Let $G$ be a simple algebraic group group over $\mathbb C$.  

Let $V$ be a self-dual representation of $G$.  

Let $\lambda$ be the highest weight of $V$.


Write $\lambda$ as a sum of fundamental weights: $\lambda=\sum \lambda\_i\omega\_i$ for $\lambda\_i\in\mathbb N$.


Is there a simple formula for the [Frobenius-Schur indicator](https://en.wikipedia.org/wiki/Frobenius%E2%80%93Schur_indicator#Invariant_bilinear_forms) of $V$ in terms of the numbers $\lambda\_i$?




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For the reader's convenience, I recall the definition of the Frobenius-Schur indicator. It is $1$ if the trivial rep occurs inside $Sym^2(V)$, and it is $-1$ if the trivial rep occurs inside $Alt^2(V)$.



Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
The Frobenius-Schur indicator (of a self-dual finite dimensional representation) is 
$$
\chi\_\pi(\exp(2\pi i\rho^\vee))
$$
where $\chi\_\pi$ is the central character of $\pi$, $\rho^\vee$ is half the sum of the positive coroots, so $\exp(2\pi i\rho^\vee)$ is an element of order $2$ in the center of $G$. Equivalently if $\lambda$ is the highest weight then this equals
$$
e^{2\pi i\langle\lambda,\rho^\vee\rangle}
$$
This is, of course, equivalent to the other formulas cited, but is conceptually simpler. In particular: if $G$ is adjoint every (irreducible, finite dimensional, self-dual) representation is orthogonal.
See Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Chapter IX, Section 7.2, Proposition 1. The proof is included, and is the one sketched by Borovoi. For a simpler proof based on the Tits group see [this preprint](http://arxiv.org/abs/1203.1901).


