NIL-THETA FUNCTION S 5
n
xl
V
y
l V
t
l
) ( a
i ' - - - ' V
b
l ' - - - '
b
n '
t
2
) = ( v
l
+ v
2 ' V
t
2
+ I : x
i V
i=l
where v = (x . . . ,x y , . . . , y ) an d v = (a , . . . , a b , . . . ,b ) .
1 1 n l n £ l n l n
We will no w discuss ou r justificatio n fo r callin g N(D ) th e dua l pairin g
presentation o f th e Heisenber g group . Le t I R b e th e abelia n grou p con -
sisting o f n copie s o f th e real s an d I R b e it s dua l group . Le t x ] R an d
y c I R . The n D(x,y ) = exp2*ix y i s th e pairin g o f I R X IR - % wher e
T T is th e circl e group , give n by the dualit y o f locall y compac t abelia n groups .
n A n
Let V - I R © IR an d defin e a grou p G o n V X TT where multiplicatio n i s
given by
27rixi y
2
((x
1
,y
1
),6
1
)((x
2
,y
2
),s
2
) = ((x
1
+ x
z
»y
1
+y
2
)^
1
£
2
*e )
where ((x,y),£ ) I R Q IR XlT . I t i s easil y verifie d tha t th e universa l
covering grou p o f G ha s th e dua l pairin g presentatio n o f th e Heisenberg group .
n ?
The basic presentatio n i s intimatel y relate d t o considerin g C = V
Let ( c , . . . , c ) C . I f we defin e
1 n
«c! c
n
),^Hici,...,c'n),t2)=((cl+c'l,...,cn+c'n).tin2
+
^üaic
l
^ + . .. + c^;) )
we obtai n th e basic presentatio n o f the Heisenber g group .
Computations involvin g th e Heisenber g grou p ar e usuall y easies t i n
its basi c presentation . W e will lis t a few fact s abou t th e basi c presentation .
B.l. Th e straigh t line s throug h th e origi n i n V X IR ar e th e l-paramete r
groups i n th e basi c presentation , i . e. , th e one-paramete r group s g(t ) i n
N(A ) hav e th e for m
g(t) = {(vt,ct)|v V 2n, c IR al l t IR}.
B.2. I f [g
r
g
2
] denote s g ^ g j g
2

[(v1,t1),(v2,t2)]=(0,2A0(v1,v2)).
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