Canad. J. Math. 51(1999), 771-791
Printed: Aug 1999
This paper starts by introducing a bi-periodic summation formula
for automorphic forms on a group $G(E)$, with periods by a subgroup
$G(F)$, where $E/F$ is a quadratic extension of number fields. The
split case, where $E = F \oplus F$, is that of the standard trace
formula. Then it introduces a notion of stable bi-conjugacy, and
stabilizes the geometric side of the bi-period summation formula.
Thus weighted sums in the stable bi-conjugacy class are expressed
in terms of stable bi-orbital integrals. These stable integrals
are on the same endoscopic groups $H$ which occur in the case of
The spectral side of the bi-period summation formula involves
periods, namely integrals over the group of $F$-adele points of
$G$, of cusp forms on the group of $E$-adele points on the group
$G$. Our stabilization suggests that such cusp forms---with non
vanishing periods---and the resulting bi-period distributions
associated to ``periodic'' automorphic forms, are related to
analogous bi-period distributions associated to ``periodic''
automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$.
This offers a sharpening of the theory of liftings, where periods
play a key role.
The stabilization depends on the ``fundamental lemma'', which
conjectures that the unit elements of the Hecke algebras on $G$ and
$H$ have matching orbital integrals. Even in stating this
conjecture, one needs to introduce a ``transfer factor''. A
generalization of the standard transfer factor to the bi-periodic
case is introduced. The generalization depends on a new definition
of the factors even in the standard case.
Finally, the fundamental lemma is verified for $\SL(2)$.
11F72 - Spectral theory; Selberg trace formula
11F70 - Representation-theoretic methods; automorphic representations over local and global fields
14G27 - Other nonalgebraically closed ground fields
14L35 - Classical groups (geometric aspects) [See also 20Gxx, 51N30]