Can someone please help me with the basic properties of the little group of the 1st kind?
Let $K$ be an invariant subgroup of a group $G$:
$$K \lhd G\;.$$
If $\Delta (K)$ is an irreducible representation of $K$, then $$\Delta_g(K) \equiv \Delta (g^{-1} K g)$$ is an irreducible representation also.
All representations $$ \{\;\Delta_g(K)\;|\;g\in G \;\} $$ make an orbit. The orbit can be parameterised by any representation belonging to it.
The entire set of irreducible representations of $K$ can be presented as a set of non-intersecting orbits.
${\underline{\mbox{Definition.}}}~~$
For an arbitrary irreducible representation $\Delta$, the little group of the 1st kind is defined as $$ H = \left\{\;h\in G\;|\;\Delta(K) \sim \Delta_h (K) \right\}\;. $$
${\underline{\mbox{Lemma 1.}}}~~$
Representations belonging to the same orbit have their little groups interrelated as $H' = g H g^{-1}$ and therefore isomorphic to one another.
${\underline{\mbox{Proof:}}}~~$
If we define $H'$ as $$ H' = \left\{\;h'\in G\;|\;\Delta_g(K) \sim \left(\Delta_g\right)_{h'}(K) \;\right\}\;,$$ this definition implies, for an arbitrary $k_1\in K$: $$ \Delta(g^{-1} k_1 g) \sim \Delta (g^{-1} h'^{-1} k_1 h' g) = \Delta \left(\;(g^{-1} h'^{-1} g) \; (g^{-1}k_1g)\;(g^{-1} h' g)\;\right)\;. $$ Recalling that $K \lhd G$, we see that $k = g^{-1} k_1 g \in K$, so we may write the above equation as $$ \Delta(k) \sim \Delta \left(\;(g^{-1} h'^{-1} g)\;k\;(g^{-1 }h g)\;\right)\;. $$ Comparing this with the definition of $H$, we see that $(g^{-1} h' g)\in H$, whence $H' = g H g^{-1}$.
QED
${\underline{\mbox{Lemma 2.}}} \qquad K< H$
${\underline{\mbox{Proof.}}} ~~$
For an arbitrary fixed $k\in K$ and for all $k'\in K$, we have: $$ \Delta_{k'}(k) = \Delta( k'^{-1}\,k\,k') = \Delta^{-1}(k') \Delta(k) \Delta(k')\quad \Longrightarrow\quad \Delta_{k'}(k) \sim \Delta(k) $$ QED
Now a question: is the subgroup $K$ invariant $\,\underline{\mbox{in}\;H}\,$?
Can we claim that $K\lhd H\,$?
Many thanks!