  We compute generating functions for the sum of the real-valued character
degrees of the finite general linear and unitary groups, through symmetric
function computations. For the finite general linear group, we get a new
combinatorial proof that every real-valued character has Frobenius-Schur
indicator 1, and we obtain some q-series identities. For the finite unitary
group, we expand the generating function in terms of values of Hall-Littlewood
functions, and we obtain combinatorial expressions for the character degree
sums of real-valued characters with Frobenius-Schur indicator 1 or -1.
