Given a collection of automorphic distributions and an ambient group
which acts with an open orbit on the product
of their (generalized) flag varieties, one can also define a holomorphic
This condition is related to the uniqueness principal in Reznikovs talk
Main difference: we insert distribution vectors into the multilinear
functionals (and justify).
These pairings can be used to obtain the analytic continuation of
L-functions which have not been obtained by the Langlands-Shahidi or Rankin-Selberg methods.
Theorem (Miller-Schmid, 2005). Let F be a cusp form on GL(n) over Q, and S any finite set of places containing the ramified
nonarchimedean places.Then Langlands
partial L-function LS(s,Ext2F) is fully holomorphic, i.e. holomorphic on all of C, except perhaps for simple poles at s = 0 or 1 which occur for
In particular, if F is a cuspidal Hecke eigenform on GL(n,Z)n GL(n,R), the completed
global L-function L(s,Ext2F) is fully holomorphic.
The main new contribution is the archimedean theory, which
seems difficult to obtain using the Rankin-Selberg method.Similarly, the Langlands-Shahidi method gives the correct functional
equation, but has difficulty
eliminating the possibility of poles.
Pairings (formally, at least) also can be set up for nonarchimedean
places also.Thus, this method
represents a new, third method
for obtaining the analytic properties of L-functions.It requires other models of unitary irreducible representations, such as the Kirillov
Two main reasons this works:
Ability to apply pairing theorem (which holds in great generality)
Ability to compute the pairings (so far in all cases reduces to
one-dimensional integrals, but the reason for this is not understood).