Duality for Regular Systems of Weights: A Précis

The theory of a regular system of weights was originally developed in order to understand the flat structure in the period maps for primitive forms [S9]. In the present work, the theory is applied to understand the self-duality of ADE and the strange duality of Arnold. Beyond the original purpose, the theory yields further a class of dual weight systems, which seem to have close connection with the Conway group and to be interesting to be studied yet. On the other hand, the duality of weight systems has an interpretation in terms of the Dedekind eta function. But its meaning is not yet clear.