Local Langlands Correspondence for Classical Groups and Affine Hecke Algebras

Using the results of J. Arthur on the representation theory of classical groups with additional work by Colette Moeglin and its relation with representations of affine Hecke algebras established by the author, we show that the category of smooth complex representations of a split $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). A statement of this kind had been conjecture by G. Lusztig. All classical groups (general linear, orthogonal, symplectic and unitary groups) appear in this context. We get also parameterizations of representations of affine Hecke algebras, which seem not all to be in the literature yet. All this should also shed some light on what is known as the stable Bernstein center.Comment: 29 pages; this is an updated version (among others, the statement of the definition of the inertial orbit of a Langlands parameter W_F->^LG was corrected and the proposition 1.3 improved

A note on Standard Modules and Vogan L-packets

Let $F$ be a non-Archimedean local field of characteristic $0$, let $G$ be the group of $F$-rational points of a connected reductive group defined over $F$ and let $G'$ be the group of $F$-rational points of its quasi-split inner form. Given standard modules $I(\tau ,\nu )$ and $I(\tau ',\nu ')$ for $G$ and $G'$ respectively with $\tau '$ a generic tempered representation, such that the Harish-Chandra's $\mu$-functions of a representation in the supercuspidal support of $\tau$ and of a generic essentially square-integral representation in some Jacquet module of $\tau '$ agree (after a suitable identification of the underlying spaces under which $\nu =\nu '$), we show that $I(\tau ,\nu )$ is irreducible whenever $I(\tau ',\nu ')$ is. The conditions are satisfied if the Langlands quotients $J(\tau ,\nu )$ and $J(\tau ',\nu ')$ of respectively $I(\tau ,\nu )$ and $I(\tau ',\nu ')$ lie in the same Vogan $L$-packet (whenever this Vogan $L$-packet is defined), proving that, for any Vogan $L$-packet, all the standard modules whose Langlands quotient is equal to a member of the Vogan $L$-packet are irreducible, if and only if this Vogan $L$-packet contains a generic representation. The result for generic Vogan $L$-packets of quasi-split orthogonal and symplectic groups was proven by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these Groups.Comment: 15 page

On the generic local Langlands correspondence for GSpin groups

In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of Levi subgroups of $GSpin$, by constructing Langlands parameters for general generic irreducible admissible representations of $GSpin$ from the one for generic irreducible supercuspidal representations of its Levi subgroups.Comment: 17 pages, To appear in Transactions of the AM

On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras

Let $\pi$ be an irreducible smooth complex representation of a general linear $p$-adic group and let $\sigma$ be an irreducible complex supercuspidal representation of a classical $p$-adic group of a given type, so that $\pi\otimes\sigma$ is a representation of a standard Levi subgroup of a $p$-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate $p$-adic classical group obtained by (normalized) parabolic induction from $\pi\otimes\sigma$ does not depend on $\sigma$, if $\sigma$ is "separated" from the supercuspidal support of $\pi$. (Here, "separated" means that, for each factor $\rho$ of a representation in the supercuspidal support of $\pi$, the representation parabolically induced from $\rho\otimes\sigma$ is irreducible.) This was conjectured by E. Lapid and M. Tadi\'c. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set $I$ of inertial orbits of supercuspidal representations of $p$-adic general linear groups, the category \CC _{I,\sigma} of smooth complex finitely generated representations of classical $p$-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by $\sigma$ and $I$, show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type $A$, $B$ and $D$ and establish functoriality properties, relating categories with disjoint $I$'s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato's exotic geometry.Comment: 21 pages, the results of the paper have been improved thanks to the remarks and encouragements of the anonymous refere

Mischwälder im Vergleich zu Monokulturen : eine Garantie für besseren Forstschutz und höhere Artendiversität? Makroheterocera (Lepidoptera) in Buchen- und Fichtenwäldern sowie Buchen/Fichten-Mischwäldern

The current forest policy in Germany is to change forest monocultures into mixed forests. This is based on the assumption that monocultures are less robust against climatic influences (e.g. storm, drought), more susceptible to pest organisms (JACTEL et al. 2002, WOODS 2003, BURTON et al. 1992) and are for several insect taxa known to show a lower species richness in comparison to mixed forests (YOUNG 1986, BARKMAN 1992, BURKHART & THAM 1992, DENNIS 1997, BRAGANCA et al. 1998). This investigation wanted to verify this thesis in forests consisting of European beech (Fagus sylvatica), Norway spruce (Picea abies) and of both tree species. The area of investigation was in the Solling region, a large woodland in Lower Saxony (Germany).Mischwälder im Vergleich zu Monokulturen : Eine Garantie für besseren Forstschutz und höhere Artendiversität? ; Makroheterocera (Lepidoptera) in Buchen- und Fichtenwäldern sowie Buchen/Fichten-Mischwäldern Die vorliegende Untersuchung stellt Ergebnisse aus einer vierjährigen Studie über die Nachtfalterfauna in Buchen- (Fagus sylvatica) und Fichtenreinbeständen (Picea abies), sowie Buchen/Fichten- Mischwäldern des Sollings (südliches Niedersachsen, Deutschland) vor. Ein besonderes Augenmerk wurde auf das Vorkommen forstschutzrelevanter Schädlingsarten gelegt. Die Erfassung der Nachtfalter erfolgte durch den Einsatz von Lichtfallen. Die Ergebnisse zeigen, dass die Buchenreinbestände die höchsten und die Fichtenmonokulturen die geringsten Falterzahlen aufwiesen. Ein Vergleich der vorgefundenen Artenzahlen ergibt in den Buchenreinbeständen und Buchen/Fichten-Mischbeständen gleich viele Arten, wohingegen in den Fichtenmonokulturen eine etwas geringe Artenzahl festgestellt wurde. Vermutungen, dass die Mischbestände die höchsten Artenanzahl aufweisen, konnte nicht bestätigt werden. Eine Aufstellung der Artendominanzen verdeutlicht, dass die hohe Anzahl der Nachtfalter innerhalb der Buchenreinbestände hauptsächlich auf das individuenreiche Auftreten der beiden Arten Calliteara pudibunda L. und Colocasia coryli L. beruhte. Innerhalb der Fichtenmonokultur war der Fichtenschädling Lymantria monacha L. und im Buchenreinbestand der Buchenschädling C. pudibunda mit sehr hohen Individuendichten auffällig gewesen – ein Zeichen für die potentielle Gefährdung dieser Waldtypen durch Kahlfraß dieser beiden Schadfalterarten. Resümierend kann zusammengefasst werden, dass Buchen/Fichten-Mischwälder die Artendiversität nicht erhöhen (dies gilt für Falterarten, die mit Lichtfallen erfasst werden können), aber aus Sicht des Forstschutzes bzgl. des Buchenschädlings C. pudibunda und des Fichtenschädlings L. monacha das Risiko von Kalamitäten verringern könnnen

On the unramified spherical automorphic spectrum

For a split connected reductive group $G$ defined over a number field $F$, we compute the part of the spherical automorphic spectrum which is supported by the cuspidal data containing $(T,1)$, where $T$ is a maximal split torus and $1$ is the trivial automorphic character. The proof uses the residue distributions which were introduced by the third author (in joint work with G. Heckman) in the study of graded affine Hecke algebras, and a result by M. Reeder on the weight spaces of the (anti)spherical discrete series representations of affine Hecke algebras. Note that both these ingredients are of a purely local nature. For many special cases of reductive groups $G$ similar results have been established by various authors. The main feature of the present proof is the fact that it is uniform and general.Comment: We warn the reader that we discovered a gap in the contour shift argument at the end of the paper. We are hopeful to solve it, but this may add technical complications and/or conditions. In any case, we are convinced that the paper is still important and believe in addition that the approach is still possible. We will update the text of the paper soo