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HARMONIC ANALYSIS OF INVARIANT DISTRIBUTIONS 384. James Arthur*
 

Summary: HARMONIC ANALYSIS OF INVARIANT DISTRIBUTIONS 384.
James Arthur*
Suppose t h a t G i s a reductive Lie group, with Lie
algebra .@J . To be s a f e , l e t us assume t h a t G is t h e set
of r e a l points of a reductive algebraic group defined over
IR. Harish-Chandra has defined t h e Schwartz space, C ( G ) , on
G. I t is a c e r t a i n space of functions on G such t h a t
In each case t h e inclusion is a continuous map from one space
onto a dense subspace of t h e second. A d i s t r i b u t i o n , F, on
G i s c a l l e d tempered i f i t extends t o a c o n t i n ~ o i ! ~l i n e a r
functional on C i G ) . F is s a i d t o be i n v a r i a n t i f f o r fixed
f e c¡(G , i s independent of y e G. Here
EXAMPLE: Suppose t h a t T i s a Cartan subgroup of G with
Lie algebra 4 . I f y E Treg, t h e s e t o f points i n T whose
c e n t r a l i z e r i s T, define
Harish-Chandra has shown t h a t t h e map which sends f e C ( G )
t o
i s well defined, and i s a tempered i n v a r i a n t d i s t r i b u t i o n on
G. The d i s t r i b u t i o n s so obtained, known a s t h e o r b i t a l
i n t e g r a l s of f , play a c e n t r a l r o l e i n t h e harmonic analysis

  

Source: Arthur, James G. - Department of Mathematics, University of Toronto

 

Collections: Mathematics