Superpolynomials are formed with $N$ commuting and anti-commuting (skew) variables. By considering the space of skew variables of fixed degree as a module of the symmetric group $\mathcal{S}_{N}$ the theory of generalized Jack polynomials constructed by S Griffeth can be used to define nonsymmetric Jack superpolynomials. We present the theory, give details about the structure and derive norm formulas. Denote the parameter by $\kappa$ then the norm is positive-definite for $-\frac{1}{N}<\kappa<\frac{1}{N}$. Analogously there is a structure as Hecke algebra $\mathcal{H}_{N}(t)$-module on the skew polynomials and this allows the use of the theory of vectorvalued $(q, t)$-Macdonald polynomials studied by J-G Luque and the author. We outline the theory and present norm formulas and evaluations at special points. The norm is positive-definite for $q>0$ and min $(q^{1 / N}, q^{-1 / N}) < t < max (q^{1 / N}, q^{-1 / N} )$. As in the scalar case the evaluations use $(q, t)$-hook products.[-]

Matrix spherical functions associated to the symmetric pair $(G, K)=$ $\left(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2) \times \mathrm{U}(m))\right.$, having reduced root system of type $\mathrm{BC}_{2}$ are studied. We consider a $K$-representation $\left(\pi, V_{\pi}\right)$ arising from the $\mathrm{U}(2)$-part of $K$, then the induced representation $\operatorname{Ind}_{K}^{G} \pi$ is multiplicity free. The corresponding spherical functions, i.e. $\Phi: G \rightarrow \operatorname{End}\left(V_{\pi}\right)$ satisfying $\Phi\left(k_{1} g k_{2}\right)=\pi\left(k_{1}\right) \Phi(g) \pi\left(k_{2}\right)$ for all $g \in G, k_{1}, k_{2} \in K$, are studied by studying certain leading coefficients. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading coefficients. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder's 1970 s two-variable orthogonal polynomials, which are Heckman-Opdam polynomials for $\mathrm{BC}_{2}$. In particular, we find explicit orthogonality relations and the polynomials being eigenfunctions to a second order matrix partial differential operator. This is joint work with Jie Liu (Radboud $\mathrm{U}$ ).[-]