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2. Fundamental properties


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      Problem 2.1.

      [W. Gan] Determine notion of stable conjugacy when stable conjugacy classes in $\widetilde{G}_F$ and $\widetilde{H}_L$ naturally are related.


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          Problem 2.2.

          [J. Adams] In the real case, given a cover of $G_{\mathbb{R}}$, describe Brylinski-Deligne data.

            • ABV for covering groups


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              Conjecture 2.3.

              [P. Trapa] There is a canonical isomorphism as $K$-groups \[ KRep \widetilde{G}_{\mathbb R} \cong \Big( KPer_{H} (X) \Big)^*, \] where $X$ is a $\mathbb C$-algebraic variety and $H$ is a $\mathbb C$-algebraic group (for the trivial cover, $H={^{\vee}}G$).

                • Lurie Conjecture


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                  Problem 2.4.

                  [S. Lysenko] Twisted Whittaker models by D. Gaitsgory

                    • Covering group of tori


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                      Problem 2.5.

                      [M. Weissman and S. Lysenko] Let $T$ be a (not necessary split) torus over a $p$–adic field. Construct $\widetilde{T}$ \[ 1 \longrightarrow \mu_F \longrightarrow \widetilde{T} \longrightarrow T \longrightarrow 1 \] explicitly within Brylinski-Deligne framework.
                          Nice subcases: when $T$ is anisotropic or unramified.


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                          Problem 2.6.

                          [J. Adams] What is special about $2$-fold covers?
                              Maybe, $\pm \in \mathbb Q$?

                              Cite this as: AimPL: Automorphic forms on covering groups, available at http://aimpl.org/autoformcovergp.