AI 中文总结
研究\(\mathbb{Q}_p\)有限扩张上约化群\(G\)的局部代数表示\(V\)上非阿基米德范数集的扩展戈德曼 - 岩堀距离，通过固定参考范数研究其有限距离分量，证明\(G\)轨道有界与不变范数的关系并给出相关条件，还对\(G = GL_n(F)\)进行了专门研究。
AI 中文摘要
设\(F\)和\(E\)是\(\mathbb{Q}_p\)的有限扩张，\(\mathbb{G}\)是\(F\)上的约化群，\(G = \mathbb{G}(F)\)。设\(V = \pi_{\mathrm{sm}}\otimes_E\sigma_{\mathrm{alg}}\)是局部代数表示，其中\(\pi_{\mathrm{sm}}\)是光滑可允许的，\(\sigma_{\mathrm{alg}}\)是有限维代数的。研究\(V\)上非阿基米德范数集上的扩展戈德曼 - 岩堀距离。固定参考范数\(\alpha_0\)后，其有限距离分量\(\mathscr{N}_{\alpha_0}(V)\)是与\(V_K=\pi_{\mathrm{sm}}^K\otimes_E\sigma_{\mathrm{alg}}\)相关的扩展布鲁哈特 - 蒂茨建筑的有界射影极限，对所得一致上确界度量是完备的，该度量是\(\ell^\infty\)型且一般不是CAT(0)。直接证明\(\mathscr{N}_{\alpha_0}(V)\)中的\(G\)轨道有界当且仅当该分量包含\(G\)不变范数，不变范数是轨道的逐点上确界。给出不变范数所需的积分群代数和型 - 赫克条件。对于\(G = GL_n(F)\)，专门研究\(V=\operatorname{BS}(r)=\pi_{\mathrm{gen}}(r)\otimes_E \pi_{\mathrm{alg}}(r)\)。
英文摘要
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 $V=π_{\mathrm{sm}}\otimes_Eσ_{\mathrm{alg}}$, where $π_{\mathrm{sm}}$ is smooth admissible and $σ_{\mathrm{alg}}$ 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 $α_0$, its finite-distance component $\mathscr N_{α_0}(V)$ is the bounded projective limit of the extended Bruhat--Tits buildings attached to $V_K=π_{\mathrm{sm}}^K\otimes_Eσ_{\mathrm{alg}}$. 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 $\mathscr N_{α_0}(V)$ 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 $V=\operatorname{BS}(r)=π_{\mathrm{gen}}(r)\otimes_E π_{\mathrm{alg}}(r)$.
