Kirillov orbit method for finite Lazard Lie group

Statement

The Kirillov orbit method is a method that can be used to determine the irreducible representations of a finite Lazard Lie group. The procedure is as follows:

• Let $L$ be the Lazard Lie ring of $G$.
• Denote by ${\hat {L}}$ the Pontryagin dual of $L$, viewed only as an additive abelian group. Note that ${\hat {L}}$ is isomorphic to $L$, but there is no natural isomorphism.
• The natural action of $G$ on $L$ (called the adjoint representation) induces a natural action of $G$ on ${\hat {L}}$ (called the coadjoint representation). The orbits under this action correspond to the irreducible representations. Moreover, the size of any orbit is the square of the degree of the irreducible representation to which it corresponds. Note that this is consistent combinatorially with the fact that sum of squares of degrees of irreducible representations equals order of group.

Explanation

A full proof is beyond our scope here, but we can explain what is going on.

In terms of group rings, duals, and bases

The underlying vector space of the group ring $\mathbb {C} [G]$ can be thought of as the set of all set maps $G\to \mathbb {C}$ with pointwise addition and scalar multiplication. The multiplication is defined in terms of convolution.

Since $G$ has an element-to-element bijection with $L$, its Lazard Lie ring, this bijection induces a $\mathbb {C}$-vector space isomorphism between $\mathbb {C} [G]$ and $\mathbb {C} [L]$, where the latter is the group ring for $L$, and can be thought of as set maps from $L$ to $\mathbb {C}$.

The ring $\mathbb {C} [L]$ has another choice of basis: the one-dimensional characters of $L$. With this basis, the multiplication by convolution simply becomes coordinate-wise multiplication (this follows from the orthonormality of characters). In other words, the ring $\mathbb {C} [L]$ is isomorphic to the ring $\mathbb {C} ^{\hat {L}}$, i.e., a direct product of ${\hat {L}}$ many copies of the algebra $\mathbb {C}$.

It is important to note that for this analysis, $L$ and ${\hat {L}}$ are being treated as basis sets rather than as vector spaces or modules themselves.

The adjoint action of $G$ on $L$ induces a coadjoint action of $G$ on the set ${\hat {L}}$. We get orbits for this action. Each orbit is invariant under the $G$-action. Thus, the subspace of $\mathbb {C} [L]$ spanned by each orbit is a $G$-invariant subspace of $\mathbb {C} [L]$. We have thus obtained a decomposition of $\mathbb {C} [L]$ as a direct sum of a bunch of $G$-invariant subspaces.

The claim is that, under the identification of $\mathbb {C} [L]$ with $\mathbb {C} [G]$ explained above, these subspaces correspond to the minimal two-sided ideals of $\mathbb {C} [G]$. Based on knowledge of the representation theory of $\mathbb {C} [G]$, we know that the minimal two-sided ideals are the matrix algebras for the irreducible representations of $G$. We thus have:

Size of an orbit for the $G$-action on ${\hat {L}}$ = Dimension of corresponding matrix algebra for an irreducible representation of $G$ = Square of degree of that irreducible representation of $G$

References

• Enumerating classes and characters of p-groups by E. A. O'Brien and C. Voll, (Year 2012): ^{ArXiVMore info}: Section 3.2 of this paper is the most relevant, with Theorem 3.1 (stated without proof) being the relevant theorem.
• Kirillov's orbit method for p-groups and pro-p-groups by Jon Gonzalez-Sanchez, Commutative Algebra, Volume 37,Number 12, Page 4476 - 4488(Year 2009): ^{Unofficial copy on author page (PDF)More info}: Theorem 4.4
• The orbit method for profinite groups and a p-adic analogue of Brown's theorem by Mitya Boyarchenko and Maria Sabitova, Israel Journal of Math, Volume 165, Page 67 - 91(Year 2008): ^{ArXiV copyMore info}: Theorem 2.6