We study the wave-front set of an element in a $p$-adic reductive Lie algebra (for $p\gg\operatorname{rank}$), namely the set of maximal nilpotent orbits appearing in its Shalika germ expansion. By adapting an algorithm of Waldspurger that computes orbital integrals, we obtain an inductive algorithm to compute an invariant that determines the wave-front set. This gives an algorithm to compute the wave-front sets for regular supercuspidal representations and reveals examples whose wave-front sets are not contained in a single geometric orbit, for arbitrarily large $p$ within a fixed rank.
Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible p-adic Banach space (respectively locally analytic) representations of p-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible representation.
This paper gives a $p$-adic analogue of the Mackey theory, which relates representations of a group of type $G=H\times_{t} A $ to systems of imprimitivity.
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.
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.
Let K be a locally compact nonarchimedean field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We study the category C_g of coadmissible modules over the nonarchimedean Arens-Michael envelope of U(g). Let p be a parabolic subalgebra of g. The main result identifies a certain explicitly given highest weight category inside C_g with the classical parabolic BGG category of g relative to p. This paper is in final form, replaces and expands the former preprint 'BGG reciprocity for p-adic Arens-Michael envelopes of semisimple Lie algebras' and appears in Journal o…
We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's theory of arithmetic $D$-modules should give a $p$-adic analogue of Kashiwara's theory of $D$-modules for real Lie groups i.e. it should give a realization of the $p$-adic representations of a $p$-adic Lie group as spaces of overconvergent solutions of arithmetic $D$-modules which will come equipped with an action of the Galois group. We shall discuss the ca…
We discuss Euler characteristics for finitely generated modules over Iwasawa algebras. We show that the Euler characteristic of a module is well-defined whenever the 0th homology group is finite if and only if the relevant compact p-adic Lie group is finite-by-nilpotent and that in this case all pseudo-null modules have trivial Euler characteristic. We also prove some other results relating to the triviality of Euler characteristics for pseudo-null modules. We also prove some analogous results for the Akashi series of Coates et al.
We give a generalization of Kostant's theorem on Lie algebra cohomology of finite dimensional highest weight representations to some infinite dimensional cases over a $p$-adic family of highest weight distribution representations. For proving this, we develop a theory of eigen orthonormalizable Banach representations of $p$-adic torus over an affinoid algebra, and we construct an eigen orthonormalizable weight completion of the distribution representations.
The Harish-Chandra--Howe local character expansion expresses the characters of reductive, $p$-adic groups in terms of Fourier transforms of nilpotent orbital integrals on their Lie algebras, and Murnaghan--Kirillov theory expresses many characters of reductive, $p$-adic groups in terms of Fourier transforms of semisimple orbital integrals (also on their Lie algebras). In many cases, the evaluation of these Fourier transforms seems intractable; but, for $\operatorname{SL}_2$, the nilpotent orbital integrals have already been computed. In this paper, we use a variant of Huntsinger's integral for…
We prove that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a non-zero coadjoint orbit. To do this we establish analogues for $p$-adically completed enveloping algebras of Bernstein's inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen's Lemma about the endomorphism ring of a simple module over an enveloping algebra.
Given a compact p-adic Lie group we show that its distribution algebra is Frechet-Stein with Auslander regular Banach algebras. As an application, we show that nonzero coadmissible modules coming from smooth or, more general, U(g)-finite representations have maximal codimension.
Let G be a group SO(2n+1) defined over a p-adic field. We compute the wave front set of the anti-tempered irreducible representations of G(F) which are of unipotent reduction. It is the orthogonal orbit dual to the symplectic orbit appearing in the Arthur's parametrization of the representation.
The concept of wave front set was introduced in 1969-1970 by M. Sato in the hyperfunctions context and by L. Hörmander in the $\mathcal C^{\infty}$ context. Howe used the theory of wave front sets in the study of Lie groups representations. Heifetz 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 $k((t))$-setting with $k$ a characteristic zero field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness …
Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.
Let G be a special orthogonal group SO(2n+1) defined over a p-adic field F. Let $π$ be an admissible irreducible representation of G(F) which is tempered and of unipotent reduction. We prove that $π$ has a wave front set. In some particular cases, for instance if $π$ is of the discrete series, we give a method to compute this wave front set.
Let $G$ be a profinite group. A strongly admissible smooth representation $ρ$ of $G$ over $\mathbb{C}$ decomposes as a direct sum $ρ\cong \bigoplus_{π\in \mathrm{Irr}(G)} m_π(ρ) \, π$ of irreducible representations with finite multiplicities $m_π(ρ)$ such that for every positive integer $n$ the number $r_n(ρ)$ of irreducible constituents of dimension $n$ is finite. Examples arise naturally in the representation theory of reductive groups over non-archimedean local fields. In this article we initiate an investigation of the Dirichlet generating function \[ ζ_ρ(s) = \sum_{n=1}^\infty r_n(ρ) n^{-…
We develop an approach to the character theory of certain classes of finite and profinite groups based on the construction of a Lie algebra associated to such a group, but without making use of the notion of a polarization which is central to the classical orbit method. Instead, Kirillov's character formula becomes the fundamental object of study. Our results are then used to produce an alternate proof of the orbit method classification of complex irreducible representations of p-groups of nilpotence class less than p, where p is a prime, and of continuous complex irreducible representations o…
This is an extended version of the first part of a forthcoming paper where we will study the local Zeta functions of the minimal spherical series for the symmetric spaces arising as open orbits of the parabolic prehomogeneous spaces of commutative type over a p-adic field. The case where the ground field is $\mathbb{R}$ has already been considered by Nicole Bopp and the second author ([7]). If $F$ is a p-adic field of characteristic $0$, we consider a reductive Lie algebra $\widetilde{\mathfrak{g}}$ over $F$ which is endowed with a short $\mathbb{Z}$-grading: $\widetilde{\mathfrak{g}} = \mathf…
Let $H$ be a compact $p$-adic analytic group without torsion element, whose Lie algebra is split semisimple and $\mathfrak{N}_H(G)$ be the full subcategory of the category of finitely generated modules over the Iwasawa algebra $Λ_G$ that are also finitely generated as $Λ_H$-modules, where $G = \mathbb{Z}_{p} \times H$. We show that if the class of a module $N$ in the Grothendieck group of $\mathfrak{N}_H(G)$ equals to the class of a completely faithful module, then $q(N)$ is also completely faithful, where $q(N)$ denotes the image of $N$ via the quotient functor modulo the full subcategory of …
Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.
In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups.
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…
Vogan and Barbasch-Vogan attach two similar invariants to representations of a reductive Lie group, one by an algebraic process, the other analytic. They conjectured that the two invariants determine each other in a definite manner. Here we prove the conjecture. Our arguments involve two finer invariants -- the characteristic cycles of representations -- which are interesting in their own right.
If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $τ$ is a unitary representation of $H$, then the authors give a sufficient condition on $ξ\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^Gτ$. In the special case where $τ$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $π$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\operatorname{WF}(π)$ in terms of the direct integral decomposition of $π$ int…
The Langlands-Shelstad transfer factor is a function defined on some reductive groups over a p-adic field. Near the origin of the group, it may be viewed as a function on the Lie algebra. For classical groups, its values have the form q^c s, where s is -1, 0, or 1, q is the cardinality of the residue field, and c is a rational number. The function s partitions the Lie algebra into three subsets. This article shows that this partition into three subsets is independent of the p-adic field in the following sense. We define three universal objects (virtual sets in the sense of Quine) such that for…
Let L be a p-adic local field and g a finite dimensional Lie algebra over L. We show that its hyperenveloping algebra F(g) is a stably flat completion of its universal enveloping algebra. As a consequence the relative cohomology for the locally convex algebra F(g) coincides with the underlying Lie algebra cohomology. Final version. Some minor items corrected. Appeared in Journal of Algebra (2010).
