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
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