Hodge–Riemann bilinear relation for the unitary invariant hermitian form

Let $V$ be an irreducible module with real infinitesimal character, let $F_pV$ be its filtration, let $a$ be the integer appearing in the filtration construction, and let $(\,\ )_{\mathfrak u_{\mathbb R}}$ be the preferred $\mathfrak u_{\mathbb R}$-invariant hermitian form on $V$. For a nonzero vector $v\in F_pV\cap(F_{p-1}V)^\perp$, one has

Hodge–Riemann conjecture.

$$(-1)^{p-a}(v,v)_{\mathfrak u_{\mathbb R}}>0.$$

This is a positivity assertion for the hermitian form on the primitive graded pieces of the filtration; the supplied text does not indicate whether it has been proved or disproved.