In the direction of academic development plans, the fellow attended many training courses, workshop, and conferences.
In the direction of research, the fellow worked on development of the mathematical theory of covariant functions from different perspectives. A website is created which informed about project’s targets and achieved progress. In line with the policy for dissemination of our results, we make our papers online on arXiv once they submitted to journals for review process.
For compact subgroups, the developed harmonic analysis methods for covariant functions of characters based on an operator theoretic approach. The results include interesting characterizations for classical Banach spaces on covariant functions of characters of compact subgroups. We classified classical Banach spaces of covariant functions of characters of compact subgroups as quotient spaces of classical Banach spaces of functions on the group. These classifications imply a unified and constructive characterizations for the dual spaces of classical Banach spaces of covariant funcions of characters of compact subgroups. We also investigated different algebraic and analytic properties of these spaces including module properties and convolutions.
In the case of normal subgroups, we developed a unified operator theoretic approach related to Banach covariant function spaces of characters of normal subgroups including a theory for structure of Banach convolution modules induced by the group algebras on Banach covariant function spaces of characters of normal subgroups and covariant convolutions (convolution of covariant functions). For general characters, we studied properties of convolution module actions induced by group algebras on classical Banach spaces of covariant functions. In the case that the character is invariant, we introduced a well-defined notion of convolution for covariant functions.
