By Prof. Louis Auslander, Prof. Richard Tolimieri (auth.)

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2~i(~-~ y- ~ x) e e = ~(X,y,z) First • -2~iz e[:,](~,i) onto itself. onto itself and it is readily checked that A ~(x,y,z) We will ~ (-x,-y,z) s ¢ ¢¢t R(~--- , ~ , T)*(x,Y,Z) e I ¢ e[ ,](-C,i) = e • I e S ¢s e , %)~(-x,-y,z) e . Now el R('~-, ~ , - ~ ) ~ ( - x , - y , z ) Let y = (-¢',-¢,0) by elements of F E F = ~(-x I -y + ~ , z + ~ and use the invariance and the fact that ¢ ¢ R(-~-, ~ , - - C - ) ~ ( - x , - y , x ) This implies that i • R(-~, ~, ----~)*(x,y,z) Using formula 1 we + ~-, of ~(-x,-y,z) = ~ ( x + ~--, 2~i(~ y + -~-, ~ -y ~-) .

Define a(T) Hence k(~)(T) Note: a(T) function on ~ * (a)(T)(a(T) = )-1 a E C×(]~) a E cX(]~) ~(o,o,o[~) such that by . and we have proven our assertion. is U s u a l l y called the theta series associated w i t h the Jacobi theta ~(a,y,zlT ) . Let 2 = ~ a function B X ( ~ 2) never denote the group of bounded measurable taking the value 0 . A2: Fix T E ~ . Let ~ E L p E B × ( w 2) L into group and define aut(Bk(~2)) 1 H (L, BX(~7 2) ~ by This gives 42 XT(a) (x,y) Then k ~(~(x,y,z)IT) *(x,y,zjT) is a mapping of T Further L into B×(~ 2) k T = k i mod BI(L, BX(~ 2) ,~) and i s in and so kT ZI(L, BX(~2) ,~) .

A bijection between ~S n a s s o c i a t e d to each super d i s t i n g u i s h e d subspace group 4# E 8# n the Aj of J bijection (actually and J E ~# n Aj/I"'Z); i n t h e o t h e r corresponds the eigenvalue one space direct sum 8# . n w e have direction of In to L(A #) on n Next H calculation n for A we associate an orthogonal into R - i n v a r i a n t and i r r e d u c i b l e subspaces. each A S = A/FZ extending, ch(4 S) each J 6 ~# n to each ~n ' and to denote character E 8S n the k the A of of extending I-'Z n 4 ( a l , b l , b 2) A #.

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