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**

**Example text**

TTiT(m 8[a](~lT) The on the remainder spaces 2~i(m • e mEZ of this An[~](T) . An[~](T) spaces : a b + "~)(~ + '~) , ~ = x + iy E e section For is devoted convenience we to investigating consider the this action solely for I of B n the . a+2s. ,Lb+2r J a result in Chapter , = An[0](a(T)) A cr(H 1) (x,y,z • IT) 3. 53 0 $(~[o](X,y,zITl) where [0O 7. $(~) O~ = Mn[OJ((Y(T))~(~ ) is the first order theta function O"( ' r ) of period and character given by ~(~) = e~ib(bT+d)~ 2 @[~]((bT+d)~tT ) • Proof: It s u f f i c e s given clearly to prove is in o(*[~](x,y,z[T)) = Mn[~](~(T))~(~) ~(~) since ®I[~](~(T),A ) .

IV will functions. DISTINGUISHED e x a m p l e we could the i n t e r - r e l a t i o n s Chapter AND III w e c o n c e n t r a t e d first o r d e r theta simplest FUNCTIONS IV character- in that at that time. SL2(Z) as a g r o u p of a u t o m o r p h i s m s saw that this action induced a unitary 45 r e p r e s e n t a t i o n of O ~ n ~ O mod 2 of SL2(Z) L on L2(F\N) and on e a c h H It is easy to see that for n this r e p r e s e n t a t i o n extends to a u n i t a r y r e p r e s e n t a t i o n of all on Hn The group GL2(Q) , Q the rationals, acts as a g r o u p of a u t o m o r p h i s m s of N by the following formula (1) Z(x,y,z) g = ( ca where bd ) We s h a l l actions to the = (cy + dx, E GL2(Q), now d e f i n e structure set of the all natrices ~ Each theory a subgroup o f the subsets llO~ll = n.

Let us b e g i n by recalling the following basic definitions f r o m the c o h o m o l o g y theory of groups. Let G be a group, A p l i c a t i o n w r i t t e n multiplicatively, of A . a . an abelian group, w i t h g r o u p multiaut(A) denote the g r o u p of a u t o m o r p h i s m s p: G ~ aut(A) and denote p(g)(a), g ~ G, A map : G ~ A is called a 1 -cocycle if k(glg2) We d e n o t e is the collection a subgroup of A mapping the k: of group G ~ A all of is such all the of a is subset fixed.