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Authors
Martin H. Weissman
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We incorporate covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. We work with all covers that arise from extensions of quasisplit reductive groups by $mathbf{K}_2$ -- the class studied by Brylinski and Deligne. We use this L-group to parameterize genuine irreducible representations in many contexts, including covers of split tori, unramified representations, and discrete series for double covers of semisimple groups over $mathbb R$. An appendix surveys torsors and gerbes on the etale site, as they are used in the construction of the L-group.
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In one article, the author has defined an L-group associated to a cover of a quasisplit reductive group over a local or global field. In another article, Wee Teck Gan and Fan Gao define (following an unpublished letter of the author) an L-group associated to a cover of a pinned split reductive group over a local or global field. In this short note, we give an isomorphism between these L-groups. In this way, the results and conjectures discussed by Gan and Gao are compatible with those of the author. Both support the same Langlands-type conjectures for covering groups.
We incorporate nonlinear covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. This L-group is an extension of the absolute Galois group of a local or global field $F$ by a complex reductive group. The L-group depends on an extension of a quasisplit reductive $F$-group by $mathbf{K}_2$, a positive integer $n$ (the degree of the cover), an injective character $epsilon colon mu_n rightarrow {mathbb C}^times$, and a separable closure of $F$. Our L-group is consistent with previous work on covering groups, and its construction is contravariantly functorial for certain well-aligned homomorphisms. An appendix surveys torsors and gerbes on the etale site, as they are used in a crucial step in the construction.
In this joint introduction to an Asterisque volume, we give a short discussion of the historical developments in the study of nonlinear covering groups, touching on their structure theory, representation theory and the theory of automorphic forms. This serves as a historical motivation and sets the scene for the papers in the volume. Our discussion is necessarily subjective and will undoubtedly leave out the contributions of many authors, to whom we apologize in earnest.
We establish the functorial transfer of generic, automorphic representations from the quasi-split general spin groups to general linear groups over arbitrary number fields, completing an earlier project. Our results are definitive and, in particular, we determine the image of this transfer completely and give a number of applications.
Let $pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${rm U}_{mathfrak n}$ defined over a number field $F$. Under the assumption that $pi$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(frac{1}{2},pitimeschi)$, with a certain automorphic character $chi$ of ${rm U}_1$, for the case of ${mathfrak n}=2,3,4$, and for the general ${mathfrak n}geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.
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