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Search: MSC category 11F70 ( Representation-theoretic methods; automorphic representations over local and global fields )

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26. 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 Langlands-Shahidi 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

27. CJM 2002 (vol 54 pp. 263)

Chaudouard, Pierre-Henri
 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 test---indexÃ©es par les sous-groupes de LÃ©vi $M$ de $G$ et les Ã©lÃ©ments semi-simples 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 Harish-Chandra. 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 semi-simples rÃ©guliers, se dÃ©composent en produits de ces nouvelles intÃ©grales locales. Categories:22E35, 11F70

28. 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

29. 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

30. 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, buildingsCategory:11F70

31. CJM 2000 (vol 52 pp. 737)

Gan, Wee Teck
 An Automorphic Theta Module for Quaternionic Exceptional Groups We construct an automorphic realization of the global minimal representation of quaternionic exceptional groups, using the theory of Eisenstein series, and use this for the study of theta correspondences. Categories:11F27, 11F70

32. 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

33. CJM 1999 (vol 51 pp. 771)

Flicker, Yuval Z.
 Stable Bi-Period Summation Formula and Transfer Factors 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 standard conjugacy. 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 $\SL(2)$. Categories:11F72, 11F70, 14G27, 14L35

34. 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

35. CJM 1999 (vol 51 pp. 164)

Tan, Victor
 Poles of Siegel Eisenstein Series on $U(n,n)$ Let $U(n,n)$ be the rank $n$ quasi-split 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

36. 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 (quasi-split) 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
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