the nil-shadow SN of S.
Again, as in the case of differential operators with periodic coefficients in
Rn
(cf.
[BLP]), the macroscopic behavior of L is described by a homogenised operator L#.
LH is a left invariant sub-Laplacian on SN and it is given by a formula involving
certain bounded smooth functions ipi, called correctors. In this case, the correctors
are defined on R.k = S/N and are finite sums of periodic functions.
Let us denote by
g~ls
and
g~lsN
respectively the *s and *sN inverses of g.
Then, there is c 1 such that for all g, h G S
-\g~ls
*s h\s
\g~lsN
*sN h\sN
c\g~ls
*
5
h\s.
c
5. Non-simply connected solvable Lie groups. Let S be a connected solvable
Lie group of polynomial volume growth and let us assume that it is not simply
connected. As in section 1.4.2, let us consider the maximal torus C of the nil-
radical N of S. Then N/C is simply connected. Since C is fully invariant in
T V it is also normal in S. So we can consider the group S/C. Since the group
(S/C)/(N/C) ^ S/N is abelian, we have that (S/C)/(N/C) ^
Rk
x T
m
(cf. [Ho
p. 39]), where T = R/Z. Let TT be the projection n : S/C - Rk x T m and let
Si =
TT-^R* 5 ).
Then Si is simply connected and solvable and (S/C)/S± ^ T m is
compact. Hence Si is a semidirect factor in S/C, i.e. S/C = Si X T m (cf. [Ho pp.
139-140, Va pp. 254-256]) .
Previous Page Next Page