NOTATION
Z,Q,R,C : ring of rational integers and the fields of rational
numbers, real numbers and complex numbers respectively.
M (Z),Mn(R),Mn(C) : rings of n x n matrices over Z,R,C
respectively.
GL (Z)/ GL (R) : general linear groups over Z,R respectively.
SL (Z),SL (R) : special linear groups over Z/R respectively.
U(n) : group of n x n unitary matrices;
U(n) = {u Mn(C) | if1 = fcU }.
Sp(n,R) : the real symplectic matrices of degree n ; specifically,
Sp(n,R) = {M 6 M2n(R)| ^JM = J, J = _E
Q
n } .
Here E is the identity of matrices ring M (C).
Sp(n,Z) = Sp(n,R) 0 M2 (Z) : the discrete modular subgroup of
degree n.
r (N) : the principal congruence subgroup of level N of Sp(n,z)/
specifically,
Tn(N) = { M £ Sp(n,Z) | M = E2n( mod N ) } .
H : Siegel upper-half space of degree n ; specifically,
Hn = { Z 6 Mn(C)| tZ = Z, Im Z 0 } .
D : the generalized disc of degree n ; specifically,
Dn = { W Mn(C) | fcW = W, E - W ^ 0}
[ S , U ] : element of Sp(n,R) of the form
L°EJ
U 0
o
V11
diag [ a,,a2,... ,a ] or [ a^a^ .. . ,aR] : the diagonal matrix
[ai-] with ai:L = ai (i= 1,2, ... ,n) and ai- = 0 (i ^ j).
VI
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