8 GEORGIOS K. ALEXOPOULOS

Let us denote by TT the quotient map n : S — S/C. Then, the homogenised

sub-Laplacian LH of a left invariant sub-Laplacian L on S is the same as the

homogenised sub-Laplacian d7r(L)# of its image dix{L) on S/C, i.e. it is a left

invariant sub-Laplacian on the nil-shadow of (Si) A T of Si. The correctors ipi will

still be finite sums of periodic functions defined this time on E^ x T

m

= S/N.

6. Simply connected Lie groups of polynomial volume growth. Every connected

Lie group G admits a Levi decomposition (cf. [Va]), i.e. it has a largest solvable

normal analytic subgroup S G called the radical of G and a maximal semisimple

analytic subgroup M, called Levi subgroup, such that G = SM. M is not neces-

sarily unique. G also has a largest nilpolent normal analytic subgroup N \G, called

the nil-radical of G. Both S and N are closed in G. Also N S and [G,G] C NM.

If G has polynomial volume growth then M is compact.

If G is simply connected, then S n M = {e} and hence G = S X M. Also

G/N ^

Rk

x M.

Now, let us assume that G is a simply connected Lie group of polynomial volume

growth. Then the homogenised sub-Laplacian LH of a left invariant sub-Laplacian

L on G is a left invariant sub-Laplacian on the nil-shadow SN of S. The correctors

ipi will be finite sums of bounded smooth functions (p(x, z) which are defined on

Wk

x M and which are periodic with respect to the variable x G l

x

.

7. Non-simply connected Lie groups of polynomial volume growth. Let G be a

connected Lie group of polynomial volume growth and let us assume that G is not

simply connected. Let G = SM a Levi decomposition of G. Since G is not simply

connected, it is not necessarily true that SC\M — {e}. To get around this difficulty,

we proceed as in section 1.4.5 and we consider the maximal torus C of N. Again,

since C is fully invariant in TV, is also a normal subgroup of G.

Let us denote by n the quotient map n : G — G/C. As in section 1.4.5

we can see that the radical S/C of G/C is isomorphic to the semidirect product

S/C = Si X T of a simply connected solvable subgroup Si S/C and a compact

abelian analytic subgroup T of G/C.

We set Mi = T (MC/C). Then Mi is a compact subgroup of G/C and

G/C — Si Mi. Since Si is simply connected, we also have Si fl Mi = e (cf. [Ho p.

138]) and hence G/C ^ Si X Mi.

Note also that G/N ^ (G/C)/(N/C) ^

Rk

x Mx.

Let us denote by *(51)JV the group product of

(SI)N

and by X~1{S^N the *(51)JV-

inverse of x G (Si)AT. Then, there is c 1 such that for all g, h G G with n(g) =

(x, z), 7r(ft) = (y, w) G Si X Mi

- ^ " ^ I G k~ 1(Sl) ^ *(51)iV 2/l(Si)iv C ^ - ^ I G -

The homogenised sub-Laplacian LH of a left invariant sub-Laplacian L on G is

the same as the homogenised sub-Laplacian d7r(L)# of its image dir{L) on G/C, i.e.

L^ is a left invariant sub-Laplacian on the nil-shadow (Si)AT of Si. The correctors

tyi will be finite sums of bounded smooth functions (j)(x, z) which are defined on

Rfc

x Mi and which are periodic with respect to the variable x G l

f e

.

Since CG there are Haar measures dc/c9

a n

d dcg on G/C and C respectively

so that the Haar measure dg on G desintegrates as follows (cf. [Boul]):

/ f(g)dg= / / f(gz)dczdG/C7r(g).

JG JG/C JC