Distinction of the Steinberg representation II: An equality of characters

Abstract

Abstract In a previous paper, Broussous and the author prove that for symmetric spaces of the form G(E)/G(F), where G is a reductive group defined and split over a p-adic field F and E is an unramified quadratic extension of F, when the residual field of F is large enough, the Steinberg representation is ε-distinguished for a unique quadratic character ε of G(F). This proves a conjecture stated by Dipendra Prasad in 2001, under the assumption that ε is the same character as the character χ used in the statement of that conjecture. This article completes the proof of Prasad's conjecture in the unramified case assuming that G is split over F, by proving that we actually have ε = χ ${\varepsilon =\chi }$ , and also extends the aforementioned distinguishedness result by removing the condition on the residual field.