Introduction

Let D bea differential Operator on a manifold V with a specifled volume element

dv and Ô a distribution on V satisfying the differential equation AEÃ=0 . Suppose D

and Tf respeetively, are invariant under some Lie transformation groups L and Ç of the

manifold Ê The differential equation will then eontain some inessential variables, and gen-

eral theories for differential Operators may not be effective until after these variables are

somehow eliminated. Consider for example the differential equation

(1) (3

2

/3x

2

-d

2

/3j

2

)r=0,

where Ô is a distribution invariant under the orthogonal group 0(1, 1) for the quadratic foiro

x2-y2.

In the quadrant x\y\ the change of variables ÷ = r eoshö, y = rsinh è shows

Ã ( Ã , 0 ) = Ã ( Ã )

annihilated by the elliptic Operator

d2/3r2+ r~l

d/dr. Thus Ô is analytic

in the complement of the lines y = ±x.

This simple example illustrates the principie in Harish-Chandra's proof in [15 (c)] that

the characters of quasisimple irreducible representations of a semisimpie Lie group are analy-

tic functions on the open set of regulär elements in the group, (For the examples of unitary

irreducible representations of the complex classical groups known at the time this was clear

from the work [12] of Gelfand-Naimark.)

The theme of these lectures is treatment of invariant differential equations by Separation

of variables techniques. To be more specific let V be an arbitrary manifold of dass C°°.

Let £ be a Lie transformation group of V and D(F) the set of differential Operators on

V invariant under L. As usual let t?'(F) denote the set of distributions on V and if

HCL is any Lie subgroup we make the following defimtion.

Definition. Let PHL{V) denote the set of distributions Ô on V satisfying the

following two conditions:

(i) Ô is an eigendistribution of each DGB(V).

(ii) Ô is invariant under Ê

This set of distributions is the principal subject of study in these lectures. Chapter I

deals with general techniques useful for this study. We use a submanifold W C V transver-

sal to the //-orbits to establish some general properties about distributions on V invariant

under Ç and to express an arbitrary differential Operator on V as a polynomial in "orbit-

al Operators" with "transversal Operators" as coefficients. Special attention is given to the

case when V is a Riemannian (or a pseudo-Riemannian) manifold and some geometric proper-

ties of the Laplace-Beltrami Operator are established.

1

http://dx.doi.org/10.1090/cbms/014/01