168 J.W. COGDELL, £-FUNCTIONS FOR GLn
Proposition 6.2 (Multiplicativity of '!'-factors). If 'Irv is a generic irreducible ad-
missible representation of H( kv) and if 7r~ is a generic irreducible admissible repre-
sentation of GLm(kv) with 7r~ = Ind(7r~,v @7r2,v), with 1ri,v an irreducible admissible
representation of GLr, (k), then
There is a similar multiplicativity in the first variable, i.e., if the representation 'Irv
of H(kv) is a full induced representation.
The second property is t he stability of the local '!'-factors under highly ramified
twists. The knowledge of t his property is more limited. It is known in the case
we need, namely H = S0 2 n+l [11], and there is progress in establishing it in
general [8, 82].
Proposition 6.3 (Stability of '!'-factors). If 7r 1 ,v and 7r 2 ,v are two irreducible ad-
missible generic representations of H(kv) then for every sufficiently highly ramified
character T/v of k:; we have
'Y(S,7r1,v X T/v,'l/Jv ) = 'Y(S,7r2,v X T/v,'l/Jv )
and
L(s, 7r1,v X rJv) = L(s, 7r2,v X rJv) = l.
Hence the local c:-factor stabili zes as well.
To see how these function as a replacement for the Local Langlands Conjec-
ture at the places is S, first recall from Section 3. 1.6 that we also know the local
multiplicativity and stability of '!'-factors for GL 2 n(kv)· If we use the multiplica-
tivity of t he '!'-factor in the first variable then we can actu ally compute the stable
form of the '!'-factors 'Y(s, 'Irv x rJv, 'l/Jv) with 'Irv a generic representation of H(kv)
and 'Y(s, Ilv x rJv, 'l/Jv) with Ilv a representation of GL2n(kv) by taking 7r2,v or II2,v
to be full induced representations in the statement of stability. Multiplicativity in
the first variable then reduces both '!'-factors to a product of 2n one dimensional
Artin '!'-factors. This then allows for a comparison of the stable forms on these two
different groups. As a result we find the following proposition.
Proposition 6.4 (Comparison of stable forms). Let 'Irv be a generic irreducible
admissible representation of H(kv) and let Ilv be a gen eric irreducible admissible
representation of G L 2 n ( kv) having trivial central character. Then for every suffi-
ciently ramified character T/v of k:; we have
'Y(s, 'Irv X rJv, 'l/Jv) = 'Y(s, ITv X T/v, 'l/Jv)and
L(s, 'Irv X rJv) = L(s, Ilv X rJv) = l.
Hence the local c:-factor are stably equal as well.
This equ ality, combined with t he multiplicativity of the '!'-factors, lets us make
the following definition of a local lift. If v is a place in S then we take the local
lift of 1rv to be any irreducible generic representation Ilv of GL 2 n(kv) having trivial
central character. We can then establish the following analo gue of Proposition 6.1.