Remarks on Diffeological Frobenius Reciprocity

Abstract

A recent paper [R22] 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.

v3 to appear in Trans. AMS: added table row (5.7h) and reference [H19], both inadvertently omitted from v2