Stable unipotent orbital integrals and their Satake transforms.
(Intégrales orbitales unipotentes stables et leurs transformées de Satake.)

*(French)*Zbl 1112.22005Summary: We are concerned with some questions arising from harmonic analysis on \(p\)-adic groups. More precisely, we are interested in Satake transforms of stable unipotent distributions in the case of split groups. This problem is motivated, on one hand, by M. Assem’s work on the computation of unipotent orbital integrals, and on the other hand, by J.-L. Waldspurgers’ on the determination of the space of stable unipotent distributions. This question is easy for general linear groups but unkown in general. In this work, we deal with the groups \(\text{Sp}(2n)\). For \(n=2\), we show that these Satake transforms are regular functions over the rank-2 unitary real torus. We then show that these functions can be recovered by the Satake transform of some distributions of a totally different kind: the twisted compact traces of an explicit familly of representations of \(\text{GL}(5)\). This phenomenon may be explained by twisted endoscopy between \(\text{Sp}(2n)\) and \(\text{GL}(2n+1)\) as remarked by Arthur. For \(n > 2\), we show, in some cases, that the Satake transforms of these traces are actually regular functions, of a common form, over the rank-\(n\) unitary real torus. In particular, we have verified it when \(n\leq 4\). We expect that it is true in general. Thanks to these computations, we then propose a quite precise conjecture, that describes the Satake transforms of stable unipotent distributions on \(\text{Sp}(2n)\).

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

22E35 | Analysis on \(p\)-adic Lie groups |