# Kirillov orbit method for finite inner-Lazard Lie group

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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.