This paper develops various foundational results in the locally analytic representation theory of p-adic groups. In particular, we define the functor ``pass to locally analytic vectors'', which attaches to any continuous representation of a p-adic analytic group on a locally convex p-adic topological vector space the associated space of locally analytic vectors. Using this functor, and the point of view that its construction suggests, we establish some basic facts about admissible locally analytic representations (as defined by Schneider and Teitelbaum). We also introduce the related notion of essentially admissible locally analytic representations.
We develop the theory of locally analytic representations of compact $p$-adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard's isomorphisms between continuous, locally analytic and Lie algebra cohomology to solid representations. We also prove a comparison result between the group cohomology of a solid representation and of its analytic vectors.
A Jacquet-Langlands functor for $p$-adic locally analytic representations
Université de Clermont-Auvergne（克莱蒙奥弗涅大学）；Institute of Mathematics ”Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米恩·斯托伊洛瓦数学研究所）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）
Let $G$ be a $p$-adic reductive group and $\mathfrak{g}$ its Lie algebra. We construct a functor from the extension closure of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ associated to $\mathfrak{g}$ into the category of locally analytic representations of $G$, thereby expanding on an earlier construction of Orlik-Strauch. A key role in this new construction is played by $p$-adic logarithms on tori. This functor is shown to be exact with image in the subcategory of admissible representations in the sense of Schneider and Teitelbaum. En route, we establish some basic results in the the…
Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\math…
We prove that the canonical dimension of an admissible Banach space or a locally analytic representation of an arbitrary semisimple p-adic Lie group is either zero or at least half the dimension of a non-zero coadjoint orbit. This extends the results of Ardakov-Wadsley and Schmidt in the split semisimple case.
Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{g}$. In analogy to the translation functors introduced by Bernstein and Gelfand on categories of $U(\mathfrak{g})$-modules we consider similarly defined functors on the category of coadmissible modules over the locally analytic distribution algebra $D(G)$ on which the center of $U(\mathfrak{g})$ acts locally finite. These functors induce equivalences between certain subcategories of the latter category. Furthermore, these translation functors are naturally related to those on category $\mathcal{O}$ via the functors from cate…
Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $σ$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $σ.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.
In [BS07] Breuil and Schneider formulated a conjecture on the equivalence of the existence of invariant norms on certain $p$-adically locally algebraic representations of $GL_n(F)$ and the existence of certain de-Rham representations of $Gal(\bar{F}/F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. In [Bre03b, DI13] Breuil and de Ieso proved that in the case $n = 2$ and under some restrictions, the existence of certain admissible filtrations on the $ϕ$-module associated to the two-dimensional de-Rham representation of $Gal(\bar{F}/F)$ implies the existence of invariant norms on the corre…
Let $k$ be an algebraically closed field of characteristic $l\neq p$. We construct maximal simple cuspidal $k$-types of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$, when $F$ is a non-archimedean locally compact field of residual characteristic $p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$ is unique up to $\mathrm{M}'$-conjugation, when $F$ is either a finite field of characteristic $p$ or a non-archimedean locally compact field of residual characteristic $p$.
We show that the continuous cohomology groups of a $ p $-adic reductive group with coefficients in the locally analytic vectors of an admissible $ \mathbb{Q}_p $-Banach space representation are homeomorphic to those with coefficients in the Banach space representation itself. Moreover, we deduce that the canonical topologies on those continuous cohomology groups are Hausdorff and are the uniquely determined finest locally convex topologies.
On local integrability results for $p$-adic reductive groups
Institute of Mathematics, Academia Sinica（中央研究院数学研究所）；Department of Applied Mathematics, National Sun Yat-Sen University（国立中山大学应用数学系）；Department of Mathematics, National Taiwan University（国立台湾大学数学系）
The Emerton-Jacquet functor is a tool for studying locally analytic representations of p-adic Lie groups. It provides a way to access the theory of p-adic automorphic forms. Here we give an adjunction formula for the Emerton-Jacquet functor, relating it directly to locally analytic inductions, under a strict hypothesis that we call non-critical. We also further study the relationship to socles of principal series in the non-critical setting.
We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.
Let $G$ be the group of points of a split reductive group over a finite extension of ${\mathbb Q}_p$. In this paper, we compute the dimensions of certain classes of locally analytic $G$-representations. This includes principal series representations and certain representations coming from homogeneous line bundles on $p$-adic symmetric spaces. As an application, we compute the dimensions in Colmez' unitary principal series of ${\rm GL}_2({\mathbb Q}_p)$.
