Null little-group unitary duals, including low dimensions

Suppose $D\ge 4$ and put $n=D-2$. By [F1] the stabilizer group is $E={\mathbb{R}}^n⋊C$ with $C$ acting through rotations on ${\mathbb{R}}^n$. Under Euclidean character coordinates, the dual action is $\xi \mapsto c\xi$. It has orbits $\{0\}$ and the spheres $S_{\rho }=\{\xi :\mathrm{\mid }\xi \mathrm{\mid }=\rho \}$: norm is invariant, and $SO(n)$ acts transitively on each unit sphere by extending any unit vector to an oriented orthonormal basis; the cover $C$ has the same orbits because it maps onto $SO(n)$. The rational sublevel sets $\{\mathrm{\mid }\xi \mathrm{\mid }<r\}$, $r\in {\mathbb{Q}}_{>0}$, together with $\{0\}$ form a countable invariant Borel family separating these orbits. At $\xi =\rho e_1$, its stabilizer is ${\pi }^{-1}(SO(n-1))$ for the covering rotation map $\pi :C\to SO(n)$. If $C=SO(n)$ this is $SO(n-1)$. If $C=\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)$ and $n\ge 4$, the inclusion $SO(n-1)\hookrightarrow SO(n)$ induces an isomorphism on ${\pi }_1$ by the sphere-fibration calculation in [F1], so this preimage is connected and is the universal cover $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n-1)$. For $n=3$, its preimage over $SO(2)$ is the connected double circle $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(2)$ (the lifted $2\pi$ rotation ends at the nontrivial deck element); for $n=2$, $SO(n-1)=SO(1)=\{1\}$ and the preimage is the two-element kernel $\{\pm 1\}$.

Suppose $D=3$. By [F1] the null stabilizer is $A=\mathbb{R}\times C_N$, where $C_N=\mathbb{Z}\mathrm{/}N\mathbb{Z}$ or $\mathbb{Z}$, and the action is direct-product action because $SO(1)$ is trivial. The group $A$ is abelian. For any irreducible strongly continuous unitary representation $\pi$ of $A$, every $\pi (a)$ commutes with all $\pi (b)$; [F3] makes $\pi (a)$ scalar. Thus $\pi$ is one-dimensional and is a continuous character of each factor. By [F4], its real-line factor is $u\mapsto e^{itu}$ for a unique $t\in \mathbb{R}$ (put $t=2\pi \xi$). Since the action on $\widehat{\mathbb{R}}$ is trivial, rational intervals in the real character coordinate form a countable invariant Borel family separating all its singleton orbits, so [F2] applies to this semidirect product as well.

The hypotheses of [F2] hold for $E$: $C$ is second-countable locally compact, the action is continuous, and step 1.1 gives countably separated dual orbits. Applying [F2] therefore gives every irreducible over $\{0\}$ as an irreducible representation of $C$ with trivial ${\mathbb{R}}^n$ action, and every irreducible over $S_{\rho }$ as ${\mathrm{Ind}}_{{\mathbb{R}}^n⋊C_{\xi }}^E({\chi }_{\xi }\otimes \tau )$ for an irreducible $\tau$ of $C_{\xi }$. The equivalence criterion in [F2] says that the orbit radius and, at fixed $\xi$, the class of $\tau$ determine the representation exactly. In this induced model, a little translation $v\in {\mathbb{R}}^n$ acts over the orbit point $\eta \in S_{\rho }$ by multiplication by $e^{i\eta \cdot v}$. The application of Mackey and Schur here inherits AC through [F2], [F3] and [A1].

If $C_N=\mathbb{Z}\mathrm{/}N\mathbb{Z}$, [F5] lists its possible generator values as $e^{2\pi ik\mathrm{/}N}$ with $k=0,\dots ,N-1$, so its characters are $j\mapsto e^{2\pi i\alpha j}$ with $\alpha =k\mathrm{/}N$ modulo $1$; these are precisely the parameters satisfying $N\alpha \in \mathbb{Z}$. If $C_N=\mathbb{Z}$, a character is determined by its arbitrary value $e^{2\pi i\alpha }$ at $1$, with $\alpha \in \mathbb{R}\mathrm{/}\mathbb{Z}$. The two factors commute and every pair of factor characters multiplies to a character of the direct product, so these formulas give all and only the displayed ${\theta }_{t,\alpha }$. They are inequivalent unless both parameters agree in their respective parameter spaces.

For $n=2$, $SO(2)$ fixes a nonzero vector only at the identity, whereas in its connected double cover $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(2)$ the full inverse image of that identity is its two-element kernel. This stabilizer is abelian, so [F3] makes every irreducible unitary representation one-dimensional; its generator must act by a scalar $\lambda$ with ${\lambda }^2=1$, and [F5] gives exactly $\lambda =\pm 1$. The base stabilizer is trivial and has only its identity representation. For every $n\ge 2$, $C$ is compact: this is immediate for $SO(n)$ and for $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(2)$, and for $n\ge 3$ the universal cover $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to SO(n)$ has two sheets by [F1]; a finite-sheeted cover of compact $SO(n)$ is compact, since finitely many relatively compact neighborhoods whose closures lie in evenly covered charts cover $SO(n)$, and their compact closures lift to finitely many compact sheets. The closed subgroup $C_{\xi }$ is compact, and the continuous bijection $C\mathrm{/}{C}_{\xi }\to S_{\rho }$ is a homeomorphism because its domain is compact and its codomain Hausdorff. Since ${\mathbb{R}}^n$ is in the inducing subgroup, $E\mathrm{/}({\mathbb{R}}^n⋊C_{\xi })\to C\mathrm{/}{C}_{\xi }$, $(v,c)({\mathbb{R}}^n⋊C_{\xi })\mapsto c{C}_{\xi }$, is a homeomorphism; hence the full-support Radon measure of [F2] is a full-support finite measure on the positive-dimensional sphere. Choose countably many pairwise disjoint nonempty open spherical caps along an arc converging to one point; each has positive measure and its indicator is a nonzero $L^2$ function. For a unit vector $w$ in the nonzero space $V_{\tau }$, the functions $1_{U_j}w$ are pairwise orthogonal, so the induced Hilbert space is infinite-dimensional.

The discrete factor acts trivially on the real translation dual, so it cannot map the character $u\mapsto e^{itu}$ to $u\mapsto e^{-itu}$. For $t\mathrm{\ne }0$, the two signs are distinct by uniqueness in [F4]. The parameter $\alpha$ belongs to the independent second direct-product factor. Each displayed character has one-dimensional carrier by definition.

Fix $v=\pi e_1\mathrm{/}\rho$. At $\eta =\rho e_1$, the multiplier from step 2.1 equals $e^{i\eta \cdot v}=-1$. By continuity it is different from $1$ on an open neighborhood of that point in $S_{\rho }$, and the full-support measure of the induced model assigns positive measure to this neighborhood. Thus the little translation representation is not the trivial representation for every $\rho >0$.

If $D=2$, [F1] says the nonzero future null stabilizer is the trivial group. It has exactly one irreducible unitary representation, the one-dimensional trivial representation. In light-cone coordinates the connected boost of rapidity $s$ sends $(p^0+p^1,p^0-p^1)$ to $(e^s(p^0+p^1),e^{-s}(p^0-p^1))$. The two future rays $(r,r)$ and $(r,-r)$, $r>0$, have respectively the second and first coordinate zero, so no boost carries one to the other. Combining the $D\ge 4$ cases in steps 1.1, 2.1, 3.1 and 4.1, the $D=3$ cases in steps 1.2, 2.2 and 3.2, and the $D=2$ case just proved yields the stated null little-group classification in every dimension.