论文标题
部分可观测时空混沌系统的无模型预测
On generalization of Breuil--Schraen's $\mathscr{L}$-invariants to $\mathrm{GL}_n$
论文作者
论文摘要
令$ p $为质数，$ k $为$ p $ - adic领域。我们系统地计算了$ \ mathrm {gl} _n（k）$的局部分析通用Steinberg表示（滞后）之间的较高$ \ Mathrm {ext} $ - 通过某些组合序列的新组合序列，由所谓的Tits tits Complex产生。这样的频谱序列在第二页和每个$ \ mathrm {ext} $ - 组允许的规范过滤中，其分级件是相应频谱序列的第二页中的术语。对于每对滞后，我们特别感兴趣的是他们的$ \ mathrm {ext} $ - 在底部的两个非散态度中。我们写下了此类$ \ mathrm {ext} $组的每个分级件（在规范过滤下）的明确基础，然后描述使用这些基础的$ \ mathrm {ext} $组之间的杯子产品图。作为一个应用程序，我们将Breuil的$ \ Mathscr {l} $ - $ \ Mathrm {gl} _2（\ Mathbb {q} _p）$和Schraen的较高$ \ m nathscr {l} $ for $ \ mathrm { $ \ mathrm {gl} _n（k）$。一路上，我们还建立了伯恩斯坦 - Zelevinsky几何引理对Orlik-trauch构建的可允许的本地分析表示形式的概括，从而推广了Schraen论文的$ \ Mathrm {gl} _3（\ MathBb {Q} _p} _p）$的结果。
Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute the higher $\mathrm{Ext}$-groups between locally analytic generalized Steinberg representations (LAGS for short) of $\mathrm{GL}_n(K)$ via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each $\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their $\mathrm{Ext}$-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an $\mathrm{Ext}$-group, and then describe the cup product maps between such $\mathrm{Ext}$-groups using these bases. As an application, we generalize Breuil's $\mathscr{L}$-invariants for $\mathrm{GL}_2(\mathbb{Q}_p)$ and Schraen's higher $\mathscr{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ to $\mathrm{GL}_n(K)$. Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for $\mathrm{GL}_3(\mathbb{Q}_p)$.
