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## 2. Fundamental properties

1. #### 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.
• #### Problem 2.2.

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

#### 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

#### Problem 2.4.

[S. Lysenko] Twisted Whittaker models by D. Gaitsgory
• ### Covering group of tori

#### Problem 2.5.

[M. Weissman and S. Lysenko] Let $T$ be a (not necessary split) torus over a $p$â€“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.
• #### 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.