# p-adic representations and arithmetic D-modules

Runned by: Prof. Francesco Baldassarri and Andrés Sarrazola Alzate.

The aim of the seminar is to give an introduction to the theory of representations of p-adic locally analytic groups and the theory of D-cap-modules for smooth rigid analytic spaces developed by Ardakov-Wadsley. We also aim to study localisation of admissible (locally analytic) representation in the sens of Huyghe-Patel-Schmidt-Strauch. The foundations in these subjects were done by P. Schneider, J. Teitelbaum, K. Ardakov, S. Wadsley, C. Huyghe, D. Patel, T. Schmidt , M. Strauch and S. Orlik whose articles will be our main references.

• Clicking on the title of the presentation the interested reader may find the slides of the lecture.

Notes (still under construction!)

Talks

Algebraic groups and representation theory of complex semi-simple Lie algebras

In this talk we review the definition of the Grassmannian (dimension, the grassmannian as a projective variety and several examples). We also introduce the flag variety.

Abstract of the first two lectures of Pr. Giovanna Carnovale:

These talks are meant to give a very basic introduction to the notions that are needed for talking about the flag variety and its properties. After defining algebraic groups and giving some examples, I will show how to embed each algebraic group into some general linear group, how to equip homogeneous spaces with the structure of a quasiprojective variety. Next, I will define semisimple and unipotent elements, tori and unipotent groups.

Detailed summary of the lectures.

We begin by defining the Lie algebra associated to a linear algebraic group G, that corresponds to the tangent space to G at 1. This will be a sort of motivation in order to study the structure of Lie algebras. We give the first definitions, properties and examples, studying then in detail the behaviour of nilpotent and solvable Lie algebras.

We define and study the structure of semisimple Lie algebras, using the killing form. We define the abstract Jordan decomposition for elements of semisimple Lie algebras and we look at its properties. Then we skip to the study of maximal toral subalgebras, introducing the decomposition of a semisimple Lie algebra into weight spaces. Finally, as an example, we characterize the representations of the simple Lie algebra sl_2(K).

Notes for the second talk.

In this presentation we review the classical theory of root systems and Weyl groups. We also introduce Coxeter graphs and Dynkin diagrams. As an application we sketch how to classify semisimple complex Lie algebras.  At the end, we define the Bruhat order and we give some motivations for the forthcoming presentations.

In this seminar we illustrate the geometry of the complete flag variety by means of the powerful structure theory of reductive
algebraic groups. We introduce the decomposition of the flag variety into Schubert cells  and we define Schubert varieties. The latter yield a stratification of the flag variety into finitely many strata whose geometry can be nicely explained in combinatorial terms. These topics are the basic ideas of Schubert calculus, a branch of  algebraic geometry introduced in the nineteenth century by Hermann  Schubert, in order to solve problems of enumerative geometry.

• 9. Introduction to the theory of representations (part I). December 2, 2020. Francesco Esposito. University of Padova.
• 10. Introduction to the theory of representations (part II). December 10, 2020. Francesco Esposito. University of Padova.

Abstract of the talks of Pr. Francesco Esposito:

The lecture is intended to be an introduction to the theory of  representations of semisimple algebraic groups and of their Lie
algebras. The goal is to describe the classification of irreducible representations and their structure, by algebraic means by introducing  the universal enveloping algebra and the theory of weights, and by geometric means by considering global sections of line bundles on the flag variety.

Notes of the talks.

Algebraic D-modules

The aim of this talk is to give a basic introduction to the theory of  algebraic D-modules. We start with the definitions of  differential operator, D-module (left and right), integrable connection (with examples). We introduce the notion of good filtration for a coherent D-module  which leads us to the notion of characteristic variety and holonomicity. Holonomic D-modules form an abelian category which permits us to develop the tool of six Grothendieck operations, but this theory has a price to pay, we need derived categories. After a pedestrian introduction to derived categories, we introduce some Grothendieck opeartions. In particular the duality functor for D-modules, permits us to describe holonomic D-modules as coherent D-modules whose dual is holonomic too.

In the first part of this presentation we will prove that holonomicity is preserved under inverse and direct image. To make the reasoning as clear as possible, we will work on the special case where X is the affine space, and therefore D_X is the Weyl algebra. In fact, we will see that a direct computation proves that holonomicity is preserved under the inverse images of a projection, and the direct image of a standard embedding. Then we will prove the Kashiwara's equivalence (Weyl algebra edition), and using a trick due to Grothendieck, we will generalize the preceding preservation for any polynomial map between two affine spaces.

In the second part, we will explain the general statement of the Kashiwara's equivalence.

* In the notes of the presentation, the interested reader can find the classification of the irreducible representations of the first Weyl algebra given by R.E Block in his article "The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra". We use the language of minimal extensions to achieve this classification.

Recall that every integral weight gives rise to a G-equivariant line bundle on the flag variety. To begin with, we will define a sheaf of differential operators acting on a line bundle and illustrate with an example in the SL_2 case. Then we will describe a
family of sheaves of twisted differential operators parametrized by general weights. The global sections are given by a central reduction of the universal enveloping algebra U(Lie(G)).

The localization theorem establishes an equivalence between Lie algebra representations and (twisted) D-modules on the flag variety. After identifying the key elements of the proof, I will discuss explicitly the example of sl_2. We also lift the localization theorem to the equivariant level (i.e., with compatible group actions). Using a holonomicity result for suitable K, we then classify irreducible K-equivariant D-modules.

I will model this presentation on Schneider's "Non-archimedean functional analysis" (Springer 2002).

Topics: Non-archimedean fields. Example:  C_p is not maximally complete.

Locally convex  spaces: lattices vs. seminorms. Limits and colimits of locally convex spaces.  Banach and Fréchet spaces.

Examples from 1-variable p-adic analysis. Boundedness and completeness.

Topological tensor products and internal Hom's. Duality. Various notions of compactness.  Spaces of compact type.

Fréchet-Stein algebras. Examples: spaces of locally analytic functions and distributions on a p-adic locally analytic group.

In this talk, we will define the notion of locally analytic functions and use this to define locally analytic manifolds and Lie
groups. We will then describe the topology on the space of locally analytic functions on a manifold and define the corresponding dual space of distributions, recalling the necessary concepts from functional analysis. Throughout we will illustrate these definitions with examples. Time permitting, we will also define locally analytic representations of p-adic Lie groups and give examples again.

In the 1960s, Amice carried out a study of the representations of the space of K-valued, locally analytic functions on Zp and achieved a complete description of its dual, the ring of K-valued, locally Qp-analytic distributions on Zp (when K being a complete subfield of Cp). She established a topological isomorphism (by using the p-adic Fourier transform) between the ring of distributions and the space of holomorphic functions on a rigid analytic variety over K parameterizing K-valued, locally analytic characters of Zp. This rigid variety is in fact the open unit disk. This description of the ring of distributions was complemented by Lazard, who introduced a theory of divisors on the open disk and proved that if K is spherically complete, then the classes of closed, finitely generated, and principal ideals in this ring coincide.

In the first part of this talk, we will review the work of Lazard and then we will prove Amice thorem. Time permmiting, we will discuss some ideas of Scheneider-Taitelbaum to generalize Amice theorem for the ring of integers of a finite extension L of Qp.

• 18. Iwasawa decomposition. February 17, 2021. Stefano Morra. Université de Paris Saint-Denis.

In this talk we describe classical results of Iwahori and Iwahori-Matsumoto on decompositions of the K-points of split reductive groups in terms of parahoric subgroups (Iwasawa, Iwahori, Cartan decompositions). We will follow the work of Iwahori-Matsumoto (IHES, 1965), based on the theory of Bruhat-Tits, with focus on the GLn case.

References

1. Iwahori Nagayoshi, and Hideya Matsumoto. "On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups." Publications Mathématiques de l'Institut des Hautes Études Scientifiques 25.1 (1965): 5-48.
2. Tits Jacques. "Reductive groups over local fields." Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part. Vol. 1. 1979.
3. The interested reader may also check this notes.

This talk is supposed to give an algebraic approach of the theory of locally analytic representations of a p-adic Lie group. As we have remarked in the previous talks, to make manageable this theory, it is necessary to identify a finiteness condition that include fundamental examples like principal series representations and algebraic representations, but restrictive enough to avoid pathologies. In this talk, we will introduce the notion of Fréchet-Stein algebra and its associated category of coadmissible modules. These notions are the fundamental tools to present the finiteness condition of "admissibility" for locally analytic representations.

Let G be a locally analytic p-adic group. As explained in the previous talk, by adding some topological conditions, we can study locally analytic representations via the study of (topological) modules on the algebra of distributions on G (due to the works of P. Schneider and J. Teitelbaum).  In this talk, we aim to give an overview of a class of locally analytic representations which has finer property: admissible locally analytic representations and also look at some concrete examples. More precisely, we will see how the distribution algebra plays the role in the case of the additive group Z_p and how the principal series of GL_2(Q_p) are (topologically) irreducible under certain additional condition.

The main reference will be the paper of Schneilder and Teitelbaum: Locally analytic distributions and p-adic representation theory with applications to GL_2.

Bernstein, I. Gelfand and S. Gelfand category O is a category of  representations of a semisimple Lie algebra satisfying good finiteness conditions. It contains interesting modules, including finite dimensional simple ones. After giving the definition, we will describe some of its properties and its block decomposition. The treatment will be mainly based on Chapter 1 in: J. Humphreys, "Representations of Semisimple Lie Algebras in the BGG cateogry O".

• 22. On Jordan-Hölder series of some locally analytic representations. March 16, 2021. Matthias Strauch. University of Indiana.

In this talk we will give a survey of joint work done with Sascha Orlik.  We will introduce a functor F^G_P from the parabolic Bernstein-Gelfand-Gelfand category O^p (or rather a subcategory thereof), where p  =  Lie(P), to the category of locally analytic representations. Then we will

(1) show that those functors are exact,

(2) explain how  F^G_P is related to F^G_Q when the standard parabolic subgroup Q contains P,  and

(3) sketch the proof of the irreducibility of F^G_P(M) for certain modules M.

I will give an overview of the construction and key properties of Emerton’s Jacquet functor, insisting on the analogy with the case of smooth representations of p-adic groups.

• 24. On some local properties of the functor F^G_P from Lie algebras to locally analytic representations. March 30, 2021. Sascha Orlik.  University of Wuppertal.

In  my talk  I will discuss  certain properties of the functors $\cF^G_P$ introduced  by Matthias Strauch in his talk. We consider the aspects of faithfulness, projective (and injective) objects in certain categories and compute some Ext-groups.

Sascha Orlik and Matthias Strauch have constructed a functor that builds locally analytic representations out of objects in the BGG category O. In this talk, we'll describe how we can use p-adic logarithms to expand on this construction and build locally analytic representations out of objects in the larger category O^infty, ie, the extension-closure of category O inside the category of all Lie algebra representations. This is based on joint work with Matthias Strauch.

D-modules on smooth rigid analytic spaces

• 26. Rigid analytic spaces. April 13, 2021. Francesco Baldassarri. University of Padova.

Affinoid spaces and admissible coverings. Tate's acyclicity theorem. Coherent sheaves and their cohomology.  Formal models and admissible blowing-ups. Other viewpoints: Berkovich, Huber, Raynaud. Sheaves of topological structures and problems of quasi-coherence.

In this talk we will first discuss the GAGA functor which associates to any scheme locally of finite type over a nonarchimedean field K a corresponding rigid analytic space. We will then discuss Kiehl's proper mapping theorem, which states that higher direct images along a proper morphism of coherent sheaves are coherent, and sketch its proof.

* Video of the presentation.

Let X be a smooth rigid analytic variety. We review Ardakov-Wadsley’s definition of the sheaf Dcap, a 'Frechet completion' of the sheaf of differential operators on X. We show that if X is affinoid, then Dcap(X) is a Frechet—Stein algebra. A suitable theory of localization then allows us to consider coadmissible Dcap-modules (locally given as localizations of coadmissible Dcap(U)-modules). This category, which should be thought of as the rigid analytic analogue of coherent D-modules, enjoys many of the good properties satisfied by coherent O-modules on rigid analytic spaces.

* Video of the presentation.

Abstract of the first two lectures of Pr. Konstantin Ardakov:

The algebra \wideparen{D}_X on a smooth rigid analytic space X arises  naturally as a rigid analytic quantisation of the cotangent bundle T*X on X. The category of coadmissible \wideparen{D}_X-modules is an analogue of the category of coherent D-modules on a smooth complex algebraic variety X which permits a Beilinson-Bernstein-style localisation theorem for the cotangent bundle of the full rigid analytic flag variety X := (G/B)^{an} of a split semisimple algebraic group G: the coadmissible \wideparen{D}_X-modules in this case are equivalent to coadmissible modules over the Arens-Michael envelope \wideparen{U(g)_0} of an appropriate central reduction U(g)_0 of the enveloping algebra U(g) of the Lie algebra g of the algebraic group G. This Arens-Michael envelope is, however, only a subquotient of the locally analytic distribution algebra D(G(F)) of the group G(F) if F is a finite field extension of \mathbb{Q}_p: in order to obtain a localisation theorem for the full central reduction D(G(F))_0 of G(F), it is necessary to introduce equivariance conditions. These talks will discuss these equivariance conditions, the definition of the category of coadmissible equivariant D-modules \mathcal{C}_{X/G(F)} on a smooth rigid analytic variety X equipped with a continuous G(F)-action, and the Beilinson-Bernstein-style localisation theorem for admissible locally analytic G(F)-representations.

Kashiwara's Theorem and Bernstein's inequality are foundational results in the classical theory of D-modules that were discussed earlier in this series of talks, the latter leading to the central notion of holonomicity. In this talk we will review the statements of these results and how they may be adapted when studying coadmissible D-cap modules. We will also discuss the proofs in the D-cap setting. Finally we will present the tentative notion of weak holonomicity of Ardakov, Bode and the speaker; the definition mirrors one possible definition of the classical notion of holonomicity and shares some, but not all, of its pleasant properties. We will discuss the results that can, and cannot, be obtained in this setting. Fuller details of everything in this talk can be found in 'DCap II' by Ardakov and the speaker or 'DCap III' by Ardakov, Bode and the speaker.

We will study support conditions on coadmissible G-equivariant D-modules. In the case where this support is the single G-orbit of a classical point x, it is possible to show that show that the equivariant D-module M in question is completely determined by fibre M(x) at x of the local cohomology sheaf of M at x. In fact, in this situation, M \mapsto M(x) is an equivalence of categories. This is a special case of the combination of an Induction Equivalence and an equivariant Kashiwara equivalence. When combined with the locally analytic Beilinson-Bernstein localisation theorem, these results can be used to construct new examples of irreducible admissible locally analytic representations.

Let G be a connected split reductive group over a p-adic field F. In this talk, we will give another application of equivariant D-modules on rigid-analytic spaces to locally analytic G(F)-representations. In particular, we will work on the rigid-analytic flag variety X of G. Let Y be a Zariski-closed subset in X and let G(F)_Y be its stabilizer in G(F). The geometric induction functor, as introduced by Konstantin Ardakov in the previous lecture, allows to build G(F)-equivariant D_X -modules out of equivariant D_X-modules for the „smaller“ group G(F)_Y. More precisely, it relates G(F)_Y-equivariant D_X-modules with support Y to G(F)-equivariant D_X-modules with support G(F)Y. If Y is a classical point, then this functor is even an equivalence of categories, but there are other interesting examples with dim(Y)>0, where this remains true. As an explicit example, we will look at a certain class of rigid-analytic Schubert varieties Y. Then G(F)_Y will be a parabolic subgroup P(F) of G(F). We will explain how the locally analytic Beilison-Bernstein localisation theorem relates geometric induction to the functor F_{P(F)}^{G(F)} constructed by Orlik and Strauch and discussed in some earlier lectures. This allows to reprove geometrically (and extend to nonsplit groups) some of the irreducibility results for G(F)-representations, as established by Orlik and Strauch. This is work in progress with Konstantin Ardakov.

The objective of this talk is to review the results obtained by Mihn Phuong Vu in her PhD thesis. More exactly, we aim to extend the dimension theory introduced by Ardakov-Bode-Wadsley for coadmissible D-cap-modules to the category of coadmissible equivariant D-modules, introduced by K. Ardakov.
For technical reasons (which we will explain in the talk), we will restrict our attention to the rigid analytic flag variety, and we will sketch how to prove the so-called Bernstein inequality. As in the classical case, this allows to define a full subcategory of the category of coadmissible equivariant D-modules, whose objects are moreover weakly holonomic.
The final part of the this talk is devoted to use the previous lectures to give some examples arising from the BGG category O.