By Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)
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Extra resources for Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold
K# where T h i s is however, RJL(g)J . In Hence J ( A # , 4 #) is the s trivial c o m p u t a t i o n is i n d u c e d by the c h a r a c t e r h ~ ~n[g,h~ . C_orollary: Hn is the o r t h o g o n a l direct sum of n u n i t a r i l y equivalent R- invariant and irreducible subspaces. Proof: Since the o r d e r of ........ Corollary: Aj/FZ If J is n the result follows from (3). is an i r r e d u c i b l e R - i n v a r i a n t subspace of has o r d e r Proof: A /4 # n n then there is a c h a r a c t e r k# on H n n J = J(A@,k#).
X,y,z) left m u l t i p l i c a t i o n e i e ee R(-~-, ~ , By formula (i,0,0) . E F ee t e e[ e e,](~, i) = e' e[~,](~,i) (-i) e et e + ~-, O[~,](~+i i . ee' y+~ e' , z +--~- + y~-) . The . result of this o r as theory of N SL(2,~) Some of W e l l ' s ~L2(T2,1) to produce their deep connection a computation w i t h basis by given . representations the details . the As I I e + x + 7) in [ 7 ] u s e d the u n i t a r y HI formulation theory of t r a n s f o r m a t i o n HI . We w i l l , as representation of c e r t a i n groups ideas can be given a n o t h e r by Well.
L (~) and so p0(@) 6 ®n(A) p0(@) = 0 . as is @ - p0(@) . + pM_l(0) E ®n(A) . 7. 4. 8 where Ck n%) n = i we have that ck k 2 1 * c k T 2 1) nO = ~nO Cn(T ' ~ ) + ~nO n ( 'H Since L(O'~'O)~nO = C n . ' • = ~*n . L(O'~'O)¢nO and already p r o v e n and Theorem I I . 4 is established. The proof of T h e o r e m 1 1 . 5 C ' s w i t h H ' s and noting that for Hn0 and Hn9 L(O,~,0) , 9 = 1 .... ,n-I for the r e p r e s e n t a t i o n T h u s w e have that for this module. is carried out in an a n a l o g o u s m a n n e r by replacing R of k CO -Ckn is a module o v e r are i n t e r t w i n i n g o p e r a t o r s N .