By Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)
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Extra info for Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold
Q ~ ~-I(I~Z) maps Then 0 < al,bl,b 2 ~ n a unitary representation . 8# n is established L(A #) ' and the super disting- by u s i n g t h e r e p r e s e n t a A# E g# of n' A# on H n n > I . 4 we have that the map This association A # = A/~Z E 8#n " group g e n e r a t e d on This follows to induce a family of representations F o r simplicity we fix defines ~-I(A#/q-I([~Z)#) ~-l(a) 8# . n The next series of results tion . ~-I(A)/FZ £ 8 # nm As a particular case of T h e o r e m B Furthermore, it follows that once we show that w h e n e v e r q-l(~z) E 8# nm ~ E B m, A # E ~ } ~-I(I~Z) D FZ FZ .
Hence we may assume without any loss of generality J E ~# . n Let A # = Aj/FZ then it is easy to see that pj(F) = l/n( y~EA # (L(~#)(A#)F) 0 An(T) which is in again using the previous theorem. The second statement of (5) is now obvious and the theorem is verified. In this part we give another characterization of the distinguished subspaces in terms of theta functions. Setting ~0 07 0 An[o]O-) = Mn[OJ(T) ®n[O](T,A) we know that An[~](T ) into any distinguished is an n-dimensional subspace JA[~](T) which is a One dimensional J of H n complex vector space whose projection equals = An[~](T) n J complex vector space.
T h u s it remains to module. Qe = [(x,y,0) E N I O g x < l , be a c o n t i n u o u s p a r t i t i o n n = 1 . of unity for T2 and let or ~+e g y N 1} . subordinate Let to the covering Define A:(x,y,z) = , ~ e J~ * (x,y,o) 2Eiz e 2~i(z-x) if 0 ~ y K ~ ¢ if ~ + e ~ y ~ 1 A (x,y,z) = ~6 Then of A and A b e l o n g to z 6 ~ , we have C1 . * (x,y,O) Further, e if 2~iz denotes the c o m p l e x conjugate 28 A~ + A*(A*)- = Hence given F E C I , we i . have A(~F) + A * ( A * ) - F = F Define by ~: C 1 ~ C O ~F = ~F and ~ .
Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold by Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)