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We compute generating functions for the sum of the real-valued characterdegrees of the finite general linear and unitary groups, through symmetricfunction computations. For the finite general linear group, we get a newcombinatorial proof that every real-valued character has Frobenius-Schurindicator 1, and we obtain some q-series identities. For the finite unitarygroup, we expand the generating function in terms of values of Hall-Littlewoodfunctions, and we obtain combinatorial expressions for the character degreesums of real-valued characters with Frobenius-Schur indicator 1 or -1.

Generating functions for real character degree sums of finite general linear and unitary groups