  We prove that Alvis-Curtis duality preserves the Frobenius-Schur indicators
of characters of connected reductive groups of Lie type with connected center.
This allows us to extend a result of D. Prasad which relates the
Frobenius-Schur indicator of a regular real-valued character to its central
character. We apply these results to compute the Frobenius-Schur indicators of
certain real-valued, irreducible, Frobenius-invariant Deligne-Lusztig
characters, and the Frobenius-Schur indicators of real-valued regular and
semisimple characters of finite unitary groups.
