Let $G$ be a Hausdorff, etale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a Hausdorff, am
ple groupoid, then we show that $C^*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$. We then show how this result applies to $k$-graph $C^*$-algebras. Finally, we investigate strongly purely infinite groupoid $C^*$-algebras.
We classify the finite dimensional irreducible representations with integral central character of finite $W$-algebras $U(mathfrak g,e)$ associated to standard Levi nilpotent orbits in classical Lie algebras of types B and C. This classification is gi
ven explicitly in terms of the highest weight theory for finite $W$-algebras.
