Symmetry And Group

Group representations;: A survey of some current topics by Ronald L. Lipsman

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By Ronald L. Lipsman

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The contents of this publication were used in classes given by means of the writer. the 1st used to be a one-semester path for seniors on the college of British Columbia; it used to be transparent that sturdy undergraduates have been completely in a position to dealing with basic team idea and its program to uncomplicated quantum chemical difficulties.

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I, it is necessary on Caftan subgroups be a real semisimple A subalgebra ~ Lie algebra, of and parabolic ~c its eomplexi- is called a Cartan subalgebra if i t s ~ is a Cartan subalgebra be a Cartan d e c o m p o s i t i o n In order to general- ~ and 6 of ~e" Let ~ the corresponding = ~+~ Cartan in- volution. LEMMA 7. of ~ . (Harish-Chandra Let [4]) ~ Then it is possible to conjugate be a Cartan subalgebra ~ by an element of Ad so that the resulting Cartan subalgebra is e-invariant. We assume henceforth particular that ~ is 0-invariant.

Hc it is the maximal compact the Cartan i n v o l u t i o n of [6, Lemma Let Z G = {e}. + i(~ %Z Ad ~c is a subgroup of is invariant under Z~ . By H a r i s h - C h a n d r a is connected and exp(Z~n~c ) = n /) Let b I e B0 = 2-+ and i~. b 2 = bllb b e B ~ r n K = F. When Rs P is minimal, M is compact and M = MOF. Neither is true in general for arbitrary cuspidal parabolics. How- ever the entire theory of the discrete series can be carried over to the group M (Lipsman [3]). The procedure consists of extending H a r i s h - C h a n d r a ' s theory to connected r e d u c t i v e Lie groups with compact center, then to the direct product e x t e n s i o n procedure (which is discussed discrete series for the group ferred to Lipsman EXAMPLE, N = i G MOF, M.

Is given e x p l i c i t l y in terms of the characters -- objects we will discuss in detail in sec- It is important to realize that H a r i s h - C h a n d r a does not write down the discrete series representations. In fact, a general r e a l i z a - tion of the d i s c r e t e series is still not available, been c o n s i d e r a b l e progress on this problem. this area is that of Schmid EXAMPLE, Let Then o n e - p a r a m e t e r lattice, and ~ : ~ ~f- ~' 0) ' M = (-+~ +-i :{( ~_ 0b): b ( R}, contains only 0.

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