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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.