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By Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)
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Additional info for Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold
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.
E j th to he any c o n t i n u o u s [~ satisfying the following functional equations. %o(C+i) = ( - z ) a ~ ( C ) n and , ~ E ~ . 52 Recall we denoted the space of all and characteristic [~] by ®n[~](T) n th and order theta functions of period the subspace of entire T functions a (argument In by ~) by ®n[b](T,A ) . II Chapter it w a s proved t h a t Mn[~](T) of Chili(T) defined the formula a ~-vector in H space y - ~a x + n z ) 2~i(~ a (Mn[b](T)~)(x,y,z) is the map ~iTny 2 = e e isomorphism ®n [~](T) of ~(x+Ty), onto (x,y,z) E N , the continuous functions Cn n Set sical ~[~](T) ®n[~](T,A) An[~](T) result in Chapter that II.
Of the From of the H n formula A l s o it is easy to check 51 L(g)R(h) = R(h)L(g), g ~ A and n h ~ N it follows that L(g) is an i n t e r t w i n i n g o p e r a t o r for RJJ p a r t i c u l a r since L(g)J J is i r r e d u c i b l e and R-invariant, and is also. w e are done as soon as w e know that the c o l l e c t i o n of spaces same as the c o l l e c t i o n of spaces and in fact of A L(g)J = J ( A # , X #) L(g)J . 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~ .
Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold by Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)