UNITARY REPRESENTATIO N THEORY FOR SOLVABLE LIE GROUPS CHAPTE R I Introduction The object of this pape r is to apply the a b s t r a c t theor y of induced unitar y representation s to the explicit study of the unitar y representatio n theor y of solvable Lie groups . The a b s t r a c t machiner y was built by G. Wo Mackey ([20], [21], and [22]) and C . C . Moore ([5], Chapter II). The firs t application to solvable Lie groups was mad e by Takenouchi in [27], wher e it is proved that a solvable Lie group of exponential type is type I. Le t G be a Lie group , let G be the Lie algebr a of G , and let ^ be a linea r functional on G. If H is a subalgebr a of G such that / ([H, H]) = 0, then / d e t e r m i n e s , in an obvious fashion, a c h a r a c t e r ^ °f the connected subgroup H of G correspondin g to H : X (h) = exp (#(h)) y for all h H,cp being the h o m o m o r p h i s m from H into the r e a l n u m b e r s such that d(j) is the r e s t r i c t i o n of (f to H. y being a h o m o m o r p h i s m from H into the circl e group , can be viewed as a unitar y representatio n of H . We shall use ind$), H G) to denote the (unitary equivalence clas s of the) unitar y representatio n of G induced by \ . The crucia l step in the application of the a b s t r a c t m a c h i n e r y of Mackey (and, later , Moore) was the methodica l utilization of the c o r r e s p o n - dence (/ H ind (j , H G) (1) 1. By Lie group we shall always m e a n a connected, simpl y connected Lie group . 1
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