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.
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