REMARKS ON DIFFEOLOGICAL FROBENIUS RECIPROCITY

A recent paper of T. S. Ratiu and the third author established “Frobenius reciprocity” as a bijection t between certain symplectically reduced spaces (which need not be manifolds), and conjectured: (1^◦) t is a diffeomorphism when these spaces are endowed with their natural subquotient diffeologies, (2^◦) t respects the reduced diffeological 2-forms they may (or might not) carry. In this paper, we prove both this conjecture and a similar one on prequantum reduction, and also give new sufficient conditions for the reduced forms to exist. We stop short of proving that they always exist.