Symmetry And Group

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By Ronald L. Lipsman

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Extra info for Group representations;: A survey of some current topics

Example text

Ind L = of the diagonal k(M,L)L.

23 (ii) (Bruhat [i]) ~(s,~) ~ ~(ss,sT), (iii) (Kostant [i]) If o = IM, (Wallach If G s E W. then is irreducible W(IM,T) V'T ~ i . (iv) Cartan subgroups [i]) (see no. is irreducible ~(a,T) has only one conjugacy class isfied by complex groups and We mention that the condition is sat- SOe(2n+l,l). Thus we see that "almost all" of the r e p r e s e n t a t i o n s irreducible. then 3 for the m e a n i n g of this phrase), V o,~. g. Knapp and Stein [i]). An i n t e r e s t i n g c o n j e c t u r e that is still not settled is: ~(o,T) = ~ ~i ~ ~i ~ ~j' i # j.

That Schwartz series is, space REMARK, For semiJ(G) (see either a prin- space [8] for the definition). or d i s c r e t e tribution, ~G" For abelian analysis that simple groups H a r i s h - C h a n d r a employs his S c h w a r t z the the space Let [8,9]) Then representation. it e x t e n d s ~ ~ G to a c o n t i n u o u s g z be is a t e m p e r e d dis- linear functional on ~(G). It has been c o n j e c t u r e d that a converse of T h e o r e m 6 should hold, that is, if 07, ~ r G, is tempered, in the support of the regular representation.

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