Space of norms on locally algebraic representations

Let $F$ and $E$ be finite extensions of $\mathbb Q_p$, let $\mathbb G$ be a reductive group over $F$, and put $G=\mathbb G(F)$.

Let $V$ be a locally algebraic representation of the form $V=\pi_{\mathrm{sm}}\otimes_E\sigma_{\mathrm{alg}}$, where $\pi_{\mathrm{sm}}$ is smooth admissible and $\sigma_{\mathrm{alg}}$ is finite-dimensional algebraic.

We study the extended Goldman--Iwahori distance on the set of non-Archimedean norms on $V$.

After fixing a reference norm $\alpha_0$, its finite-distance component $\mathscr N_{\alpha_0}(V)$ is the bounded projective limit of the extended Bruhat--Tits buildings attached to $V_K=\pi_{\mathrm{sm}}^K\otimes_E\sigma_{\mathrm{alg}}$.

It is complete for the resulting uniform sup metric; this metric is of $\ell^\infty$ type and is generally not CAT(0).

We prove directly that a $G$-orbit in $\mathscr N_{\alpha_0}(V)$ is bounded if and only if this component contains a $G$-invariant norm.

The invariant norm is the pointwise supremum of the orbit.

We formulate an integral group-algebra and type-Hecke condition necessary for an invariant norm.

For $G=GL_n(F)$ we specialise to $V=\operatorname{BS}(r)=\pi_{\mathrm{gen}}(r)\otimes_E \pi_{\mathrm{alg}}(r)$.