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We compute univariate and multigraded Hilbert series of invariants andcovariants of representations of the circle and orthogonal group$\operatorname{O}_2$. The multigradings considered include the maximal gradingassociated to the decomposition of the representation into irreducibles as wellas the bigrading associated to a cotangent-lifted representation, orequivalently, the bigrading associated to the holomorphic and antiholomorphicparts of the real invariants and covariants. This bigrading induces a bigradingon the algebra of on-shell invariants of the symplectic quotient, and thecorresponding Hilbert series are computed as well. We also compute the firstfew Laurent coefficients of the univariate Hilbert series, give samplecalculations of the multigraded Laurent coefficients, and give an example toillustrate the extension of these techniques to the semidirect product of thecircle by other finite groups. We describe an algorithm to compute each of theassociated Hilbert series.

Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups