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 𝜖L∕F is the character of F×
corresponding to the extension L∕F, and let GL2(F)+ be the subgroup of GL2(F)
consisting of elements with 𝜖L∕F(detg) = 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 GL2(F) associated with 1,𝜖L∕F. As the second result, we
show that we can deduce from this a theorem of D. Prasad on the restrictions to
GL2(F)+ of irreducible supercuspidal representations of GL2(F) associated to
characters of L×.
