T σρ(G) theories and their Hilbert series

We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3d superconformal gauge theories T (rho) (sigma) (G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3 planes. Here G is a classical group, sigma is a partition of G and rho a partition of the dual group G (a). In deriving such a formula we make use of the recently discovered formula for the Hilbert series of the quantum Coulomb branch of superconformal theories. The result can be expressed in terms of a generalization of a class of symmetric functions, the Hall-Littlewood polynomials, and can be interpreted in mathematical language in terms of localization. We mainly consider the case G = SU(N) but some interesting results are also given for orthogonal and symplectic groups.