THE BASIC INTEGRAL FORMULA 7

This follows from (2-3) with p = Ra*. By convention a(V, W) = 1 if V = {0} or W = {0}.

2.3 We now define the modula r function A of G. Let JE?I,..., En (n = dimen-

sion of G) be any basis for the left invariant vector fields on G, Then, for each g £ G,

Rg-i + Ex,... ,Rg-i*En is also a basis for the left invariant vector fields and thus

(2-7) \\Rg-i.Ex A •-. A Rg-i*En\\ = A($)||£i A - • • A En\\

for some positive real number A(g). From this definition it follows that A is a smooth

homomorphism of G into the multiplicative group of positive real numbers. The following

equivalent definition will also be used in the sequel. If £ is any point of G and u i , . . . ,un

any basis of TG{ then

(2-8) A(0)||tn A • • • A t*n|| = HJVuii ! A -. . A Rg-i*un\\

This follows from (2-7) by extending each U{ to a left invariant vector field on G.

2.4 R e m a r k . A Lie group G is called unimodular if A = 1. It is well known that all

compact groups, all semisimple groups and all nilpotent groups are unimodular.

2.5 Recall that if M and iV are immersed submanifolds of some manifold S then M and

N intersect transversely if and only if x € M D N implies TMX -f- TNX = TSX (here

TMX + TNX is the subspace of TSX generated by TMX and TNX). If S has a Riemannian

metric this is the same as requiring

T±MX

O

T±NX

= {0}. If M and N have nonempty

intersection and intersect transversely then M D N is a smooth submanifold of S whose

dimension is dim M -f dim N — dim S.

2.6 A r e m a r k on notation . For any Riemannian manifold M we will denote the

volume density on M by SIM- Then f l ^ can be thought of either as a measure on M

or as the absolute value of one of the two locally defined volume forms on M . (See [25]

page 53 for a more detailed discussion of densities.) In particular QM and integration with

respect to QM are defined without any assumption about the orientablity of M. Despite

this it will often be useful when doing calculations to assume that M is oriented and that

Q.M is one of the two volume forms on M. In all cases where it is convenient to do this

the calculation is local, and thus we can restrict down to an oriented subset of M, do the

calculation just as if H M was a form and then take absolute values when we are done. This

will be done without mention in the sequel and hopefully no confusion will result.

2.7 Basic integral formula. Let G be a Lie group with a left invariant metric (,) .

Let M and N be immersed submanifolds (possibly with boundary) of G with dim(M) -f-

dim(iV)

dim(G?).

Then for almost all g G G the submanifolds M and gN intersect

transversely and if h is any Borel measurable function on M X N such that the function

(£,77) 1— » h(£,rj)A(r]) is integrable on M x TV, then

(2-9) / / hopgilMngNSlG{g)= [[ hti^AinWT^M^N^nMxN&T])

JGJMngN JJMXN

where (pg : M 0 N — M x N is given by

(2-10)

P.(*) = « , f f

_ 1

« .