A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$

Question

Consider $(\mathbb{C}^2, \omega)$ where $\omega$ is a non-degenerate complex skew-symmetric bilinear form on $\mathbb{C}^2$. Let us write

$V = (\mathbb{C}^2, \omega)$

There are many spaces one can construct from $V$. For instance $Sym^n(V)$ is a vector space endowed with a non-degenerate complex bilinear form (respectively a non-degenerate complex skew-symmetric bilinear form) if n is even (respectively odd). Thus, we have realized $O(2k+1,\mathbb{C})$ and $Sp(2k,\mathbb{C})$ as groups of symmetry of a natural space constructed from $V$.

A more complicated example is $G_2$, which is the group of symmetries of $Sym^6(\mathbb{C}^2)$, endowed with a complex skew-symmetric trilinear form which can be defined using $\omega$ only. See for instance Theorem 1.1 (page 2) in https://arxiv.org/abs/1107.2813, though I am sure it goes back way before (not sure who is the first).

I am still struggling to get $O(2n,\mathbb{C})$ that way, as the group of symmetries of a natural space constructed from $V$, and I wonder whether the remaining exceptional Lie groups can be viewed this way.

I remark that there are more complicated spaces one can build from $V$, such as the kernel of the "symmetrization map":

$Sym^k(\mathbb{C}^2) \otimes Sym^l(\mathbb{C}^2) \to Sym^{k+l}(\mathbb{C}^2)$

or, the kernel of:

$Sym^k(\mathbb{C}^2) \otimes Sym^l(\mathbb{C}^2) \to Sym^{k-l}(\mathbb{C}^2)$

if $k \geq l$, the map being defined by contracting using the symplectic form $\omega$. (I believe that the right term for "natural space constructed from $V$" is a Schur functor, a term I have just met in one of Prof. Robert Bryant's answers to another post).

Edit 1: Allow me please to rephrase a little my question. Prof. @RobertBryant, while your answer is quite helpful, and I particularly thank you for the very nice description of $E_6$ that you have provided, yet what I would like to achieve, if possible, is different, though related.

Does there always exist, for any complex semisimple Lie algebra, a construction/definition/description of a corresponding compact real form, as a group of symmetries of $(W, \sigma)$, where $W$ is a complex vector space and $\sigma$ is a set of structures on $W$, under the conditions that $W$ can be constructed from the standard representation $V = \mathbb{C}^2$ of $SU(2)$ using (symmetrized, skew-symmetrized,...) tensor products of copies of $V$, and $\sigma$ can be constructed using the standard structures on $V$, namely the complex symplectic form $\omega \in \Lambda^2V^*$ and the quaternionic structure $j$ on $V$, which maps $(u,v)$ to $(-\bar{v}, \bar{u})$.

I already know that $Sp(m, \mathbb{R})$, $O(2m+1, \mathbb{R})$ and $G_2$ can be defined as groups of symmetries of some corresponding $(W, \sigma)$, where the latter is constructed from $(V, (\omega, j))$. We note also that $O(4, \mathbb{R})$ can also be defined as the group of symmetries of $(V \otimes V, (\sigma \otimes \sigma, j \otimes j))$. What about $O(2m, \mathbb{R})$ or $Spin(2m, \mathbb{R})$ and the other $4$ exceptional cases?

Edit 2: I would like to add that the complex $E_6$, using Prof. Robert Bryant's answer below, can also be defined using $SU(2)$ data. Indeed, consider first $(S^7(V), (S^7(\omega)))$. Note that $S^7(\omega)$ can play the role of what Prof. Bryant in his answer called $\omega$. Then one can form $\Lambda^2(S^7(V^*))$ and proceeds just as in his answer. While it is true that the $27$-dimensional complex space defined as the space of elements $\phi \in \Lambda^2(S^7(V^*))$ such that $\phi \wedge S^7(\omega)^3 = 0$, while it is not defined as a (symmetrized, skewsymmetrized,...) tensor product of copies of $V = \mathbb{C}^2$, yet the previous equation is defined only using $\omega$. I also allow that.

I have a question by the way. $S^7(j)$ is a quaternionic structure on $S^7(V)$, and $S^7(j) \wedge S^7(j)$ induces a real structure on $\Lambda^2(S^7(V^*))$. Is the group of symmetries of the previous $27$-dimensional complex space together with the cubic form and the real structure that I have just defined the compact real form $E_6$, or is it some other non-compact real form of $E_6^\mathbb{C}$? I am hoping it is the compact real form!

Answer 1

I think that the kind of question you are asking is one that was treated by Dynkin back in the 1950s (see Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72), (1952), though it is available in the AMS Translation series, and, if I understand correctly, there are more recent articles in which missing entries in the tables are added and some work has been done on the corresponding questions for the real forms.)

The basic problem is this: If $G$ is a (connected) semi-simple complex Lie group and $V$ is an irreducible (complex) representation of $G$, i.e., one has a homomorphism $\rho:G\to \mathrm{GL}(V)$, then, by semi-simplicity, the image subgroup $\rho(G)\subset \mathrm{GL}(V)$ actually lies in $\mathrm{SL}(V)$. The question is whether there is any proper (connected) Lie subgroup $H\subset \mathrm{SL}(V)$ that properly contains $\rho(G)$.

Nearly always, the answer is either (a) 'No', (b) 'Yes, $\rho(G)$ preserves a nondegenerate inner product $h$ on $V$ whose symmetry group $\mathrm{SO}(h)\subset\mathrm{SL}(V)$ is the only connected Lie subgroup between $\rho(G)$ and $\mathrm{SL}(V)$', or (c) 'Yes, $\rho(G)$ preserves a nondegenerate symplectic form $\omega$ on $V$ whose symmetry group $\mathrm{Sp}(\omega)\subset\mathrm{SL}(V)$ is the only connected Lie subgroup between $\rho(G)$ and $\mathrm{SL}(V)$.'

Dynkin classified the exceptions to this 'nearly always' rule.

For example, when $G = \mathrm{SL}(2,\mathbb{C})$, the only exception is the irreducible representation of $G$ on $V = S^6(\mathbb{C}^2)$, i.e., the $7$-dimensional representation. In that case, $\mathrm{SL}(2,\mathbb{C})$ preserves an inner product $h$ on $V$, but it also preserves a nondegenerate $3$-form $\phi$ on $V$ whose symmetry group (in $\mathrm{SL}(V)$) is $\mathrm{G}_2$, so that we have the chain of Lie groups $$ \rho(G)\subset \mathrm{G}_2\subset \mathrm{SO}(h)\subset\mathrm{SL}(V). $$

Other exceptional groups do show up in analogous ways from time to time. For example, if $\omega$ is a nondegenerate $2$-form on $W = \mathbb{C}^8$, its stabilizer is $G = \mathrm{Sp}(4,\mathbb{C})$. This $G$ acts on $\Lambda^2(W^*)$ reducibly, since it preserves the line spanned by $\omega$. The irreducible decomposition is $$ \Lambda^2(W^*) = \mathbb{C}\omega\oplus \Lambda^2_0(W^*), $$ where $V = \Lambda^2_0(W^*)$ is the $27$-dimensional representation of $G$ that consists of the $2$-forms $\phi$ on $W$ that satisfy $\phi \wedge \omega^3 = 0\in \Lambda^8(W^*)\simeq\mathbb{C}$. Clearly, $G$ does not preserve any symplectic form on $V$ (since it has odd dimension). $G$ does preserve a quadratic form $h$ defined so that $\phi^2\wedge\omega^2 = h(\phi)\,\omega^4$ for all $\phi\in V$. However, it also preserves a cubic form $c$, defined so that $\phi^3\wedge\omega = c(\phi)\,\omega^4$. It turns out that this cubic form in 27 variables is just the cubic form discovered by Cartan whose symmetry group is the exceptional group $\mathrm{E}_6$. Thus, in addition to the 'expected' inclusion $\rho(G)\subset \mathrm{SO}(h)\subset\mathrm{SL}(V)$, there is the 'exceptional' inclusion: $$ \rho(G)\subset \mathrm{E}_6\subset \mathrm{SL}(V). $$ This may be the kind of thing you are thinking about.

For more such examples, see Dynkin's paper listed above (available in translation, if you don't read Russian).