By Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)
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Additional resources for Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold
We will now assume facts about the g e n e r a l i z e d L e g e n d r e symbol Gauss ~ (Q) and the in S. Lang [ 5 ] pages 76 to 90. Then = k(UIT) , 3 ( x , y , z t o ( T ) ) 37 ~ ) i-d/2 ~ i/(Tb+d) k(~IT ) ) i (b'l)/2 ~ i/(Tb+d) Proof: The case Assume b > 0 . 2. l ~3(q(x,Y,Z)IT) = e 2~iz Z g(y+m) mEZ e 2~imx where 1 g(y) = j 21~iq~(x,y) ~iT(ay+bx+m) 2 + 217im(cy+dx) e dx Z e 0 where the convergence a T~i ~ y g(y) is uniform convergence. ~ = e 2 e n mod Hence d odd b odd k(qlT) = ~ O(- -~,b) ~/ Tb+d we have g(y) = k(ClT) e~i~(T)Y 2 It then follows from the definition of ,3(~(x,y,z)IT) The theorem for b > 0 ~3(x,y,zl~(T)) that = k(C,T) ~3(x,y,zlq(T)) then follows by applying the basic facts about Gauss SUmS.
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 . ' • = ~*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.
Exists in F\N . It follows that ~%) = @ f f E cO ( T 2 ,~I) . that and further ~(g%)) 29 But since w e have f f M is then b o u n d e d in the complex plane and is a constant. Hence n-i M=O and w e have o u r assertion, ~ and @ are analytic, CKAPTER III ELEMENTARY TRANSFOP3~ATION THEORY We h a v e shown that the spaces H , n ~ Z+ , v = 0 ..... nv n-1 in Chapters I and II have two different Hilbert space basis, e2Nivx e2~i(nax+by) e2~inz and the first involving discontinuous functions, and Jacobi theta functions.