Exercise 15, § 4

Let $q$ be a power of an odd prime number and F a finite field with $q$ elements. Consider the Heisenberg group $H_3(F)$ of matrices of size 3 with coefficients in F which are upper triangular and unipotent. For $(a, b, c)\in F^3$, write

0 0 1

The mapping $\sigma$ is therefore a bijection.

a) The center C of $H_3(F)$ is identified with F by the group homomorphism such that $\sigma (a, b, c)\mapsto c$.

b) There exist $q^2$ unitary characters of dimension 1 of $H_3(F)$; these characters are trivial on C.

c) Let X be the subgroup of index $q$ of F formed by the elements $\sigma (0, b, c)$ for $(b, c)\in F^2$. The subgroup X is identified with $F^2$. Let $\psi_1$ and $\psi_2$ be unitary characters of F and put $\psi (\sigma (0, b, c)) =\psi_1(b)\psi_2(c)$. The unitary representation $\varrho$ of $H_3(F)$ induced by the character $\psi$ of X is irreducible if and only if $\psi_2$ is nontrivial; it depends only on $\psi_2$ up to isomorphism, and its central character is equal to $\psi_2$. The support of the character of $\varrho$ is equal to C.

d) The irreducible representations constructed in the two preceding questions are all the irreducible unitary representations of $H_3(F)$. e) Let $\varrho$ be one of the irreducible unitary representations of dimension $q$ of $H_3(F)$. The representation $\varrho$ is faithful and satisfies $M_4(\varrho ) =q^2$.