  An extended local Lorentz symmetry in four-dimensional (4D) theory is
considered. A source of this symmetry is a group of general linear
transformations of four-component Majorana spinors GL(4,M) which is isomorphic
to GL(4,R) and is the covering of an extended Lorentz group in a 6D Minkowski
space M(3,3) including superluminal and scaling transformations. Physical
space-time is assumed to be a 4D pseudo-Riemannian manifold. To connect the
extended Lorentz symmetry in the M(3,3) space with the physical space-time, a
fiber bundle over the 4D manifold is introduced with M(3,3) as a typical fiber.
The action is constructed which is invariant with respect to both general 4D
coordinate and local GL(4,M) spinor transformations. The components of the
metric on the 6D fiber are expressed in terms of the 4D pseudo-Riemannian
metric and two extra complex fields: 4D vector and scalar ones. These extra
fields describe in the general case massive particles interacting with an extra
U(1) gauge field and weakly interacting with ordinary particles, i.e.
possessing properties of invisible (dark) matter.
