LECTURE 2. EISENSTEIN SERIES AND L-FUNCTIONS 3132.3. Examples
We shall now give a number of important examples of £-functions which appear in
constant terms for appropriate pairs (G, M). We refer to [Lal,Sh2] for the complete
list.
2.3.1. Let G = GLn+t, M = GLn x GLt, and let 7r = ©v1rv cusp form on
GLn(AF) and 7r^1 = ©v7r~ one on GLt(AF ). Then m = 1 and we get L(s, 7r x 7r'), the
Rankin-Selberg product £ - function for the pair ( 7r, 7r') (cf. [JPSSl J and [ShlO]). It
will be discussed by Cogdell in more length.
2.3.2. Let G be a classical group, split over F and let M = GLn x G', where
G' is a classical group of the same type, but of lower rank. Let 7r and 7r^1 be cuspidal
representations of GLn(AF) and G'. Then m = 2. One gets L(s, 7r xii"') as its first
£-function. For i = 2, we get L(s, 7r, p), where p = A^2 if LG is orthogonal and p =
Sym^2 if LG is symplectic (cf. [CKPSSl,2]).
2.3.3. Let G = GSpinn+t, M = GLn x GSpint, 7r and 7r^1 cusp forms on
GLn(AF) and GSpint(AF ), respectively. Then m = 2. Again we get L(s, 7r xii"') as
our first £-function. The second £-function is then an appropriate twist of either
L(s, 7r, A^2 ) or L(s, 7r, Sym^2 ).
2.3.a. Let G = GSpin5+2n, M = GLn x Gpins = GLn x GSp4, and let (7r, 7r^1 )
be a cusp form on GLn(AF) x GSp 4 (AF ). Again we get L(s, 7r x 7r') as our first
£-function. This is very important.
2.3.b. G = GSpin6+2n, M = GLn x GSpin6. We have 0 ---> {±1} --->
GL 4 (C) ---> GS06(C) ---> 0. Suppose 7r is on GLn(AF) and 7r^1 on GL4(AF)· We
then get L(s, 7r 0 7r', Pn 0 A^2 p 4 ) as our first £-function (cf. [K4]).
2.4. Let G be a simply connected group of either type E6 or E1. Choose
M such that MD, the derived group of M, is either SL3 x SL2 x SL3 or SL3 x
SL 2 x SL 4 , respectively. There exist F-rational injections from M into GL3 x
GL 2 x GLt, t = 3 or 4, which are identity on SL3 x SL2 x SLt. Let 7r107r20 17
be a cuspidal representation of GL3(AF) x GL2(AF) x GLt(AF)· Then m = 3 or
4 according as if G = E6 or E1. The first £-function is then L(s, 7r1 x 7r2 x 0-)
(cf. [KS2]).
All these £-functions will be revisited later in connection with functoriality.