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On irreducible representations of \(\text{SO}_{2n+1}\times \text{SO}_{2m}\). (English) Zbl 0829.22031

Let \(F\) be a local field; the authors consider certain groups of the form \(G = SO(V) \times SO(W)\) over \(F\) where \(V\) is of the form \(W \oplus X \oplus X' \oplus \langle a\rangle\); here \(X\) is isotropic, \(X'\) is the dual of \(X\) and \(\langle a\rangle\) is 1-dimensional. The authors construct a spherical subgroup \(H\) of \(G\) and a homomorphism \(H \to \mathbb{G}_a\); \(H\) is a twisted product of \(SO(W)\) with a unipotent group and, if \(G\) is quasi-split, it has an open orbit, with trivial stability subgroup, on the variety of Borel subgroups of \(G\). From the homomorphism the authors construct a character \(\Theta\) and they consider the dimension of \(\text{Hom}_H (\pi, \Theta)\) for irreducible admissible representations \(\pi\) of \(G(F)\). They make a number of conjectures about this space, which, would then be of dimension 0 or 1 and they make a precise conjecture as to when it is 1 in terms of symplectic root numbers. This conjecture was partially motivated by work of Ginzburg, Piatetski-Shapiro and Soudry which affirms the first part of the conjecture when \(F\) is non-archimedean. They also give further evidence for its validity in various cases.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11R39 Langlands-Weil conjectures, nonabelian class field theory
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