Classical invariants for representations of one Lie group can
often be related to invariants of some other Lie group. Physics suggests
that the right objects to consider for these questions are certain
refinements of classical invariants known as conformal blocks. Conformal
blocks appear in algebraic geometry as spaces of global sections of line
bundles on the moduli stack of parabolic bundles on a smooth curve.
Rank-level duality connects a conformal block associated to one Lie
algebra to a conformal block for a different Lie algebra.

In this talk, we will first discuss  rank-level duality for the pair
orthogonal and spin groups on the projective line and then for the pair
(G_2, F_4) over a smooth curve of genus g. If time permits, we will
discuss  rank-level duality for odd orthogonal groups over curves of
positive genus.  The later is joint with Richard Wentworth.