Twisted signs for p-adic linear groups

Let G be a reductive p-adic group and let θ be an automorphism of G of order at most two. Let π be an irreducible smooth representation of G that is taken to its dual by θ. The space V of π then carries a non-zero bilinear form ( , ) ( unique up to scaling) with the invariance property (π(g)v, π(g)w) = (v, w), for g ∈ G and v, w ∈ V . The form is easily seen to be symmetric or skew-symmetric and we set εθ(π) = ±1 accordingly. We apply Cassleman’s pairing (in certain commonly observed circumstances) to obtain a descent result that expresses the sign εθ(π) in terms of the Jacquet modules of π. For F a p-adic field and D a quaternion division algebra over F , the groups GLn(F ) and GLn(D) admit involutions that take each irreducible smooth representation to its dual (and are the only general linear groups over p-adic division algebras with this property). We use our descent result along with special features of the representation theory of GLn(F ) and GLn(D) to study the associated signs.