,.
320 FREYDOON SHAHID!, LANGLANDS-SHAHID! METHODand
(3.20) c:(s,7r,ri) =II c(s,7rv,ri,'l/Jv)·
v
We then have:
Theorem 3.11 (Functional Equation [Shl]). For each i, 1::::; i::::; m ,
(3.21) L(s, 7r, ri) = c:(s, 7r, ri)L(l - s, 7r, i'i)·
Exercise 1. Use the pair (G,M), G = GL3 and M = GL2 x GL1, to get the
standard £-function for G £ 2. Determine £ - functions using our method and show
that they are equal to those of J acquet-Langlands.
Exercise 2. Let G = E~c and M be such that MD= SL 3 x SL2 x SL3.
Fact 1. There exists a F- rational map (injection) f: M ____, GL3 x GL2 x GL3
whose restriction to MD is identity.
Fact 2. m = 3 and if 7r 2 0 7r 1 0 CJ is an unramified representation of GL3(F) x
GL 2 (F) x GL 3 (F), where Fis a local field, then
L(s, 7r2 x 7r1 x 0-) = L(s, (7r2 0 7r1 0 CJ)· f , r1).
Define 'Y(s, 7r2 x 7r 1 x 0-) to be 'Y(s, (7r2 0 7r 1 0 CJ)· f, r1), using our method for
arbitrary local representations 7r2 0 7r 1 0 CJ. Assume
(3.22) a = Ind (μ1 0 μ2 0 μ3) 0 i.
(F•)^3 xUTGL3(F)
Show that multiplicativity implies:
3
'Y(s, 7r2 x 7r1 x 0-) =II 'Y(s, 7r2 x (7r1 0 μj)),
j=l
where the "(-functions on the right are those of Rankin-Selberg for GL3(F) x
GL 2 (F). (This is crucial to Kim- Shahidi's proof of functoriality [KS2] of the in-
clusion GL 2 (1C) 0 GL 3 (1C) <-t GL 6 (1C), to be discussed later .)
Exercise 3. Let G = S0(2m+2n+ 1) and M = GLm x S0(2n+ 1). Let CJQSl7r be
an irreducible admissible x-generic representation of GLm(F) x S0 2 n+i(F), where
F is a local field. Assume
m
(3.23) CJ <-t Ind l'V\ μ.
(F•)mxUTGLm(F) ;;;_ JShow t hat multiplicativity implies:
m
'Y(S,CJ x 7r) =II 'Y(S,7r0μj)·
j=l
(This is crucial to Cogdell- Kim-Piatetski-Shapiro- Shahidi's proof [CKPSSl J of
functorial transfer from generic cusp forms on S 02n+i (AF) to G L2n (AF). )