Not quite, since the Weyl group of the quasi-split unitary group is not the symmetric group. See page 1272 of https://projecteuclid.org/euclid.pjm/1102621913

Not quite, since the Weyl group of the quasi-split unitary group is not the symmetric group. See page 1272 of Goldberg, "R-Groups and Elliptic Representations of Unitary Groups," http://www.jointmathematicsmeetings.org/proc/1995-123-04/S0002-9939-1995-1224616-6/S0002-9939-1995-1224616-6.pdf