We prove that the duality operator 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.