Multiplicity one theorem for general Spin groups: The Archimedean case

On the duality involution for p-adic general spin groups

On supports of induced representations for p-adic special orthogonal and general spin groups

Each orthogonal group $${\text {O}}(n)$$ has a nontrivial $${\text {GL}}(1)$$ -extension, which we call $${\text {GPin}}(n)$$ . The identity component of $${\text {GPin}}(n)$$ is the more familiar $${\text {GSpin}}(n)$$ , the general Spin group. We prove that the restriction to $${\text {GPin}}(n-1)$$ of an irreducible admissible representation of $${\text {GPin}}(n)$$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $${\text {GSpin}}(n)$$ . Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $${\text {O}}(n)$$ , and of Waldspurger, who proved that for $${\text {SO}}(n)$$ . We also give an explicit description of the contragredient of an irreducible admissible representation of $${\text {GPin}}(n)$$ and $${\text {GSpin}}(n)$$ , which is needed to apply their method to our situations.

Local descent to quasi-split even general spin groups

In this work we prove that the local γ \gamma -factor arising from the doubling integrals for split general spin groups is stable. This deep property of the γ \gamma -factor constitutes an important ingredient in the application of the (generalized) doubling method to the construction of a global functorial lift. We obtain our result by adapting the arguments of Rallis and Soudry who proved the stability property for symplectic and orthogonal groups.

We present an integral representation for the tensor product $L$-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. This is the first in a series of papers, treating symplectic and even orthogonal groups. Subsequent papers (in preparation) will treat odd orthogonal and general spin groups, the metaplectic covering version of these integrals, and applications to functoriality coming from combining this work with the converse theorem (and independent of the trace formula).

We give a complete description of the image of the endoscopic functorial transfer of generic automorphic representations from the quasi-split general spin groups to general linear groups over arbitrary number fields. This result is not covered by the recent work of Arthur on endoscopic classification of automorphic representations of classical groups. The image is expected to be the same for the whole tempered spectrum, whether generic or not, once the transfer for all tempered representations is proved. We give a number of applications including estimates toward the Ramanujan conjecture for the groups involved and the characterization of automorphic representations of GL(6) which are exterior square transfers from GL(4), among others. More applications to reducibility questions for the local induced representations of p-adic groups will also follow.

We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose $L$ -groups have classical derived groups. The important transfer from $\\mbox{\\rm GSp}_{4}$ to $\\mbox{\\rm GL}_{4}$ follows from our result as a special case

Abstract.In this paper we study square integrable representations and L -functions for quasisplit general spin groups over a p-adic field. In the first part, the holomorphy of L -functions in a half plane is proved by using a variant formof Casselman's square integrability criterion and the Langlands–Shahidi method. The remaining part focuses on the proof of the standard module conjecture. We generalize Muić's idea via the Langlands–Shahidimethod towards a proof of the conjecture. It is used in the work of M. Asgari and F. Shahidi on generic transfer for general spin groups.

Let ${\textrm{E}}/{\textrm{F}}$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let ${\textrm{A}}$ be an ${\textrm{F}}$-central simple algebra of even dimension so that it contains ${\textrm{E}}$ as a subfield, set ${\textrm{G}}={\textrm{A}}^\times $ and ${\textrm{H}}$ for the centralizer of ${\textrm{E}}^\times $ in ${\textrm{G}}$. Using a Galois descent argument, we prove that all double cosets ${\textrm{H}} g {\textrm{H}}\subset{\textrm{G}}$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo’s result for ${\textrm{F}}$-split ${\textrm{G}}$ [ 14], which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that ${\textrm{H}}$-distinguished irreducible representations of ${\textrm{G}}$ are self-dual and this implies that $({\textrm{G}},{\textrm{H}})$ is a Gelfand pair $$\begin{equation*}\operatorname{dim}_{\mathbb{C}}(\operatorname{Hom}_{{\textrm{H}}}(\pi,\mathbb{C}))\leq 1\end{equation*}$$for all smooth irreducible representations $\pi $ of ${\textrm{G}}$. Finally we explain how to obtain the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch ([ 1]), and we then show self-duality of irreducible distinguished representations in the archimedean case too.

The analogue of the section conjecture can be explored over a local field k. In the archimedean case only \(k = \mathbb{R}\) makes sense. In this case, the space of sections is in bijection with the set of connected components of real points, see Theorem 229. Several proofs of this fact are known and presented here.For a finite extension \(k/{\mathbb{Q}}_{p}\) the results are less complete. A section in this case gives rise to a valuation of the function field that extends the p-adic valuation on k, such that the image of the section lies in the decomposition subgroup of the valuation, see Theorem 235. However, up to now we cannot exclude that this valuation might be an exotic valuation not corresponding to a rational point.KeywordsSection ConjectureExotic ValuesDecomposition SubgroupArchimedean CaseReal PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the asymptotics of Whittaker functions on split groups and relate them to the cuspidal exponents of the representation.

In this paper, we prove, following earlier work of Waldspurger \cite{Wa1}, \cite{Wa4} a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(\pi)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields \cite{Beu1} and thus are only new when $F=\mathbb{R}$.

Following the approach in the archimedean case, we introduce the notion of admissible metrics for line bundles on curves and abelian varieties over non-archimedean local fields. Several properties of admissible metrics are considered and we show that this approach yields the same notion of admissible metrics over curves as doing harmonic analysis on the reduction graph of the curve.

Langlands classification for L-parameters