ON SOME RELATIVELY CUSPIDAL REPRESENTATIONS : CASES OF GALOIS AND INNER INVOLUTIONS ON GLn

Relatively cuspidal representations attached to a p-adic symmetric space G/H are thought of as the building blocks for all the irreducible H-distinguished representations of G. This work provides certain new examples of relatively cuspidal representations. We study three examples of symmetric spaces; GLₙ(E)/GLₙ(F), GL₂ₘ(F)/GLₘ(E), and GLₙ(F)/(GLₙ−ᵣ(F) × GLᵣ(F)) where E/F is a quadratic extension of p-adic fields. Those representations are given by induction from cuspidal distinguished representations of particular kinds of parabolic subgroups stable under the involution