  We present a closed formula, analogous to the Weyl dimension formula, for the
signature of an invariant Hermitian form on any finite-dimensional irreducible
representation of a real reductive Lie group, assuming that such a form exists.
The formula shows in a precise sense that the form must be very indefinite. For
example, if an irreducible representation of $GL(n,R)$ admits an invariant form
of signature $(p,q)$, then we show that $(p-q)^2 \le p+q$. The proof is an
application of Kostant's computation of the kernel of the Dirac operator.
