When does an irreducible G-module admit an invariant quadratic form of signature (n,n+1)

Question

Let $G$ be a connected real reductive Lie group and $V$ be a finite dimensional real irreducible $G$-module. When does $V$ admit an invariant non-degenerate quadratic form of signature $(n,n+1)$? I know of two examples, the standard representations of $SO(n,n+1)$ and the $2n^{th}$-symmetric product of the standard representation of $SL(2,\mathbb{R})$.

Answer 1

If $G$ is split then a sufficient criterion would be that the $0$-weight space is of dimension $1$. Apart from the cases already mentioned, there is only one more example of this kind, namely the $7$-dimensional representation of split $\mathsf G_2$. Further examples can be obtained as products, i.e. $G=G_1\times\ldots\times G_r$ acting on $V=V_1\otimes\ldots\otimes V_r$ where each $(G_i,V_i)$ is one of the cases above.

In general, the index of the invariant quadratic form on an irreducible representation has been calculated in the paper

Grosshans, Frank: Real orthogonal representations of algebraic groups. Trans. Amer. Math. Soc. 160 (1971) 343–352.

Skimming through that paper I got the feeling that the criterion above is also necessary whenever $G$ is split, simple, and its duality automorphism is inner (i.e., $G$ is not of type $\mathsf A_n,n\ge2$, $\mathsf D_{2n+1},n\ge2$, or $\mathsf E_6$). The restriction of the quadratic form to the $0$-weight space seems to be always definite.