4 GEORGIOS K. ALEXOPOULOS

By a multiplicative functon x on G, we shall understand a function x '• G ~~ * ^

+

satisfying x(xy) =

x(x)x{y)i

x^y € G. Note that then %

c a n

be written as

x

= 0 o 7r with 0(x) =

e^,x

where 7r is the quotient map IT : G — • Rn = G/iJ and where (6, x) = fri^i + ... + bkXk

for 6 = (6i,...,6jb),a; = (xi,...,xfc) eRk.

We have the following well known lemma (see for example [MV]):

LEMMA 1.3.1. Let L be as above. Then, there is a constant (3 0, a vector

field Y G \) and a multiplicative function x such that

(1-3.1) L =

X

-1(Lc + P)x,

where

Lc = -{E2 + ... + E2p)+Y.

If pf (x, y) is the heat kernel of LQ-, then it follows from the above lemma that

(1.3.2) pt(x, y) = e-^xix-^pfix, y)X(y), x,yeG,t0

For reasons of completeness, we give below the proof of the above lemma.

PROOF.

Let n the quotient map

IT

: G — Rk = G/H and let us chose a basis

{Xi,..., Xq} of g such that

{X

n +

i, ...,Xg} C J) and dir(Xi) = — , 1 i n.

Then L can also be written as

L = — y aijXiXj -f ~ y a{Xi

li,jq 1^7

with aij = aji,l i,j q.

It follows that the image dn(L) of L can be written as

T

/rN v ^ d d v- ^ d

lM n lzn

Note that the (n x n) matrix B = (bij) with entries b^ = a^, 1 i,j n is

positive definite.