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]) .