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Register a new userL-groups and the Langlands program for covering groups: a historical introduction
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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.
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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.
We study the dimension of the space of Whittaker functionals for depth zero representations of covering groups. In particular, we determine such dimensions for arbitrary Brylinski-Deligne coverings of the general linear group. The results in the paper are motivated by and compatible with the work of Howard and the second author, and earlier work by Blondel.
In this article we propose a geometric description of Arthur packets for $p$-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of $p$-adic groups and propose a geometric description of the transfer coefficients that appear in Arthurs main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogans work, especially the prediction that Arthur packets are ABV-packets for $p$-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
Let $E/F$ be a finite and Galois extension of non-archimedean local fields. Let $G$ be a connected reductive group defined over $E$ and let $M: = mathfrak{R}_{E/F}, G$ be the reductive group over $F$ obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from $G(E)$ to $M(F)$. We obtain a depth-comparison formula for Weil-restricted groups.
These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the CMSA Math Science Literature Lecture Series in May 2020. We attempt to give a birds-eye view of basic aspects of the theory of quantum groups.
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