Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩}),Representation Theory

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Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})
Representation Theory ( IF 0.6 ) Pub Date : 2021-05-03 , DOI: 10.1090/ert/567
Zicheng Qian

Abstract:The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43-145], [Amer. J. Math. 141 (2019), pp. 661-703], and [Camb. J. Math. 8 (2020), p. 775-951] via the construction of some interesting locally analytic representations. Let

be a sufficiently large finite extension of $\mathbf {Q}_p$ and $\rho _p$ be a

-adic semi-stable representation $\mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E)$ such that the associated Weil-Deligne representation $\mathrm {WD}(\rho _p)$ has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one $(\varphi , \Gamma )$ -modules shows that the Hodge filtration of $\rho _p$ depends on three invariants in

. We construct a family of locally analytic representations $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ of $\mathrm {GL}_3(\mathbf {Q}_p)$ depending on three invariants $\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3 \in E$ , such that each representation in the family contains the locally algebraic representation $\mathrm {Alg}\otimes \mathrm {Steinberg}$ determined by $\mathrm {WD}(\rho _p)$ (via classical local Langlands correspondence for $\mathrm {GL}_3(\mathbf {Q}_p)$ ) and the Hodge-Tate weights of $\rho _p$ . When $\rho _p$ comes from an automorphic representation $\pi$ of a unitary group over $\mathbf {Q}$ which is compact at infinity, we show (under some technical assumption) that there is a unique locally analytic representation in the above family that occurs as a subrepresentation of the Hecke eigenspace (associated with $\pi$ ) in the completed cohomology. We note that [Amer. J. Math. 141 (2019), pp. 611-703] constructs a family of locally analytic representations depending on four invariants ( cf. (4) in that publication ) and proves that there is a unique representation in this family that embeds into the Hecke eigenspace above. We prove that if a representation $\Pi$ in Breuil's family embeds into the Hecke eigenspace above, the embedding of $\Pi$ extends uniquely to an embedding of a $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ into the Hecke eigenspace, for certain $\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3\in E$ uniquely determined by $\Pi$ . This gives a purely representation theoretical necessary condition for $\Pi$ to embed into completed cohomology. Moreover, certain natural subquotients of $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ give an explicit complex of locally analytic representations that realizes the derived object $\Sigma (\lambda , \underline {\mathscr {L}})$ in (1.14) of [Ann. Sci. Éc. Norm.Supér. 44 (2011), pp. 43-145]. Consequently, the locally analytic representation $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ gives a relation between the higher $\mathscr {L}$ -invariants studied in [Amer. J. Math. 141 (2019), pp. 611-703] as well as the work of Breuil and Ding and the

-adic dilogarithm function which appears in the construction of $\Sigma (\lambda , \underline {\mathscr {L}})$ in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43-145].

中文翻译：

𝒢ℒ₃（𝐐_{𝐩}）的对数和高ℒ不变性

摘要：本文的主要目的是弄清[Ann。科学 Éc。规范。极好的。44（2011），pp.43-145]，[Amer。J.数学。141（2019），pp.661-703]和[Camb。J.数学。8（2020），p。[775-951]。设

是的足够大的有限扩展，并且是-adic半稳定表示，以使相关的Weil-Deligne表示具有第二单峰级，并且相关的Hodge过滤不重要。对秩一模的扩展的计算表明的Hodge滤波取决于中的三个不变量。我们构造了的一系列本地分析表示形式 $\ mathbf {Q} _p$ $\ rho _p$

$\ mathrm {Gal}（\ overline {\ mathbf {Q} _p} / \ mathbf {Q} _p）\ rightarrow \ mathrm {GL} _3（E）$ $\ mathrm {WD}（\ rho _p）$ $（\ varphi，\ Gamma）$ $\ rho _p$

$\ Sigma ^ {\ mathrm {min}}（\ lambda，\ mathscr {L} _1，\ mathscr {L} _2，\ mathscr {L} _3）$ $\ mathrm {GL} _3（\ mathbf {Q} _p）$ 取决于三个不变量，因此族中的每个表示都包含由（通过经典的本地Langlands对应）和的Hodge-Tate权重确定的局部代数表示。当来自一个在无穷远处紧凑的compact群的自构表示时，我们证明（在某种技术假设下）上述族中存在唯一的局部解析表示，该表示作为Hecke本征空间的子表示（与 $\ mathscr {L} _1，\ mathscr {L} _2，\ mathscr {L} _3 \ in E$ $\ mathrm {Alg} \ otimes \ mathrm {Steinberg}$ $\ mathrm {WD}（\ rho _p）$ $\ mathrm {GL} _3（\ mathbf {Q} _p）$ $\ rho _p$ $\ rho _p$ $\ pi$ $\ mathbf {Q}$ $\ pi$ ）。我们注意到[Amer。J.数学。141（2019），pp。611-703]构造了一个依赖于四个不变量的局部解析表示形式的族（请参阅该出版物的（4）），并证明该族中有一个独特的表示形式嵌入到上面的Hecke本征空间中。我们证明，如果 $\ Pi$ 布劳伊（Breuil）家族中的一个表示嵌入到上面的Hecke本征空间中，则的嵌入 $\ Pi$ 将唯一地扩展为a嵌入到Hecke本征空间中，对于的某些唯一确定。这为嵌入完整的同调学提供了一个纯粹的代表理论必要条件。此外， $\ Sigma ^ {\ mathrm {min}}（\ lambda，\ mathscr {L} _1，\ mathscr {L} _2，\ mathscr {L} _3）$ $\ mathscr {L} _1，\ mathscr {L} _2，\ mathscr {L} _3 \ in E$ $\ Pi$ $\ Pi$ $\ Sigma ^ {\ mathrm {min}}（\ lambda，\ mathscr {L} _1，\ mathscr {L} _2，\ mathscr {L} _3）$ 给出一个局部解析表示的显式复合体，该复合体实现了[Ann。[1.1] 科学 Éc。苏佩尔 44（2011），第43-145页]。因此，局部解析表示法给出了在[Amer。J.数学。141（2019），第611-703]以及布勒伊和丁和的工作出现在建设进制dilogarithm函数在[安。科学 Éc。规范。极好的。44（2011），第43-145页]。 $\ Sigma（\ lambda，\下划线{\ mathscr {L}}）$ $\ Sigma ^ {\ mathrm {min}}（\ lambda，\ mathscr {L} _1，\ mathscr {L} _2，\ mathscr {L} _3）$ $\ mathscr {L}$

$\ Sigma（\ lambda，\下划线{\ mathscr {L}}）$

更新日期：2021-05-03

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