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 of $V$ in terms of the numbers $\lambda_i$?
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 $Alt^2(V)$.
Best Answer
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.