Distinguished tame supercuspidal representations of symmetric pairs (Sp4n (F), Sp2n(E))

Let (G, H) be a symmetric pair defined over a p-adic field F and pi be an irreducible tame supercuspidal representation of G. defined by Yu (J Am Math Soc 3:579-622. 201)1). Hakim and Murnaghan (Int Math Res Pap IMRP 166(2): Art. ID rpn005 2008) give a dimension formula for the space Flom H (pi, 1) in terms of generic cuspidal data. In this paper, we consider the symmetric pair (Sp(4n)( F), Sp(2n)(E)) where E is a quadratic extension over F. and give sufficient and necessary conditions on such generic cuspidal data whose corresponding tame supercuspidal representations are H-distinguished. In addition, for the symmetric pair (U-2n F). Sp(2n), ( F)) there is no Sp(2n) ( F)-distinguished tame supercuspidal representation of U-2n (J, F).