26. CJM 2002 (vol 54 pp. 828)
 Moriyama, Tomonori

Spherical Functions for the Semisimple Symmetric Pair $\bigl( \Sp(2,\mathbb{R}), \SL(2,\mathbb{C}) \bigr)$
Let $\pi$ be an irreducible generalized principal series
representation of $G = \Sp(2,\mathbb{R})$ induced from its Jacobi parabolic
subgroup. We show that the space of algebraic intertwining operators
from $\pi$ to the representation induced from an irreducible
admissible representation of $\SL(2,\mathbb{C})$ in $G$ is at most one
dimensional. Spherical functions in the title are the images of
$K$finite vectors by this intertwining operator. We obtain an
integral expression of MellinBarnes type for the radial part of our
spherical function.
Categories:22E45, 11F70 

27. CJM 2002 (vol 54 pp. 673)
 Asgari, Mahdi

Local $L$Functions for Split Spinor Groups
We study the local $L$functions for Levi subgroups in split spinor
groups defined via the LanglandsShahidi method and prove a conjecture
on their holomorphy in a half plane. These results have been used in
the work of Kim and Shahidi on the functorial product for $\GL_2
\times \GL_3$.
Category:11F70 

28. CJM 2002 (vol 54 pp. 263)
 Chaudouard, PierreHenri

IntÃ©grales orbitales pondÃ©rÃ©es sur les algÃ¨bres de Lie : le cas $p$adique
Soit $G$ un groupe rÃ©ductif connexe dÃ©fini sur un corps $p$adique $F$ et $\ggo$
son algÃ¨bre de Lie. Les intÃ©grales orbitales pondÃ©rÃ©es sur $\ggo(F)$ sont des
distributions $J_M(X,f)$$f$ est une fonction testindexÃ©es par les
sousgroupes de LÃ©vi $M$ de $G$ et les Ã©lÃ©ments semisimples rÃ©guliers
$X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales
composantes du cÃ´tÃ© gÃ©omÃ©trique des formules des traces locale et globale d'Arthur.
Si $M=G$, on retrouve les intÃ©grales orbitales invariantes qui, vues comme fonction
de $X$, sont bornÃ©es sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un rÃ©sultat bien connu
de HarishChandra. Si $M \subsetneq G$, les intÃ©grales orbitales pondÃ©rÃ©es
explosent au voisinage des Ã©lÃ©ments singuliers. Nous construisons dans cet article
de nouvelles intÃ©grales orbitales pondÃ©rÃ©es $J_M^b(X,f)$, Ã©gales Ã $J_M(X,f)$ Ã
un terme correctif prÃ¨s, qui tout en conservant les principales propriÃ©tÃ©s des
prÃ©cÃ©dentes (comportement par conjugaison, dÃ©veloppement en germes, {\it etc.})
restent bornÃ©es quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons
Ã©galement que les intÃ©grales orbitales pondÃ©rÃ©es globales, associÃ©es Ã des
Ã©lÃ©ments semisimples rÃ©guliers, se dÃ©composent en produits de ces nouvelles
intÃ©grales locales.
Categories:22E35, 11F70 

29. CJM 2002 (vol 54 pp. 352)
 Haines, Thomas J.

On Connected Components of Shimura Varieties
We study the cohomology of connected components of Shimura varieties
$S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on
the stabilization of the twisted trace formula, due to Kottwitz and
Shelstad. More precisely, for each character $\olomega$ on
the group of connected components of $S_{K^p}$ we define an operator
$L(\omega)$ on the cohomology groups with compact supports $H^i_c
(S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual
trace of the composition of $L(\omega)$ with a Hecke operator $f$ away
from $p$ and a sufficiently high power of a geometric Frobenius
$\Phi^r_p$, can be expressed as a sum of $\omega${\em weighted}
(twisted) orbital integrals (where $\omega${\em weighted} means that
the orbital integrals and twisted orbital integrals occuring here each
have a weighting factor coming from the character $\olomega$).
As the crucial step, we define and study a new invariant $\alpha_1
(\gamma_0; \gamma, \delta)$ which is a refinement of the invariant
$\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done
by using a theorem of Reimann and Zink.
Categories:14G35, 11F70 

30. CJM 2002 (vol 54 pp. 92)
 Mezo, Paul

Comparisons of General Linear Groups and their Metaplectic Coverings I
We prepare for a comparison of global trace formulas of general linear
groups and their metaplectic coverings. In particular, we generalize
the local metaplectic correspondence of Flicker and Kazhdan and
describe the terms expected to appear in the invariant trace formulas
of the above covering groups. The conjectural trace formulas are
then placed into a form suitable for comparison.
Categories:11F70, 11F72, 22E50 

31. CJM 2000 (vol 52 pp. 1121)
 Ballantine, Cristina M.

Ramanujan Type Buildings
We will construct a finite union of finite quotients of the affine
building of the group $\GL_3$ over the field of $p$adic numbers
$\mathbb{Q}_p$. We will view this object as a hypergraph and estimate
the spectrum of its underlying graph.
Keywords:automorphic representations, buildings Category:11F70 

32. CJM 2000 (vol 52 pp. 737)
33. CJM 2000 (vol 52 pp. 172)
 Mao, Zhengyu; Rallis, Stephen

Cubic Base Change for $\GL(2)$
We prove a relative trace formula that establishes the cubic base
change for $\GL(2)$. One also gets a classification of the image
of base change. The case when the field extension is nonnormal
gives an example where a trace formula is used to prove lifting
which is not endoscopic.
Categories:11F70, 11F72 

34. CJM 1999 (vol 51 pp. 771)
 Flicker, Yuval Z.

Stable BiPeriod Summation Formula and Transfer Factors
This paper starts by introducing a biperiodic 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 biconjugacy, and
stabilizes the geometric side of the biperiod summation formula.
Thus weighted sums in the stable biconjugacy class are expressed
in terms of stable biorbital integrals. These stable integrals
are on the same endoscopic groups $H$ which occur in the case of
standard conjugacy.
The spectral side of the biperiod 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 formswith non
vanishing periodsand the resulting biperiod distributions
associated to ``periodic'' automorphic forms, are related to
analogous biperiod 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 biperiodic
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 $\SL(2)$.
Categories:11F72, 11F70, 14G27, 14L35 

35. CJM 1999 (vol 51 pp. 130)
 Savin, Gordan; Gan, Wee Teck

The Dual Pair $G_2 \times \PU_3 (D)$ ($p$Adic Case)
We study the correspondence of representations arising by
restricting the minimal representation of the linear group of type
$E_7$ and relative rank $4$. The main tool is computations of the
Jacquet modules of the minimal representation with respect to
maximal parabolic subgroups of $G_2$ and $\PU_3(D)$.
Categories:22E35, 22E50, 11F70 

36. CJM 1999 (vol 51 pp. 164)
 Tan, Victor

Poles of Siegel Eisenstein Series on $U(n,n)$
Let $U(n,n)$ be the rank $n$ quasisplit unitary group over a
number field. We show that the normalized Siegel Eisenstein series
of $U(n,n)$ has at most simple poles at the integers or half
integers in certain strip of the complex plane.
Categories:11F70, 11F27, 22E50 

37. CJM 1998 (vol 50 pp. 74)
 Flicker, Yuval Z.

Elementary proof of the fundamental lemma for a unitary group
The fundamental lemma in the theory of automorphic forms is proven
for the (quasisplit) unitary group $U(3)$ in three variables
associated with a quadratic extension of $p$adic fields, and its
endoscopic group $U(2)$, by means of a new, elementary technique.
This lemma is a prerequisite for an application of the trace
formula to classify the automorphic and admissible representations
of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$.
It compares the (unstable) orbital integral of the characteristic
function of the standard maximal compact subgroup $K$ of $U(3)$ at
a regular element (whose centralizer $T$ is a torus), with an
analogous (stable) orbital integral on the endoscopic group $U(2)$.
The technique is based on computing the sum over the double coset
space $T\bs G/K$ which describes the integral, by means of an
intermediate double coset space $H\bs G/K$ for a subgroup $H$ of
$G=U(3)$ containing $T$. Such an argument originates from
Weissauer's work on the symplectic group. The lemma is proven for
both ramified and unramified regular elements, for which endoscopy
occurs (the stable conjugacy class is not a single orbit).
Categories:22E35, 11F70, 11F85, 11S37 
