Adic Lie Groups Research Articles

Translation functors for locally analytic representations

Let [Formula: see text] be a [Formula: see text]-adic Lie group with reductive Lie algebra [Formula: see text]. In analogy to the translation functors introduced by Bernstein and Gelfand on the categories of [Formula: see text]-modules, we consider similarly defined functors on the category of modules over the locally analytic distribution algebra [Formula: see text] on which the center of [Formula: see text] acts locally finitely. These functors induce equivalences between certain subcategories of the latter category. Furthermore, these translation functors are naturally related to those on category [Formula: see text] via the functors from category [Formula: see text] to the category of coadmissible modules. We also investigate the effect of the translation functors on locally analytic representations [Formula: see text]la associated by the [Formula: see text]-adic Langlands correspondence for [Formula: see text] to two-dimensional Galois representations [Formula: see text].

Compactly Supported Distributions on $$p$$-Adic Lie Groups

Compactly Supported Distributions on $$p$$-Adic Lie Groups

Semi-invariant distribution vectors for p-adic unipotent groups

Let G be a unipotent algebraic group defined over a [Formula: see text]-adic field of characteristic zero. We denote by [Formula: see text] the set of rational points of G. It is a [Formula: see text]-adic Lie group with Lie algebra denoted by [Formula: see text]. Let [Formula: see text] be an irreducible unitary representation of [Formula: see text] in a Hilbert space [Formula: see text], [Formula: see text] be a linear form on [Formula: see text] and [Formula: see text] be a polarization at [Formula: see text]. We denote by [Formula: see text] a character of [Formula: see text] related to [Formula: see text]. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors [Formula: see text] of [Formula: see text].

Irrationality of Generic Quotient Varieties via Bogomolov Multipliers

Abstract The Bogomolov multiplier of a group is the unramified Brauer group associated with the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated with nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups of exponent $p$. We show that with fixed $n$ and increasing $p$, a positive proportion of these groups of order $p^n$ have trivial multipliers. On the other hand, we show that by fixing $p$ and increasing $n$, log-generic groups of order $p^n$ have non-trivial multipliers. Whence quotient varieties of faithful representations of log-generic $p$-groups are not stably rational, applications in non-commutative Iwasawa theory are developed.

Characters of algebraic groups over number fields

Let k be a number field, \mathbf{G} an algebraic group defined over k , and \mathbf{G}(k) the group of k -rational points in \mathbf{G} . We determine the set of functions on \mathbf{G}(k) which are of positive type and conjugation invariant, under the assumption that \mathbf{G}(k) is generated by its unipotent elements. An essential step in the proof is the classification of the \mathbf{G}(k) -invariant ergodic probability measures on an adelic solenoid naturally associated to \mathbf{G}(k) . This last result is deduced from Ratner’s measure rigidity theorem for homogeneous spaces of S -adic Lie groups; this appears to be the first application of Ratner’s theorems in the context of operator algebras.

Actions of Nilpotent Groups on Complex Algebraic Varieties

AbstractWe study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when $\textbf {k}$ is a number field, a finite $p$-subgroup of the group of polynomial automorphisms of $\textbf {k}^d$ is isomorphic to a subgroup of $\textrm {GL}_d(\textbf {k})$. In the case of infinite nilpotent group actions, we show that a finitely generated nilpotent group $H$ acting on a complex quasiprojective variety $X$ of dimension $d$ can be embedded in a $p$-adic Lie group that acts faithfully and analytically on $\textbf {Z}_p^d$. As a consequence, we show that the virtual derived length of $H$ is at most the dimension of $X$.

Twisting lemma for $\lambda$-adic modules

A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb{Z}_p[[\Gamma ]]$ with $\Gamma \cong \mathbb{Z}_p, \exists$ a continuous character $\theta: \Gamma \rightarrow \mathbb{Z}_p^\times$ such that, the $ \Gamma^{n}$-Euler characteristic of the twist $M(\theta)$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra of a general compact $p$-adic Lie group $G$. In this article, we consider a further generalization of the twisting lemma to $\mathcal{T}[[G]]$ modules, where $G$ is a compact $p$-adic Lie group and $\mathcal{T}$ is a finite extension of $\mathbb{Z}_p[[X]]$. Such modules naturally occur in Hida theory. We also indicate arithmetic application by considering the twisted Euler Characteristic of the big Selmer (respectively fine Selmer) group of a $\Lambda$-adic form over a $p$-adic Lie extension.

Equivariant $\mathcal D$-modules on rigid analytic spaces

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.

STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups$\text{Sym}(n)$(in the sofic case) or the finite-dimensional unitary groups$\text{U}(n)$(in the hyperlinear case)? In the case of$\text{U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which arenotapproximated by$\text{U}(n)$with respect to the Frobenius norm$\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$. Our strategy is to show that some higher dimensional cohomology vanishing phenomena impliesstability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple$p$-adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

Motivic Wave Front Sets

Abstract The concept of wave front set was introduced in 1969–1970 by Sato in the hyperfunctions context [1, 34] and by Hörmander [23] in the $\mathcal C^{\infty }$ context. Howe in [25] used the theory of wave front sets in the study of Lie groups representations. Heifetz in [22] defined a notion of wave front set for distributions in the $p$-adic setting and used it to study some representations of $p$-adic Lie groups. In this article, we work in the $k\mathopen{(\!(} t \mathopen{)\!)}$-setting with $k$ a Characteristic 0 field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness and compactness properties. We develop a notion of definable distributions in the framework of [13] and [14] for which we define notions of singular support and $\Lambda$-wave front sets (relative to some multiplicative subgroups $\Lambda$ of the valued field) and we investigate their behavior under natural operations like pullback, tensor product, and products of distributions.

Equidimensional adic eigenvarieties for groups with discrete series

We extend Urban's construction of eigenvarieties for reductive groups $G$ such that $G(\mathbb{R})$ has discrete series to include characteristic $p$ points at the boundary of weight space. In order to perform this construction, we define a notion of "locally analytic" functions and distributions on a locally $\mathbb{Q}_p$-analytic manifold taking values in a complete Tate $\mathbb{Z}_p$-algebra in which $p$ is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on $p$-adic Lie groups given by Johansson and Newton.

A Note on Asymptotic Class Number Upper Bounds in p-adic Lie Extensions

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic Lie group with no torsion and that it contains a closed normal subgroup $H$ such that $G/H\cong \mathbb{Z}_p$. Under various assumptions, we establish asymptotic upper bounds for the growth of $p$-exponents of the class groups in the said $p$-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that $H\cong \mathbb{Z}_p$.

LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT -MODULES

AbstractLet$p$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$\unicode[STIX]{x1D70B}$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$\unicode[STIX]{x1D70B}$for all$n$, and let$L$be the Galois closure of$K_{\infty }$. Using these field extensions, Caruso constructs the$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-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 Lie group$\operatorname{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules, we can establish the overconvergence property of the$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules.

Théorie de Sen et vecteurs localement analytiques

We generalize Sen theory to extensions $K_\infty/K$ whose Galois group is a $p$-adic Lie group of arbitrary dimension. To do so, we replace Sen's space of $K$-finite vectors by Schneider and Teitelbaum's space of locally analytic vectors. One then gets a vector space over the field of locally analytic vectors of $\hat{K}_\infty$. We describe this field in general and pay a special attention to the case of Lubin-Tate extensions.

A conductor formula for completed group algebras

Let \mathfrak o be the ring of integers in a finite extension of \mathbb Q_p . If G is a finite group and \Gamma is a maximal \mathfrak o -order containing the group ring \mathfrak o[G] , Jacobinski's conductor formula gives a complete description of the central conductor of \Gamma into \mathfrak o [G] in terms of characters of G . We prove a similar result for completed group algebras \mathfrak o [[G]] , where G is a p -adic Lie group of dimension 1. We will also discuss several consequences of this result.

The representation zeta function of a FAb compact p -adic Lie group vanishes at −2

Let G by compact p-adic Lie group and suppose that G is FAb, i.e., that H/[H,H] is finite for every open subgroup H of G. The representation zeta function Z(G,s) encodes the distribution of continuous irreducible complex characters of G. Here s denotes a complex variable and Z(G,s) is defined as the Dirichlet generating function whose nth coefficient is equal to the number of irreducible characters of G of degree n. For p greater than 2 it is known that Z(G,s) defines a meromorphic function on the complex plane. Wedderburn's structure theorem for semisimple algebras implies that ZG,-2) = |G| for finite G. We complement this classic result by proving that Z(G,-2) = 0 for infinite G, assuming that p is greater than 2.

A splitting for $K_1$ of completed group rings

For p\neq 2 and a uniform pro- p group G and its Iwasawa algebras \Lambda (G) := \mathbb{Z}_{p}[\hskip-.7pt[G]\hskip-.7pt] and \Omega[\hskip-.7pt[G]\hskip-.7pt] := \mathbb{F}_p[\hskip-.7pt[G]\hskip-.7pt] we show that the natural map K_1(\Lambda(G)) \to K_1(\Omega(G)) has a splitting provided that SK_1(\Lambda(G)) vanishes. The image of this splitting is described in terms of a generalised norm operator. This result generalises classical work of Coleman for the case G=\mathbb{Z}_p . We verify the vanishing condition for certain unipotent compact p -adic Lie groups.

$$SK_1$$ SK 1 and Lie algebras

We investigate the vanishing of the group $$SK_1(\Lambda (G))$$ for the Iwasawa algebra $$\Lambda (G)$$ of a pro- $$p$$ $$p$$ -adic Lie group $$G$$ (with $$p \ne 2$$ ). We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras.

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact $p$-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a "generalized Robba ring" for uniform pro-$p$ groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a self-dual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.

The Howe-Moore property for real and $p$-adic groups

We consider in this paper a relative version of the Howe-Moore property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or $p$-adic Lie groups, the pairs with the relative Howe-Moore property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.