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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 $\rm{CP}^{d-1}$?

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A subgroup of $PGL_n$ with only finite orbits on $P^{n-1}$ is finite. This is because $P^{n-1}$ has a finite subset $F$ such that every element of $PGL_n$ acting on $F$ as the identity is the identity everywhere (take $F$ to $\{e_1,e_2,\dots,e_n,e_1+e_2+\dots+e_n\}$ for some basis $(e_i)$). –  YCor Feb 26 '13 at 16:18
    
I think that the OP meant infinitely many orbits, not that the orbits were infinite sets(which, as you say, is obvious). –  Venkataramana Feb 26 '13 at 16:21

2 Answers 2

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

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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.

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@Aakumadula: Sorry, there were a couple typos... I've just fixed them and double checked the list -- it should be fine now. Thanks! –  Renato G Bettiol Feb 28 '13 at 15:05

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