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
where Ô is a distribution invariant under the orthogonal group 0(1, 1) for the quadratic foiro
In the quadrant x\y\ the change of variables ÷ = r eoshö, y = rsinh è shows
à ( à , 0 ) = à ( à )
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.
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