LECTURE 6. CONVERSE THEOREMS AND FUNCTORIALITY 1716.6. Complements
As we noted above, Theorem 6.1 is the beginning point for a more complete analysis
of the these liftings. We would like to mention two examples of this.
In the case of the lifting from S02n+1 to GL2n, combining this lift with their
descent theory Ginzburg, Rallis and Soudry were able to completely characterize
the image locally and globally [29] and thus show that the local components II~ at
those v E S' are completely determined by the global lift, so there is no ambiguity
at these places. This is true for the liftings from the other classical groups as
well [8,88]. Once one knows that these lifts are rigid, then one can begin to define
and analyze the local lift for ramified representations by setting the local lift of 'lrv
to be the Ilv determined by the global lift. This is the content of the papers of Jiang
and Soudry [48, 49] for the case of H = S0 2 n+l· In essence they show that this
local lift satisfies the relations on L-functions that one expects from Functoriality
and then deduce the Local Langlands Conjecture for generic representation S0 2 n+l
from that for GL 2 n. We refer to their papers for more detail and precise statements.
Related results and further applications can be found in the papers of Kim [51, 52].
In the case of the tensor lifting G 12 x G 13 to G 16 , Kim and Shahidi also showed
that in fact this lift is completely determined at the places v E S and in fact is the
local Langlands lift at those places as well [55,84]. They also characterize when the
image is cuspidal, etc. Kim is able to do the same for his exterior square lift from
GL 4 to GL 6 , except possibly for places lying above 2 and 3 [53, 84]. In addition,
combining these two lifts, they are able to deduce and analyze the symmetric cube
and fourth power lifts for GL 2 [53, 55, 56]:H i,H u : i,H ---+ 1,G 1,G GGL2 GL2(<C) Sym^3 GL4(<C) GL4GL2 GL2(<C) Sym^4 GLs(<C) GLsz_From these they were able to deduce the estimates towards the Generalized
Ramanujan Conjecture for GL 2 mentioned in Section 4.5. For more details, we
refer the reader to the original papers [53, 55, 56] as well as Shahidi's article in
these proceedings [ 84].
- Concluding remarks
What further cases of Functoriality can we expect from this method? The table
in Section 6.3 gives all of the cases of split H which are attainable. Given that
the Converse Theorem requires the control of a large family of twisted L-functions,
this table covers all cases where the Langlands-Shahidi method is able to supply
that control [60, 79]. There are cases of quasi-split H and similitude groups such
as GSpin that should also be attainable and Shahidi, Kim, and their students are
currently pursuing these.
If we stay within the general Langlands-Shahidi philosophy of controlling ana-
lytic properties of L-functions through analyzing the Fourier coefficients of Eisen-
stein series there are two possible extensions of the method. The first possibility
for extending the method would be to relax the requirement of 7r being globally