Kirillov orbit method for finite inner-Lazard Lie group

This is a slight variation on the Kirillov orbit method for finite Lazard Lie group that works for finite inner-Lazard Lie groups. The range of applicability of this method is slightly larger. For instance, it includes all finite $p$-groups of nilpotency class exactly $p$, most of which are not Lazard Lie groups.

It is capable of providing the degrees of irreducible representations. However, it does not provide as much canonical information about irreducible representations.

Constructing the action analogous to the coadjoint representation

We start with a group $G$ that is a finite inner-Lazard Lie group. Note that iff $G$ is already a Lazard Lie group, it would be preferable use the usual Kirillov orbit method for finite Lazard Lie group. So the discussion here is useful mostly for the situations where $G$ barely misses being Lazard.

By using the Lazard correspondence up to isoclinism, we can find a Lie ring $L$ that serves as a Lie ring "up to isoclinism" for $G$. This Lie ring is not unique, but it is uniquely determined up to isoclinism of Lie rings.

We now have to define an action of $G$ on $L$ by Lie ring automorphisms. We'll do this by composing the quotient map $G\to \operatorname {Inn} (G)$ with an action of the inner automorphism group $\operatorname {Inn} (G)\cong G/Z(G)$ on $L$, constructed as follows. There is a Lazard correspondence with $\operatorname {Inn} (G)$ the Lazard Lie group and $\operatorname {Inn} (L)\cong L/Z(L)$ the Lazard Lie ring. For any $u\in \operatorname {Inn} (G)$, the corresponding element $\log u\in \operatorname {Inn} (L)$ defines an inner derivation of $L$. Take the formal exponential of this, and use the fact that exponential of derivation is automorphism under suitable nilpotency assumptions to argue that this is an automorphism of $L$. This is the automorphism of $L$ associated with $u\in \operatorname {Inn} (G)$. Basic facts of the Lazard correspondence now show that this defines an action of $\operatorname {Inn} (G)$ on $L$ by automorphisms.

Identification of coadjoint orbits with irreducible representations

Having constructed an action of $G$ on $L$ by Lie ring automorphisms, we proceed the same way: we use the action to construct a group action by abelian group automorphisms on the Pontryagin dual ${\hat {L}}$ of $L$ as an additive group. The square roots of the orbit sizes for this action correspond to the irreducible representations.

However, there no longer seems to be a canonical bijection between the irreducible representations and these orbits.