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Howe duality and the trace formula. (English) Zbl 1008.22008

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 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.

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F27 Theta series; Weil representation; theta correspondences

Citations:

Zbl 0827.11032
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