  We give a solution to the long-standing problem of constructing the
generators of hidden symmetries of the quantum Higgs oscillator, a particle on
a d-sphere moving in a central potential varying as the inverse cosine-squared
of the polar angle. This superintegrable system is known to possess a rich
algebraic structure, including a hidden SU(d) symmetry that can be deduced from
classical conserved quantities and degeneracies of the quantum spectrum. The
quantum generators of this SU(d) have not been constructed thus far, except at
d=2, and naive quantization of classical conserved quantities leads to deformed
Lie algebras with quadratic terms in the commutation relations. The nonlocal
generators we obtain here satisfy the standard su(d) Lie algebra, and their
construction relies on a recently discovered realization of the conformal
algebra, which contains a complete set of raising and lowering operators for
the Higgs oscillator. This operator structure has emerged from a relation
between the Higgs oscillator Schroedinger equation and the Klein-Gordon
equation in Anti-de Sitter spacetime. From such a point-of-view, the
construction of the hidden symmetry generators reduces to manipulations within
the abstract conformal algebra so(d,2).
