Representations with $\mathsf{Sp}(1)^k$-reductions and quaternion-Kähler symmetric spaces

Abstract

We classify non-polar irreducible representations of connected compact Lie groups whose orbit space is isometric to that of a representation of a finite extension of $\mathsf{Sp}(1)^k$ for some $k>0$. It follows that they are obtained from isotropy representations of certain quaternion-Kähler symmetric spaces by restricting to the “non-$\mathsf{Sp}(1)$-factor”.

Appendix

Here we check the claim stated in Sect. 3.6. We run through the cases listed in the table therein.

In case 4, $\dim W=56$, so we need $\dim H\ge 56-7=49$. The only closed subgroups of $\mathsf{U}(8)$ in this dimension range are $\mathsf{SU}(8)$ and $\mathsf{U}(7)\times \mathsf{U}(1)$; however the first group acts with cohomogeneity 5 and the second one does not act irreducibly on W.

In case 5, we need $\dim H\ge 65$. The only possibilities are $\mathsf{SU}(9)$ which acts with cohomogeneity 4 (it is orbit-equivalent to $\mathsf{U}(9)$), and $\mathsf{U}(8)\times \mathsf{U}(1)$ which acts reducibly.

In case 6, $\hat{H}_{princ}$ is finite so H would have to have dimension $\dim W-7=13$, but $\mathsf{U}(4)$ admit no closed subgroups of dimension 13.

Lemma

Let $\xi :K\rightarrow \mathsf{O}(U)$ be an representation of a compact Lie group K with $\dim K>0$. Then $c(2\xi )\ge 2c(\xi )+1$.

Proof

It follows from the fact that the action of $K_{princ}$ on U preserves the decomposition of U into the tangent and normal spaces to a principal orbit. $\square$

Consider now case 1. Take a maximal subgroup of $\hat{H}$ of the form $H_1\times \mathsf{SO}(4)$. Note that $\mathsf{U}(\frac{n}{2})\times \mathsf{SO}(4)$ with n even has cohomogeneity at least 9 if $n\ge 6$ and 6 if $n=4$. Therefore, in case $W_1$ admits an $H_1$-invariant complex structure, the only possibility is $n=4$ and $H=\mathsf{SU}(2)\times \mathsf{SO}(4)$, which is already reduced (see end of Sect. 3.3). Assume next $W_1$ does not admit an $H_1$-invariant complex structure. Using the lemma above, for $c(H_1)\ge 3$ we have

$$\begin{aligned} c(H_1\otimes \mathsf{SO}(4))\ge c(4H_1)-6\ge 15-6=9, \end{aligned}$$

which is too big. If $c(H_1)=2$, we can follow ideas from [9, Lem. 14.1] to see that $(H_1)_{princ}$ acts on the tangent space U to a principal orbit in $W_1$ with at least 3 invariant subspaces, so taking a slice representation at a pure tensor $w_1\otimes w_2$ with the $w_i$ regular points, we have

$$\begin{aligned} c(H_1\otimes \mathsf{SO}(4))= & {} 2+ c((H_1)_{princ}\times \mathsf{SO}(3),(U\otimes _{{\mathbb {R}}}\mathbb {R}^3)\oplus \mathbb {R}^3) \\= & {} 3 + c((H_1)_{princ}\times \mathsf{SO}(2),U\otimes _{{\mathbb {R}}}(\mathbb {R}^2\oplus \mathbb {R})) \\\ge & {} 3+ c((H_1)_{princ},3U)-1 \\\ge & {} 2 + 3c((H_1)_{princ},U) \\= & {} 11, \end{aligned}$$

which is too big. If $c(H_1)=1$, we run through the possible representations of real type (see e.g. [7]) to find that the condition $c(H_1\otimes \mathsf{SO}(4))\le 7$ implies $H_1=\mathsf{Spin}(7)$. Now $\mathsf{Spin}(7)\otimes \mathsf{SO}(4)$ is a representation with trivial pig and cohomogeneity 5, so we need a subgroup of codimension 2 of $\mathsf{SO}(4)$. We finally get the first candidate $H=\mathsf{Spin}(7)\times \mathsf{U}(2)\subset \mathsf{SO}(8)\times \mathsf{SO}(4)$.

Continuing with case 1, consider a maximal subgroup $\mathsf{SO}(n)\times H_2$ which acts irreducibly on W with cohomogeneity 7 where $H_2\subset \mathsf{SO}(4)$. The form of the pig of $\mathsf{SO}(n)\otimes \mathsf{SO}(4)$ shows that $H_2$ has codimension 3 in $\mathsf{SO}(4)$; further it acts irreducibly on $W_2$ so $H_2$ must be one of the factors of $\mathsf{SO}(4)=\mathsf{SU}(2)\mathsf{SU}(2)$, as desired.

Consider next case 2. Take the maximal subgroup $\mathsf{Sp}(\frac{n}{2})\times \mathsf{U}(4)$ of $\hat{H}$ for n even. We have $c(\mathsf{Sp}(\frac{n}{2})\otimes \mathsf{U}(4))\ge 11$ if $n\ge 6$ and $c(\mathsf{Sp}(2)\otimes \mathsf{U}(4))=6$. Since $\mathsf{Sp}(2)\times \mathsf{U}(4)$ acts with trivial pig, we need a codimension one subgroup. We get our second candidate $H=\mathsf{Sp}(2)\times \mathsf{SU}(4)\subset \mathsf {S}(\mathsf{U}(4)\times \mathsf{U}(4))$.

Still in case 2, consider the maximal subgroup $\mathsf{SU}(p)\times \mathsf{SU}(q)\times \mathsf{U}(4)$ of $\hat{H}$, where $pq=n$, $p\ge q$, $p\ge 3$, $q\ge 2$. By the Monotonicity Lemma [9, Lem. 12.1], its cohomogeneity is bounded below by $c(\mathsf{U}(2)\otimes \mathsf{SU}(2)\otimes \mathsf{SU}(4))=c(\mathsf{SO}(4)\otimes \mathsf{U}(4))=10$, which is too big. The same estimate rules out the maximal subgroup $\mathsf{SO}(n)\times \mathsf{U}(4)$.

Continuing with case 2, we need also to consider maximal subgroups of $\hat{H}$ of the form $\tau _1(H_1)\times \mathsf{U}(4)$ where $H_1$ is simple and $\tau _1$ is an irreducible representation of complex type. Since

$$\begin{aligned} 7\ge c(\tau _1(H_1)\otimes \mathsf{U}(4))\ge c(\tau _1(H_1)\otimes \mathsf{U}(2))\ge 2c(\tau _1(H_1))-4, \end{aligned}$$

we deduce $2\le c(\tau _1(H_1))\le 5$. In this situation the underlying groups have dimension too small to attain the desired cohomogeneity [9, §12.8]. Next, a maximal subgroup of $\hat{H}$ of the form $\mathsf{U}(n)\times H_2$ where $H_2\subset \mathsf{Sp}(4)$ has

$$\begin{aligned} c(\mathsf{U}(n)\otimes H_2)\ge c(\mathsf{U}(4)\otimes H_2)=c(H_2\otimes \mathsf{U}(4))>7 \end{aligned}$$

by the arguments above, unless $H_2=\mathsf{Sp}(2)$. Now we have $c(\mathsf{U}(n)\otimes \mathsf{Sp}(2))=c(\mathsf{SU}(n)\otimes \mathsf{Sp}(2))=6$ for $n\ge 5$. Finally, in case $n=4$, the diagonal $\mathsf{U}(4)$-subgroup of $\mathsf{U}(1)(\mathsf{SU}(4)\times \mathsf{SU}(4))$ acts reducibly on W and this finishes case 2.

We next move to case 3. Consider maximal subgroups of $\hat{H}$ of the form $\tau _1(H_1)\times \mathsf{Sp}(4)$ where $H_1$ is simple and $\tau _1$ is an irreducible representation of quaternionic type. Since

$$\begin{aligned} 7\ge c(\tau _1(H_1)\otimes \mathsf{Sp}(4))\ge c(\tau _1(H_1)\otimes \mathsf{Sp}(2))\ge 2c(\tau _1(H_1))-10, \end{aligned}$$

we deduce $2\le c(\tau _1(H_1))\le 8$. Under this condition, all cases (see [9, §12.8]) give $c(\tau _1(H_1)\otimes \mathsf{Sp}(4))>7$ simply by counting dimensions, but $\tau _1=(\mathsf{Sp}(1),\mathbb {H}^2)$ which gives $c(\mathsf{Sp}(1)\times \mathsf{Sp}(4),\mathbb {H}^2\otimes _{{\mathbb {H}}}\mathbb {H}^4)=3$, and there are no more groups to be considered. Continuing, using the Montonicity Lemma in the estimates

$$\begin{aligned} c(\mathsf{U}(n) \otimes \mathsf{Sp}(4))\ge c(\mathsf{U}(4)\otimes \mathsf{Sp}(4))=12 \end{aligned}$$

and

$$\begin{aligned} c(\mathsf{SO}(p)\otimes \mathsf{Sp}(q)\otimes \mathsf{Sp}(4))\ge c(\mathsf{SO}(3)\otimes \mathsf{Sp}(1)\otimes \mathsf{Sp}(4))=9 \end{aligned}$$

with $pq=n$, $p\ge 3$, $q\ge 1$, we further discard two classes of maximal subgroups of $\hat{H}$. We finish this case by excluding maximal subgroups of type $\mathsf{Sp}(n)\times H_2$ and the diagonal subgroup of $\mathsf{Sp}(4)\times \mathsf{Sp}(4)$ in a similar manner to case 2.

The remaining cases (7 to 10) are associated to quaternion-Kahler symmetric spaces and give examples as described in Sect. 2. Note also that replacing the $\mathsf{Sp}(1)$-factor by $\mathsf{U}(1)$ gives a representation of cohomogeneity 6. In the following we show that subgroups of $\hat{H}$ of the form $H_1\mathsf{Sp}(1)$, where $H_1$ is a closed subgroup of the “non-$\mathsf{Sp}(1)$-factor” give no examples.

In case 7, $\dim W=28$ and $\hat{H}$ has finite pig, so we need a subgroup of $\mathsf{Sp}(3)$ of dimension $28-\dim \mathsf{SU}(2)-7=18$, but it does not exist.

In case 8, $\dim W=40$ and we need closed subgroups of $\mathsf{SU}(6)$ of dimension at least 30, however they do not exist.

In case 9, $\dim W=64$, so we need closed subgroups of $\mathsf{Spin}(12)$ of dimension at least 54. The only closed maximal subgroup in this dimension range is $\mathsf{Spin}(11)$, however it acts reducibly on W.

In case 10, $\dim W=112$, so we need closed subgroups $H_1$ of $\mathsf{E}_{7}$ of dimension at least 102. According to [14, §8], there are no such subgroups.