In this paper we continue the study of locally analytic representations of a $p$-adic Lie group $G$ in vector spaces over a spherically complete non-archimedean field $K$, building on the algebraic approach to such representations introduced in our paper "Locally analytic distributions and p-adic representation theory, with applications to GL_2." In that paper we associated to a representation $V$ a module $M$ over the ring $D(G,K)$ of locally analytic distributions on $G$ and described an admissibility condition on $V$ in terms of algebraic properties of $M$.
In this paper we determine the relationship between our admissibility condition on locally analytic modules and the traditional admissibility of Langlands theory. We then analyze the class of locally analytic representations with the property that their associated modules are annihilated by an ideal of finite codimension in the universal enveloping algebra of G, showing under some hypotheses on G that they are sums of representations of the form $X\otimes Y$, with X finite dimensional and Y smooth. The irreducible representations of this type are obtained when X and Y are irreducible.
We conclude by analyzing the reducible members of the locally analytic principal series of $SL_2(\Qp)$.
Let G be a locally analytic group and H < G - a locally analytic subgroup. The main result is the condition (similar to Frommer-Orlik-Strauch theorem) for induction of locally analytic H-representation to G to be irreducible. Also this paper contains a (new) series of locally analytic representations which do not arise in this way.
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…
Let G be a p-adic Lie group. This paper is about the Jordan-Hoelder series of locally analytic G-representations which are induced from locally algebraic representations of a parabolic subgroup.
We propose a definition of characters in the context of Schneider-Teitelbaum's theory of locally analytic representations of p-adic reductive groups. This character will be a function on a compact subgroup of a maximal torus of the reductive group in question. As an example we treat the locally analytic principal series of SL(2,Q_p).
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.
We determine the composition factors of a Jordan-Hölder series including multiplicities of the locally analytic Steinberg representation. For this purpose we prove the acyclicity of the evaluated locally analytic Tits complex giving rise to the Steinberg representation. Further we describe some analogue of the Jacquet functor applied to the irreducible principal series representation constructed by Orlik and Strauch.
We study some closed rigid subspaces of the eigenvarieties, constructed by using the Jacquet-Emerton functor for parabolic non-Borel subgroups. As an application (and motivation), we prove some new results on Breuil's locally analytic socle conjecture for $\mathrm{GL}_n(\mathbb{Q}_p)$.
Let G be a connected split reductive group over a p-adic field. In the first part of the paper we prove, under certain assumptions on G and the prime p, a localization theorem of Beilinson-Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of G. In doing so we take up and extend some recent methods and results of Ardakov-Wadsley on completed universal enveloping algebras to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith's theorem on the canonical di…
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.
We study the problem of constructing a contragredient functor on the category of admissible locally analytic representations of a p-adic analytic group G. A naive contragredient does not exist. As a best approximation, we construct an involutive "duality" functor from the bounded derived category of modules over the distribution algebra of G with coadmissible cohomology to itself. On the subcategory corresponding to complexes of smooth representations, this functor induces the usual smooth contragredient (with a degree shift). Although we construct our functor in general we obtain its involuti…
Let $p>2$ be a prime number, and $L$ be a finite extension of $\mathbb{Q}_p$, we prove Breuil's locally analytic socle conjecture for $\mathrm{GL}_2(L)$, showing the existence of all the companion points on the definite (patched) eigenvariety. This work relies on infinitesimal "R=T" results for the patched eigenvariety and the comparison of (partially) de Rham families and (partially) Hodge-Tate families. This method allows in particular to find companion points of non-classical points.
Let L be a finite extension of Qp, and let K be a spherically complete non-archimedean extension field of L. In this paper we introduce a restricted category of continuous representations of locally L-analytic groups G in locally convex K-vector spaces. We call the objects of this category "admissible" representations and we establish some of their basic properties. Most importantly we show that (at least when G is compact) the category of admissible representations in our sense can be algebraized; it is faithfully full (anti)-embedded into the category of modules over the locally analytic dis…
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…
The p-adic local Langlands correspondence for GL2(Qp) attaches to any 2-dimensional irreducible p-adic representation V of the absolute Galois groups of Qp an admissible unitary representation Pi(V) of GL2(Qp). The unitary principal series of GL2(Qp) are those Pi(V) corresponding to trianguline representations. In this article, for p>2, using the machinery of Colmez, we determine the space of locally analytic vectors for all non-exceptional unitary principal series of GL2(Qp) by proving a conjecture of Emerton.
Let G be a connected reductive quasisplit algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.
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 L be a finite field extension of Q_p and let G be the group of L-rational points of a split connected reductive group over L. We view G as a locally L-analytic group with Lie algebra g. We define a functor from admissible locally analytic G-representations with prescribed infinitesimal character to a category of equivariant sheaves on the Bruhat-Tits building of G. For smooth representations, the corresponding sheaves are closely related to the sheaves constructed by S. Schneider and U. Stuhler. The functor is also compatible, in a certain sense, with the localization of g-modules on the f…
For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we examine functorial properties of the functors ${\mathcal F}^G_P$ introduced in \cite{OS2}. We discuss the aspects of faithfulness, projective and injective objects, Ext-groups and some kind of adjunction formulas.
Let $F$ be a finite extension of $Q_p$, $p>2$. We construct admissible unitary completions of certain representations of $GL_2(F)$ on $L$-vector spaces, where $L$ is a finite extension of $F$. When $F=Q_p$ using the results of Berger, Breuil and Colmez we obtain some results about lifting 2-dimensional mod $p$ representations of the absolute Galois group of $Q_p$ to crystabelline representations with given Hodge-Tate weights.
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 $H$ be a uniform pro-$p$ group. Associated to $H$ are rigid analytic affinoid groups $\bbH_n$, and their "wide open" subgroups $\bbH_n^{\circ}$. Denote by $D^\la(H)= C^\la(H)'_b$ the locally analytic distribution algebra of $H$ and by $\DHnn$ Emerton's ring of $\bbH_n^{\circ}$-rigid analytic distributions on $H$. If $V$ is an admissible locally analytic representation of $H$, and if $V_{\bbH_n^\circ-\an}$ denotes the subspace of $\bbH_n^\circ$-rigid analytic vectors (with its intrinsic topology), then we show that the continuous dual of $V_{\bbH_n^\circ-\an}$ is canonically isomorphic to $…
For V a 2-dimensional p-adic representation of G_Qp, we denote by B(V) the admissible unitary representation of GL_2(Qp) attached to V under the p-adic local Langlands correspondence of GL_2(Qp) initiated by Breuil. In this article, building on the works of Berger-Breuil and Colmez, we determine the locally analytic vectors B(V)an of B(V) when V is irreducible, crystabelian and Frobenius semi-simple with Hodge-Tate weights (0,k-1) for some integer k>=2; this proves a conjecture of Breuil. Using this result, we verify Emerton's conjecture that dim Ref^{η\otimesψ}(V)=dim Exp^{η|\cdot|\otimes xψ}…
For a local field $F$ we consider tamely ramified principal series representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module $V^{I_0}$ admits an integral structure, then such an integral structure can be chosen in a particularly well organized manner, in particular its modular reduction can be made completely explicit.
We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight $k$, we show its corresponding Galois representation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.
We prove an Induction Equivalence and a Kashiwara Equivalence for coadmissible equivariant D-modules on rigid analytic spaces. This allows us to completely classify such objects with support in a single orbit of a classical point with co-compact stabiliser. As an application, we use the locally analytic Beilinson-Bernstein equivalence to construct new examples of large families of topologically irreducible locally analytic representations of certain compact semisimple p-adic Lie groups.
This is a continuation of our previous work on the locally analytic vectors of the completed cohomology of modular curves. We construct differential operators on modular curves with infinite level at p in both "holomorphic" and "anti-holomorphic" directions. As applications, we reprove a classicality result of Emerton which says that every absolutely irreducible two dimensional Galois representation which is regular de Rham at p and appears in the completed cohomology of modular curves comes from an eigenform. Moreover we give a geometric description of the locally analytic representations of …
For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we introduce a category ${\mathcal O}^P$ which is equipped with a forgetful functor to the parabolic category ${\mathcal O}^{\mathfrak p}$ of Bernstein, Gelfand and Gelfand. There is a canonical fully faithful embedding of a subcategory ${\mathcal O}^{\mathfrak p}_{\rm alg}$ of ${\mathcal O}^{\mathfrak p}$ into ${\mathcal O}^P$, which 'splits' the forgetful map. We then introduce functors from the category ${\mathcal O}^P$ to the category of locally analytic representations, t…
Given a compact p-adic Lie group G over a finite unramified extension L/Q_p let G_0 be the product over all Galois conjugates of G. We construct an exact and faithful functor from admissible G-Banach space representations to admissible locally L-analytic G_0-representations that coincides with passage to analytic vectors in case L=Q_p. On the other hand, we study the functor "passage to analytic vectors" and its derived functors over general basefields. As an application we determine the higher analytic vectors in certain locally analytic induced representations.
In this paper we study certain sheaves of $p$-adically complete rings of differential operators on semistable models of the projective line over the ring of integers in a finite extension $L$ of ${\mathbb Q}_p$. The global sections of these sheaves can be identified with (central reductions of) analytic distribution algebras of wide open congruence subgroups. It is shown that the global sections functor furnishes an equivalence between the categories of coherent module sheaves and finitely presented modules over the distribution algebras. Using work of M. Emerton, we then describe admissible r…
The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number. The proof is based on elementary facts from $p$-adic analysis.
Let $Π$ be an irreducible unitary completion of a locally algebraic ${\rm GL}_2(\qp)$-representation. We describe those first-order deformations of $Π$ which are themselves completions of a locally algebraic representation. This answers a question of Paskunas and has direct applications to the Breuil-Mézard conjecture.
We define coadmissible equivariant $\mathcal{D}$-modules on smooth rigid analytic spaces and relate them to admissible locally analytic representations of semisimple $p$-adic Lie groups.
Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, and let $G_K$ be the Galois group. Let $π$ be a fixed uniformizer of $K$, let $K_\infty$ be the extension by adjoining to $K$ a system of compatible $p^n$-th roots of $π$ for all $n$, and let $L$ be the Galois closure of $K_\infty$. Using these field extensions, Caruso constructs the $(φ, τ)$-modules, which classify $p$-adic Galois representations of $G_K$. In this paper, we study locally analytic vectors in some period rings with respect to the $p$-adic …
We prove a comparison theorem between locally analytic group cohomology and Lie algebra cohomology for locally analytic representations of a Lie group over a nonarchimedean field of characteristic 0. The proof is similar to that of van-Est's isomorphism and uses only a minimum of functional analysis.
We establish duality theory of p-adic unitary Banach representations of locally profinite groups. This is an extension of Iwasawa theory for profinite groups by P. Schneider and J. Teitelbaum. We also establish a criterion for an irreducibility of a unitary representation by certain simpleness of the Iwasawa module dual to it. Through the duality, the continuous induction of a unitary Banach representation of a closed subgroup is interpreted as a certain induction of Iwasawa module. It gives an explicit description of the dual of a continuous parabolic induction on GL_n(Q_p).
