LECTURE 4. HOLOMORPHY AND BOUNDEDNESS; APPLICATIONS 323L(s, n.,,, ri) is entire. Then, given a finite real interval I, each L(s, n.,,, ri) remains
bounded for all s with Re( s) E I.
Remark. One expects the assertion of Assumption 4.1 to be valid even for Re(s) >
0 (cf. [MW2] and for reasons related to determination of residual spectrums). But
that is much harder to prove, even in special cases.4.2. Examples of Functoriality with Applications
Consider the embedding
i: GL2(C) ® GL3(C) <:.......+ GL6(C).
This is a homomorphism from LGL 2 xL GL 3 into LGL 6. Accordingly functoriality
predicts a map
Aut (GL2(AF)) x Aut (GL3(AF)) __, Aut (GL6(AF)).
More precisely, let 7r1 = ®vn1v and n2 = ®v1r2v be cusp forms on GL2(AF) and
GL3(AF), respectively, with 7r1v given by tiv E GL2(C) and n2v by t2v E GL3(C)
for almost all v. Let I1v be the irreducible admissible spherical representation of
GL6(Fv) defined by {t1v ® t2v} C GL6(C). Then we can state the functoriality in
this case as:
Functoriality. There exists an automorphic representation
II= ®v II
v
of GL6(AF) such that IJ: = f1 v for almost all v.
More precisely, let Piv: W~v __, GLi+ 1 (C), i = 1, 2, parametrize niv (Harris-
Taylor [HT], Henniart [Hell) for all v. Let Piv ® p 2 v be the six dimensional repre-
sentation of w~v' i.e., the homomorphism
P1v ® P2v: W~v __, GL6(C).Denote by n 1 v ~n 2 v the irreducible admissible representation of GL 6 (Fv) attached
to Plv ® P2v· Let 7r1 ~ 7r2 = ®v(7r1v ~ 7r2v)·
Theorem 4.4 (Kim-Shahidi [KS2]). The irreducible admissible representation
7r1 ~ 7r2 of GL6(AF) is automorphic. Thus GL2(C) ® GL3(C) <:.......+ GL6(C) is func-
torial.
For the proof, one appli es an appropriate version of the converse theorem [CPS2]
to the following cases of our method. In each case G and MD, the derived group
of M, are given as follows.
(1) G = SL(5)(orGL(5)), MD = SL2 x SL3
(2) G = Spin(lO), MD = SL3 x SL2 x SL2
(3) G = E~c, MD= SL3 x SL2 x SL3
(4) G = Eic, MD= SL3 x SL2 x SL4
We then get the necessary analytic properties of the highly ramified twisted
£ - functions L( s, n 1 x n 2 x ( u ® 77)), u ® 77 = u ® 77 · det, 7] a highly ramified
grossencharacter, where u's are appropriate cusp forms on GLj(AF), j = 1,2,3,4,
respectively. Observe that
L(s, 7r1v X 7r2v X O'v) = L(s, (n1v ~ 7r2v) X O'v)