On the basis of Brylinski's work, we introduce a notion of equivariant smooth Deligne cohomology group, which is a generalization of both ordinary smooth Deligne cohomology and ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant gerbes with connection.
"Equivariant smooth Deligne cohomology." Osaka J. Math. 42 (2) 309 - 337, June 2005.