Fake character degrees of an algebra group

Definition
Suppose $$F$$ is a finite field of size $$q$$, a prime power with underlying prime $$p$$. Suppose $$N$$ is a nilpotent associative finite-dimensional algebra over $$F$$. Suppose, further, that $$G$$ is the algebra group corresponding to $$N$$.

The fake character degrees of $$G$$ are as follows. First, note that $$G$$ is a multiplicative subgroup of the unitization $$F + N$$, hence has an action by conjugation on $$F + N$$, under which $$N$$ is invariant. Thus, $$G$$ acts by conjugation on $$N$$ (this corresponds to the $$G$$-action on itself by conjugation). This induces an action of $$G$$ on the dual vector space to $$N$$. The square roots of the sizes of the orbits on the dual vector space are the fake character degrees.

Facts

 * The number of fake character degrees (counting repetitions) equals the number of conjugacy classes. This follows from the fact that number of orbits for finite group action on finite vector space equals number of orbits on dual vector space, and that the orbits for the action of $$G$$ on $$N$$ correspond directly to the conjugacy classes.
 * The fake character degrees are all integers that are powers of $$q$$, and the sum of their squares is the order of the group $$G$$.
 * Fake character degrees equal character degrees if algebra has nilpotency index p or less
 * Fake character degrees need not equal character degrees

Related notions

 * Kirillov orbit method for finite Lazard Lie group uses a similar approach