Two remarks on a theorem of Dipendra Prasad

摘要

We show two results on local theta correspondence and restrictions of irreducible admissible representations of GL(2) over p-adic fields. Let F be a nonarchimedean local field of characteristic 0, and let L be a quadratic extension of F. Let epsilon(L/F) is the character of F-x corresponding to the extension L/F, and let GL(2)(F)(+) be the subgroup of GL(2)(F) consisting of elements with epsilon(L/F)(det g) = 1. The first result is that the theorem of Moen-Rogawski on the theta correspondence for the dual pair (U(1), U(1)) is equivalent to a result by D. Prasad on the restriction to GL2(F)+ of the principal series representation of GL(2)(F) associated with 1, epsilon(L/F). As the second result, we show that we can deduce from this a theorem of D. Prasad on the restrictions to GL(2)(F)(+) of irreducible supercuspidal representations of GL(2)(F) associated to characters of L-x.