I would like to learn the basic theory of unitary representations of finite groups over finite fields. Here, the unitary group $\operatorname{GU}(n,\mathbb{F}_{q^2})$ consists of all invertible transformations of $\mathbb{F}_{q^2}^n$ that preserve the Hermitian form $\langle x, y \rangle = \sum_{i \in [n]} x_i y_i^q$, and "unitary representation" means a group homomorphism $\rho \colon G \to \operatorname{GU}(n,\mathbb{F}_{q^2})$. This is a special case of the usual notion of a representation $\rho \colon G \to \operatorname{GL}(n,\mathbb{F}_{q^2})$.

Over the complex numbers, every representation $\rho \colon G \to \operatorname{GL}(n,\mathbb{C})$ of a finite group $G$ is similar to a unitary representation $\rho' \colon G \to \operatorname{GU}(n,\mathbb{C})$, in the sense that there is an invertible operator $M$ such that $\rho'(g) = M\rho(g) M^{-1}$ for every $g \in G$. In this sense and others, the theory of unitary representations over $\mathbb{C}$ is essentially the same as that of ordinary representations.

However, over finite fields the notions are distinct. If $G$ is a finite group and $\rho \colon G \to \operatorname{GL}(n,\mathbb{F}_{q^2})$ is a representation, there might not be an invertible operator $M$ such that $M \rho(g) M^{-1} \in \operatorname{GU}(n,\mathbb{F}_{q^2})$ for every $g \in G$. For example, $\mathbb{Z}_5$ has a faithful 2-dimensional representation over $\mathbb{F}_{3^2}$ that is not similar to any unitary representation, since 5 divides $|\operatorname{GL}(2,\mathbb{F}_{3^2})|$ but not $|\operatorname{GU}(2,\mathbb{F}_{3^2})|$.

**Question**:
Have unitary representations of finite groups over finite fields been systematically studied, and if so where can I learn the basics?

Here is one example of what I want to learn to do:

- Describe all the unitary representations of the dihedral group of order 8 when $q=11$.

At the moment I do not even know how to:

- Describe all the unitary representations of $\mathbb{Z}_2 \times \mathbb{Z}_2$ when $q=3$.

Some other things I want to learn include:

Where Maschke's Theorem holds (i.e. $(|G|,q) = 1$ so that $\mathbb{F}_{q^2}[G]$ is semisimple), does every unitary representation decompose as an

*orthogonal*direct sum of irreducible unitary subrepresentations?Again where Maschke's Theorem holds, is there any analogue of the Peter-Weyl Theorem to give the space $L^2(G)$ of functions $f \colon G \to \mathbb{F}_{q^2}$ an orthogonal basis consisting of matrix elements for irreducible unitary representations?

What are some conditions for a modular representation to be similar to a unitary representation? (i.e. which subgroups of $\operatorname{GL}(n,\mathbb{F}_{q^2})$ are conjugate with subgroups of $\operatorname{GU}(n,\mathbb{F}_{q^2})$?)

Bonus for answers understandable to a humble analyst.

nondegeneratesesquilinear forms. Then a "unitary subrepresentation" should mean an invariant subspace on which the form restricts nondegenerately (e.g. isotropic subspaces are out), and an "irreducible unitary representation" has no proper unitary subrepresentations. $\endgroup$ – Joey Iverson Apr 12 at 23:39