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**Extra resources for Dimensions of Spaces of Siegel Cusp Forms of Degree Two and Three**

**Example text**

For the proof, see [ 4 ] , Satz 1. REMARK : The group GL~(Z) definite symmetric 3x3 operates on the space of positive matrices by the action element in the fundamental domain for GL^(Z) Minkowski reduced. From / Y -* UY U. An is said to be [ 20 ]/ we have *1 1 Y 2 i y 3 ? 1*121'1*131 i y l '" ly23l i y 2 ; 'yi2-yi3-y231 y i+y2 , y 12- y 13 +y 23 1 y l + y 2 ; LEMMA 2. Suppose that T ; -yi2+yi3+y31 y y i+y2 12 +y 13- y 23 ± y l + y 2 '• (8) f is the projection of the fundamental F described in the dimension formula on the imaginary part and let J = JTT then the i n t e g r a l J y 1^yy2 y^ d e t i s convergent i f a + b + c < 6 , For the proof, see b + c < 5, c<3.

SL n (R) x SL^(R) •£ ; —t. ( j = 1,2 ) dg SL 2 (R)xSL 2 (R) f[g 1 xg 2 (Z t )] is a measure on / . 5 of si is given by ,. 12 Z = Z H,. CHAPTER IV. G\ H "1 [±1 = Lo a-^Lo ±lj G = 1 L € Sp(2,R) such that defined on LEMMA 2. A fundamental domain for b t^L+t^dt; SL 9 (R) H(z)y for any positive measurable function [a idg2 with , a > 0 > 0 f y 0 - y n ; > 0. ly 12 2 Proof: This follows from the fact that there exist unique L in Z G on and Z' F2 on such that Z = L(Z') for any given H2 . LEMMA 3. A fundamental domain for G' = { L 6 S p ( 2 , R ) | L = G'\H 2 with [ S, E] , S = diag [s , s'] } is given by ly Z = Z L THEOREM 13.

L,-, L 6 , L 7 , L 8 ; s ranges over nonzero integers ; v 2"" 2 3- 3 x 20 [ Pk(-/3i)+pk(/3i)+p2k+p2k] 6. 4 Contributions from Conjugacy Classes of Elements having One-Dimensional set of Fixed points (II) In this section, we shall consider conjugacy classes represented by elements of the type M = [ S, U ], with cose sin( 0 ), diag [ s, s ], U -sin9 cos0 ( sine f which have set of fixed point(s) represented by Q : Z = diag[ z , z ] if It is easy to verify that s = 0, ft' : Z = CM R I i» 0 0 if s ± 0.