Question

The special unitary group $\rm{SU}(d)$ has a canonical action on the Hilbert space of dimension $d$, and this action induces a canonical action on the projective space $\rm{CP}^{d-1}$, which is transitive. Is there any proper subgroup of $\rm{SU}(d)$ that acts transitively on $\rm{CP}^{d-1}$? If the answer is negative, is it true that any proper subgroup of $\rm{SU}(d)$ must have infinite orbits on $\rm{CP}^{d-1}$?

Answer 1

If $d$ is even, the group of symplectic matrices in $SU(d),\quad $ call it $Sp(d),\quad $ acts transitively on ${\mathbb P}^{d-1}({\mathbb C})$.

Answer 2

Groups acting transitively on $\mathbb C P^n$ (as well as on $\mathbb HP^n$ or on $\mathbb{Ca}P^2$) were classified by Oniscik; here's the original paper (if you understand Russian). As a consequence of this classification, the only group other than $SU(d)$ that acts transitively (and almost effectively) on $\mathbb C P^{d-1}$ is the example given in Aakumadula's answer: $Sp(d)$, when $d$ is even.

The full classification table of projective spaces written as homogeneous spaces $G/H$ (with the usual assumptions: $G$ connected, action is almost effective etc.) is as follows:

1. $G=SU(n+1)$, $H=S(U(1)\times U(n))$, $G/H=\mathbb C P^n$, isotropy representation is irreducible;
2. $G=Sp(n+1)$, $H=Sp(n)\times Sp(1)$, $G/H=\mathbb H P^n$, isotropy representation is irreducible;
3. $G=F_4$, $H=Spin(9)$, $G/H=\mathbb{Ca}P^2$, isotropy representation is irreducible;
4. $G=Sp(n+1)$, $H=Sp(n)\times U(1)$, $G/H=\mathbb C P^{2n+1}$, isotropy representation has $2$ irreducible summands.

Just for completeness, the list of homogeneous structures on the only remaining compact rank one symmetric spaces (i.e., spheres) was first obtained by Montgomery, Samelson and Borel and then reobtained in Oniscik's work.