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Denote by $\mathbb{F}_q$ a finite field with $q$ elements; let $\mathrm{SL}_n(\mathbb{F}_q)$ be the special linear group.

Finding all real representations of $\mathrm{SL}_n(\mathbb{F}_q)?$

The same question when instead of $\mathrm{SL}_n(\mathbb{F}_q)$ we have $\mathrm{GL}_n(\mathbb{F}_q)?$

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$\DeclareMathOperator{\SL}{SL}\DeclareMathOperator{\GL}{GL}$To determine the real representations of a finite group, it suffices to determine the complex irreducible representations and their Schur indices over $\mathbb{R}$. For $\SL_n(\mathbb{F}_q)$ this was done in [1] (see in particular Section 6), and for $\GL_n(\mathbb{F}_q)$ this had been done earlier in [2].

[1] A. Turull, The Schur Indices of the Irreducible Characters of the Special Linear Groups, J. Algebra 235 (2001), 275-314.

[2] A. Zelevinsky, Representations of Finite Classical Groups, Lecture Notes in Mathematics 869, Springer-Verlag, New York/Berlin, 1981.

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