Gequivariant

Gequivariant is a term used in differential geometry and representation theory to describe structures in which a group action interacts with geometric data in a base-dependent or fiberwise manner, generalizing the ordinary notion of G-equivariance. In a strict sense, a fiber bundle E -> B carries a gequivariant G-action if there is a smooth action of a Lie group G on E by bundle automorphisms that cover a smooth action of G on B, and the action on the fibers varies smoothly with the base point. A map f: E -> F between gequivariant bundles is gequivariant if f is G-equivariant on each fiber and respects the base action.

In another usage within the same vein, some authors describe gequivariant constructions as those in which a geometric object (such as a connection, metric, or differential operator) is compatible with a group action in a way that is not globally constant but respects the geometry of the base. This perspective connects to equivariant cohomology, where invariants are studied under a group action, and to gauge theory, where local symmetries vary over a space.

Examples include vector bundles equipped with a family of G-actions on the fibers parameterized by the base, or principal G-bundles with automorphism actions that cover a base group action. Gequivariant notions also arise in contexts where both the base geometry and the group dynamics influence symmetry properties of constructions.

See also: G-equivariant, equivariant cohomology, gauge theory, fiber bundle, representation theory.