  Discrete orthogonality relations for Hall-Littlewood polynomials are
employed, so as to derive cubature rules for the integration of homogeneous
symmetric functions with respect to the density of the circular unitary
ensemble (which originates from the Haar measure on the special unitary group
$SU(n;\mathbb{C})$). By passing to Macdonald's hyperoctahedral Hall-Littlewood
polynomials, we moreover find analogous cubature rules for the integration with
respect to the density of the circular quaternion ensemble (which originates in
turn from the Haar measure on the compact symplectic group $Sp
(n;\mathbb{H})$). The cubature formulas under consideration are exact for a
class of rational symmetric functions with simple poles supported on a
prescribed complex hyperplane arrangement. In the planar situations
(corresponding to $SU(3;\mathbb{C})$ and $Sp (2;\mathbb{H})$), a determinantal
expression for the Christoffel weights enables us to write down compact
cubature rules for the integration over the equilateral triangle and the
isosceles right triangle, respectively.
