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## 3. Lifts

1. ### Base change

• $F$ : a $p$-adic field of characteristic $0.$
• $G_F$ : an unramified linear group (quasi-split over $F$ and split over a finite unramified extension $L$ of $F.$)
• $\widetilde{G}_F$ : a covering group of degree coprime to $p.$

#### Problem 3.1.

[M. Weissman] Given a genuine irreducible representation $\widetilde{\pi}$ of $\widetilde{G}_F,$ describe $Lift_{L/F}(\widetilde{\pi})$ the base change lift of $\widetilde{\pi}.$ Note that $Lift_{L/F}(\widetilde{\pi})$ is supposed to be the virtual representation of $\widetilde{G}_L.$
Part of the problem:
• (a) Define $\widetilde{G}_L.$

• (b) List axioms to characterize $Lift_{L/F}(\widetilde{\pi}),$ including explicit description of norm map.
•     This is a preliminary problem to Problem basechange

#### Problem 3.2.

Does $Gal(L/F)$ act on $\widetilde{G}_L$?
• #### Problem 3.3.

[E. Lapid] Make sense of transferring representation from $\widetilde{G}_F$ to $\widetilde{G'}_F,$ where two linear groups $G_F$ and $G'_F$ are inner forms each other.
Possible trouble: centres of $\widetilde{G}_F$ and $\widetilde{G'_F}$ can be different.
• #### Problem 3.4.

[F. Adams] Is there a natural lifting from representations of $G_F$ to $\widetilde{G'}_F$?
• #### Problem 3.5.

What is the local character identity for $Mp(2n)$?

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