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
Fractional Now t h e linear trans- formation ~(T) aT + c - [Note: For bT + a d ff = we are really letting the transpose ~ E F The transformation group relationship generator Lemma 1 1 1 . 2 : Ira(T) > 0 " act in the usual notation]. ~3(x,y,zl~(T)) theory of Jacobi theta functions will turn out to be the study between structure For and 1 U = (2 these of two functions. ~3(~(x,y,z)IT) This study will = ¢3((x,y,z)lq(T)) = ~3((x+2y,y,z+y2)IT) = e 2~iz ~ mEZ . 3: Let on the 0 1 ) , we h a v e ~ mEZ Lena rest SL2(Z).
8 then follows by essentially the same arguments. 7: Let @ E ®n(A) . )n ° As can be seen by a direct and elementary argument. Thus our major concern must be with the proof of assertion (2). ®n(A) C ®n where ®nv Now consider consists of the eigenfunctions of exp(2~i ~), v M = 0 ..... n-i . For L* (~ I) Then ®n = ~ ~n-l~=0®nv ' with eigenvalue @ E ®n(A) , we must show that pM(@) E ®n(A) . + 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 .
Its a d v a n t a g e s irreducible n, the s i t u a t i o n under t h a n the others. were , n > l, and n th o r d e r n write theoretic finite although In C h a p t e r N. of number Brezin suhspaces paper because of J. n is facts w e used the p~ group H We have already, irreducible enlarging first are m o r e are nM projection begin we will space and first o r d e r theta f u n c t i o n s ®n[~](T,A), that w e could e x p l i c i t l y the H1 H n, n > i, is not H where First, study and second, itself w i t h the space because cated than the case w h e r e dim ® [~](T,A) BD on the H i l b e r t T h i s w a s for two reasons.
Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold by Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)