The method of little groups

The idea of the method of little groups, by Wigner and Mackey, can be summarised as follows. (Excerpted from Section 8.2 in GTM 42, Linear representations of finite groups, by J.-P. Serre)

Assume that the group $G$ is a semi-direct product of its two subgroups $H$ and $A$, with $A$ abelian. Denote $\hat{A}:=\mathrm{Hom}(A,{\mathbb{C}}^{\times })$. The group $G$ acts on $\hat{A}$ by $$(g\chi )(a)=\chi (g^{-1}ag),\forall g\in G,a\in A,\chi \in \hat{A}.$$

Let $({\chi }_i{)}_{i\in \hat{A}/H}$ be a system of representatives for the orbits of $H$ in $\hat{A}$. For each $i\in \hat{A}/H$, let Hi=StabH(χi)={h∈H∣hχi=χi}H_i = \mathrm{Stab}_H(\chi_i)=\{h \in H \mid h\chi_i = \chi_i\}Hi​=StabH​(χi​)={h∈H∣hχi​=χi​} and let $G_i=A\cdot H_i<G$. Extend ${\chi }_i$ to $G_i$ by setting $${\chi }_i(ah)={\chi }_i(a),\forall a\in A,h\in H_i$$.

Now let $\rho \in \mathrm{Irr}(H_i)$ and $p:G_i\to H_i$ the canonical projection. We thus have an irreducible representation $\tilde{\rho }=p\circ \rho$ of $G_i$. Finally, by taking the tensor product of ${\chi }_i$ and $\tilde{\rho }$ we obtain an irreducible representation $\chi \otimes \tilde{\rho }$ of $G_i$.

Denote ${\theta }_{i,\rho }={\mathrm{Ind}}_{G_i}^G{\chi }_i\otimes \tilde{\rho }$. Assume the following results:

Proposition: (a) ${\theta }_{i,\rho }$ is irreducible;
(b) If ${\theta }_{i,\rho }$ and ${\theta }_{i^{\prime },{\rho }^{\prime }}$ are isomorphic, then $i=i^{\prime }$ and $\rho \sim {\rho }^{\prime }$ (isomorphic);
© Every irreducible representation of $G$ is isomorphic to one of the ${\theta }_{i,\rho }$.

Exercise.

1. The Heisenberg group ${\mathbf{H}}_n(k)$ over a field $k$ of dimension $n$ can be construct via the exact sequence $$0\to k\to {\mathbf{H}}_n(k)\to W\to 0,$$
   where $W=V\oplus V^{\prime }$ and $V=k^n$ is a vector space of dimension $n$. The group law of ${\mathbf{H}}_n(k)$ is given by $$(x,x^{\prime },a)(y,y^{\prime },b)=(x+x^{\prime },y+y^{\prime },a+b+x{y}^{\prime }),$$
   where $(x,x^{\prime }),(y,y^{\prime })\in W$ and $a,b\in k$.

Solve the following problems.
(i) Find an embedding of ${\mathbf{H}}_n(k)$ into the group of unipotent upper-triangluar matrices of the form $$\left(\begin{array}{ccc}1& x^{\top }& t\\ 0& I_n& x^{\prime }\\ 0& 0& 1\end{array}\right).$$

(ii) Apply the method of little groups to ${\mathbf{H}}_1(k)$ in the case that $k={\mathbb{F}}_p$ (a finite field of cardinality $p$), by taking $A=\{(0,x^{\prime },a)\}$ and $H=\{(x,0,0)\}$. Find all ${\theta }_{i,\rho }$.

Assume the fact that $\hat{\mathbb{R}}\cong \mathbb{R}$.

(iii) (Challenge) Apply the method of little groups to ${\mathbf{H}}_1(k)$ in the case that $k=\mathbb{R}$. Find all ${\theta }_{i,\rho }$.

2. The Euclidean motion group of the Cartesian plane is the semi-direct product of $H=\mathbf{SO}(2)$ and $A={\mathbb{R}}^2$. Apply the method of little groups to this group and work out the representations ${\theta }_{i,\rho }$ (The action of $H$ on $A$ is obvious.)
3. Do the same thing in 2 to the group B={(ab0a−1)∣a,b∈R,a≠0}.B = \{\begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} \mid a, b \in \mathbb{R}, a \neq 0\}.B={(a0​ba−1​)∣a,b∈R,a=0}.