TITLE: Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
QUESTION [18 upvotes]: 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 $Alt^2(V)$.

REPLY [5 votes]: As Jeff Adams indicates, there is a detailed proof in Bourbaki's Chapter 8 (originally published in French in 1975, but later translated into English).   See especially Table I at the end of Chapter 8 for a summary of all simple types in terms of their fundamental weights, referring back to $\S13$ for classical types and to $\S7$, no. 5, for exceptional types.  While the Onishchik-Vinberg book is quite useful as a learning tool, it is not a reasonable choice as a reference book.   Anyway, it's worthwhile to take a somewhat broader look at what is really going on here.
(1)  The term "Frobenius-Schur indicator" goes back to the joint 1906 paper by Frobenius and his student Schur in which they focused on a finite group $G$ and its complex irreducible characters.   Such a character $\chi$ arises from an irreducible matrix representation $\rho$ of $G$ over $\mathbb{C}$.    By the usual averaging argument, there is certainly a unitary representation equivalent to $\rho$ which affords $\chi$.   A natural  further question is whether there is a real representation of $G$ affording $\chi$.  Clearly the answer is no if $\chi$ takes nonreal values.  Then the original Frobenius-Schur indicator takes value 0, by definition. But even if $\chi$ is $\mathbb{R}$-valued, it may come from either a real orthogonal representation (then the indicator is written 1) or from a symplectic representation (then the indicator is $-1$).   A major step in the Frobenius-Schur work was a formula for the indicator computable from the character values: sum the values $\chi(g^2)$ over all $g \in G$, then average by dividing out $|G|$.
There are accounts of all this in many textbooks, such as those by Isaacs and by Serre (but don't start with their indexes; in Serre, try 13.2).     
(2) Though I haven't gone far into all the subsequent history, it's clear that the underlying question arises whenever you start with finite dimensional representations over $\mathbb{C}$ and ask what happens over $\mathbb{R}$.  In particular, the Frobenius-Schur question makes sense for complex (semi-)simple Lie groups and their finite dimensional representations parametrized by highest weights $\lambda$.   Here it is certainly possible to have indicator $=0$, when the underlying vector space $V$ admits no nondegenerate invariant bilinear form.   This case is avoided by assuming that the representation is self-dual: in case $V \cong V^*$ as a $G$-module, we get End$(V) \cong V^* \otimes V \cong V \otimes V$, so there is always a 1-dimensonal trivial $G$-submodule (corresponding to the scalar endomorphisms).   Since an irredudible $G$-module has at most one such nondegenerate form, it remains to decide whether it is orthogonal or symplectic.   
The combinatorics of roots and weights leads to the formal recipe indicated in a couple of the answers, which at first looks opaque.   But as Bourbaki points out, writing $m:=\sigma(\lambda)$ (in the notation used here), the half-integer $m/2$ has a natural interpretation as the sum of coefficients in the expression for $\lambda$ as a $\mathbb{Q}$-linear combination of simple roots.   Note in particular that when the root lattice equals the weight lattice (in types $G_2, F_4, E_8$), this half-integer is actually an integer and $m$ is even (so all indicators are 1).
(3) Generalizations in a number of directions have also been worked out, for example in the case of finite dimensional Hopf algebras (not just finite group algebras) or compact Lie groups or complex semisimple algebraic groups.   The formula discussed in the answers depends only on the root system, so it can be potentially applied over many kinds of fields including local fields and those of prime characteristic:   involutive maps are lurking everywhere.