Howe duality and the trace formula
One way to study a conjectured correspondence between automorphic representations of two groups \(G, G'\) is by the comparison of two ``trace formulas''. Each of the two trace formulas has a geometric side and a representation-theoretic side. The comparison of the geometric sides amounts to the matching of local orbital integrals, with, in particular, the ``fundamental lemma'', saying that corresponding Hecke functions are matching. The authors of the present paper observed that, when \(G, G'\) form a dual reductive pair and the correspondence is the Howe duality, then Howe duality for spherical functions may be used to give an easy proof for the fundamental lemma. This is done in the paper for the dual pair \(\text{SO}(2n+1), \widetilde {\text{SL}}(2)\). The authors also consider the dual pair \(\text{SO}(2n+1), \widetilde {\text{Sp}}(2n)\), treated by \textit{M. Furusawa} in [J. Reine Angew. Math. 466, 87-110 (1995; Zbl 0827.11032)]. They introduce trace formulas involving a theta series on \(\text{SO}(2n+1) \times \widetilde{\text{Sp}}(2n)\) and prove the matching of local integrals and the fundamental lemma, using again Howe duality for spherical functions.
scientific article; zbMATH DE number 1463734