Mixed hard Lefschetz and Hodge–Riemann conjecture for non-Archimedean valuations

Let $V$ be a finite-dimensional vector space over a non-Archimedean local field, and let $Val^e(V)$ be the graded algebra of smooth translation-invariant valuations on $V$. For each lattice $\Lambda\subset V$, let $V_{\Lambda,1}\in Val_1^\infty(V)$ be the, up to scalar, unique $GL(\Lambda)$-invariant element, and let

$${\cal K}=\{V_{\Lambda,1}\mid \Lambda\subset V\text{ is a lattice}\}.$$

Mixed hard Lefschetz and Hodge–Riemann conjecture. $Val^\infty(V)$ satisfies the mixed hard Lefschetz theorem for the subset ${\cal K}$, and it satisfies the mixed Hodge–Riemann relations for the subset ${\cal K}$.

The non-mixed case of the hard Lefschetz statement was proved by the author; the corresponding non-mixed Hodge–Riemann statement is also established in the cited work. The mixed assertions are the new conjectural claims, with the analogous formulation for convolution following from the Fourier-type isomorphism.