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Extra resources for Automorphic Forms on GL (3,ℝ)
2 2. 2 2, ½ ~x3tx2Yl÷YlY 2 2. 2 2. 50) 52 If w=w4, 2 2 ½ (x2+Y 2 ) 2. 2 2. 2 2' Y½ = Y 2 Yl = y l x3÷x2YI+YlY 2+ 2 2+ 2 2 ½ x 3 x2Y I Y l Y 2 ) 2 2 x2+Y 2 w=w5' Yl = Yl 2 ) 'Y:2 = :/2 2 x ~Yl First assume that x~ = x l, x~ = 2, Y2 -- Y 2 " ( x 2 t Y 2 Then . (x2+Y2 ) x~ 2 -I (x2x3-xlx2-xlY2)-(x2+Y2) 2. - 1 ~ 2+ 2 2+ 2 2 " x 4 XlY 2 Y l Y 2 w = w 2. ½ , yl = y l - ( X 2 * Y 2 ) these -i , calculate easily that: I is an orthogonal ' 2 group, Bruhat Similarly, if ) , x~ we m a y = -x4, and now y~ we will Goldfeld .
Equations directly, However, it follows it is pleasant to prove using Cauchy's theorem. We have: ~Yl~Y2 l---i-(2~i)2 1 (2~i) 2 Si ~i. r(s), -i [Sl~~ ) the left side equals: _ Sl+8 Sl+~ (-7--)(--7--) + ( ~ ) ( ~ ) ( ~ ) s + 8 + y = O, _ V(Sl,S2)- this equals: s13 + s23 + (8y+T~+eS)(Sl+S 2) V(Sl,S2) 2 2 = (Sl-SlS2+S2-A-l)V(Sl,S2) Sl+S 2 Our previous expression now equals: o÷i~ o+i ~ f:l-- 0-- (-Sl+l)(-Sl) (2~i) 2 - (-S1+l)(-S2+1) - + (-s2+l)(-s 2) S~ + S l S 2 - S~ + ~ - -Sl+l "V(sl,s2)(~Y I) o+i® (2~i) 2 ~_ o+i~ ~i V(~,~)(~Yl ) 1 -s2+l (~Y2) dSldS 2 _Sl+ 1 _s2+ 1 (~Y2) dSldS 2 = XW(Yl,Y2) as required.
Which our , x~ we = -x 2, generate the In using for fact, all refer purposes, and g E G. 3-8) and r~ I CD ba PO v H t~ P---~8 r~ 7 r'd I r,O I PO r~ + PO F~ + r~ ,C 5- ! 16). 17-22) to the reader. 16) above. Let us turn now to the proofs of the functional equations. We require two auxiliary integral formulae: (Ax2+Bx+C) -v dx = J~" 22v-1. 54) is easily evaluated by Euler's beta integral. 55) reduces to the special case By A = C : I, B = 0, which is well known (cf. 16).