Is there a generic term for orbital groups such as $e_g$ and $t_{2g}$? I am looking for a generic term for sets of atomic orbitals (viz. spherical harmonics) which are grouped by crystal symmetry.  The most familiar examples would be $e_g$ and $t_{2g}$ (in cubic symmetry).  So I would like to be able to say something like “$e_g$ and $t_{2g}$ are examples of …”
What comes to my mind is “symmetry groups [of orbitals]”, but I am not convinced.
 A: Better late than never, right?
The standard term for this is manifold. You might say that “$e_g$ and $t_{2g}$ are examples of atomic orbital manifolds”. In the literature one often finds shorter phrases like “the $t_{2g}$ manifold” or “the $j=1/2$ manifold”. Note that the term is not unique to crystal symmetry - it is also used in phrases such as “molecular orbital manifold” or even “ground state manifold”. An alternative but less common word is subspace. At time of writing, Google finds 39,600 results for “$t_{2g}$ manifold” and only 13,000 for “$t_{2g}$ subspace”.
At first glance subspace strikes me as the more natural word. After all, in a symmetry analysis of a quantum system, we begin by writing down the Schrödinger equation. Its solution space is a Hilbert space, but we can go further and look for subspaces invariant under the relevant symmetry group. These are then reduced into irreducible invariant subspaces, which are naturally associated with irreps of the symmetry group.
In contrast, while the phase space of classical mechanics can be modeled as a (symplectic) manifold, it is not a priori clear (at least to me!) that we can constrain quantum dynamics of a system to some manifold. However, any Hilbert space is a Hilbert manifold, and supposedly a lot of functional analysis and operator theory is phrased in this language. So when group theory was first applied to quantum mechanics, the words manifold and submanifold saw heavy use. Presumably this early adoption has been a large factor in its popularity today.
Finally, make sure not to attempt abbreviating “orbital manifold” to “orbifold”!
