SUBLAPLACIANS WIT H DRIF T 5
Let / G C°°, let xix) — e^6,7r^^ be a multiplicative function on G and let
a = (ai,..., an) and b — (61,..., 6n). Then
L(xf)= E *jXXiXjf E aa(XiX)Xjf E M**/)*;X
lijq 1M? l5^?i7
+ X] aixXi(f) ^2
aijfXiXjX+
Y2
aifXiX
= ~X X^ CLijXiXjfX Yl aiJbiXjU)~X Y2 aiJh3Xif
lz,jfq l 2 n 1*9
+ x X^
a*x*^
~ fa X^
a^A'
+ fa X^
aibi
127 l ^ * ? i ^ n l i n
=  x X^
aiJxiXjf+x
X!
Ia* ~2
X!
a*A
I
x
^
+
X
E (ai2(B6)i)X4/«6,B6)ft,a»/x.
l i n
Hence
x1Lx
=
(^i2
+  + ^ ) + E
fa'2
E
a*A]*i
+ E (ai2(Bb)i)Xi((b,Bb)(b,a))
lin
We want to have
a.i2 (Bb)i = 0, 1 z n
or else that
a = 2Bb.
Since the matrix B is invertible, this is equivalent to
b= lB^a
2
For this choice of b the constant term becomes
P =  «b, Bb)  (6, a)) = (b, Bb) 0.
Also, if we set
y= E U 
2
E °*A)*
niq \ 1J™ /
then 7 El) and
X

1
L
X
= (£
1
2
+ ... + £ 2 ) + F + /3
which proves the lemma.