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We study lattices in a product
$G=G_{1}\times \cdots \times G_{n}$
of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that
$G_{i}$
is non-compact and every closed normal subgroup of
$G_{i}$
is discrete or cocompact (e.g.
$G_{i}$
is topologically simple). We show that the set of discrete subgroups of
$G$
containing a fixed cocompact lattice
$\unicode[STIX]{x1D6E4}$
with dense projections is finite. The same result holds if
$\unicode[STIX]{x1D6E4}$
is non-uniform, provided
$G$
has Kazhdan’s property (T). We show that for any compact subset
$K\subset G$
, the collection of discrete subgroups
$\unicode[STIX]{x1D6E4}\leqslant G$
with
$G=\unicode[STIX]{x1D6E4}K$
and dense projections is uniformly discrete and hence of covolume bounded away from
$0$
. When the ambient group
$G$
is compactly presented, we show in addition that the collection of those lattices falls into finitely many
$\operatorname{Aut}(G)$
-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group
$G$
is a Chabauty limit of discrete subgroups, then some compact open subgroup of
$G$
is an infinitely generated pro-
$p$
group for some prime
$p$
. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.
We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.
Let
$F$
be a non-Archimedean locally compact field of residual characteristic
$p$
with Weil group
${\mathcal{W}}_{F}$
. Let
$\unicode[STIX]{x1D70E}$
be an irreducible smooth complex representation of
${\mathcal{W}}_{F}$
, realized as the Langlands parameter of an irreducible cuspidal representation
$\unicode[STIX]{x1D70B}$
of a general linear group over
$F$
. In an earlier paper we showed that the ramification structure of
$\unicode[STIX]{x1D70E}$
is determined by the fine structure of the endo-class
$\unicode[STIX]{x1D6E9}$
of the simple character contained in
$\unicode[STIX]{x1D70B}$
, in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function
$\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$
of
$\unicode[STIX]{x1D6E9}$
. In this paper we concentrate on the fundamental Carayol case in which
$\unicode[STIX]{x1D70E}$
is totally wildly ramified with Swan exponent not divisible by
$p$
. We show that, for such
$\unicode[STIX]{x1D70E}$
, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate
$\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$
explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions
$\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$
, as
$\unicode[STIX]{x1D6EF}$
varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of
$\unicode[STIX]{x1D70E}$
to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.
We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let
$\text{F}$
be a non-Archimedean locally compact field of residue characteristic
$p$
, and let
$\text{G}$
be an inner form of the general linear group
$\text{GL}_{n}(\text{F})$
for
$n\geqslant 1$
. We consider the problem of describing explicitly the local Jacquet–Langlands correspondence
$\unicode[STIX]{x1D70B}\mapsto _{\text{JL}}\unicode[STIX]{x1D70B}$
between the complex discrete series representations of
$\text{G}$
and
$\text{GL}_{n}(\text{F})$
, in terms of type theory. We show that the congruence properties of the local Jacquet–Langlands correspondence exhibited by A. Mínguez and the first author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endo-class by the Jacquet–Langlands correspondence can be reduced to the case where the representations
$\unicode[STIX]{x1D70B}$
and
$_{\text{JL}}\unicode[STIX]{x1D70B}$
are both cuspidal with torsion number
$1$
. We also give an explicit description of the Jacquet–Langlands correspondence for all essentially tame discrete series representations of
$\text{G}$
, up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where
$\unicode[STIX]{x1D70B}$
and
$_{\text{JL}}\unicode[STIX]{x1D70B}$
are both cuspidal.
Consider the action of
$\operatorname{GL}(n,\mathbb{Q}_{p})$
on the
$p$
-adic unit sphere
${\mathcal{S}}_{n}$
arising from the linear action on
$\mathbb{Q}_{p}^{n}\setminus \{0\}$
. We show that for the action of a semigroup
$\mathfrak{S}$
of
$\operatorname{GL}(n,\mathbb{Q}_{p})$
on
${\mathcal{S}}_{n}$
, the following are equivalent: (1)
$\mathfrak{S}$
acts distally on
${\mathcal{S}}_{n}$
; (2) the closure of the image of
$\mathfrak{S}$
in
$\operatorname{PGL}(n,\mathbb{Q}_{p})$
is a compact group. On
${\mathcal{S}}_{n}$
, we consider the ‘affine’ maps
$\overline{T}_{a}$
corresponding to
$T$
in
$\operatorname{GL}(n,\mathbb{Q}_{p})$
and a nonzero
$a$
in
$\mathbb{Q}_{p}^{n}$
satisfying
$\Vert T^{-1}(a)\Vert _{p}<1$
. We show that there exists a compact open subgroup
$V$
, which depends on
$T$
, such that
$\overline{T}_{a}$
is distal for every nonzero
$a\in V$
if and only if
$T$
acts distally on
${\mathcal{S}}_{n}$
. The dynamics of ‘affine’ maps on
$p$
-adic unit spheres is quite different from that on the real unit spheres.
We develop a general procedure to study the combinatorial structure of Arthur packets for
$p$
-adic quasisplit
$\mathit{Sp}(N)$
and
$O(N)$
following the works of Mœglin. This will allow us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.
For each of the groups G = O(2), SU(2), U(2), we compute the integral and
$\mathbb{F}_2$
-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.
Soit
$\unicode[STIX]{x1D70B}$
un module de plus haut poids unitaire du groupe
$G=\mathbf{Sp}(2n,\mathbb{R})$
. On s’intéresse aux paquets d’Arthur contenant
$\unicode[STIX]{x1D70B}$
. Lorsque le plus haut poids est scalaire, on détermine les paramètres de ces paquets, on établit la propriété de multiplicité
$1$
de
$\unicode[STIX]{x1D70B}$
dans le paquet, et l’on calcule le caractère
$\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D70B}}$
(du groupe des composantes connexes du centralisateur du paramètre dans le groupe dual) associé à
$\unicode[STIX]{x1D70B}$
et qui joue un grand rôle dans la théorie d’Arthur. On fait de même pour certains modules de plus haut poids unitaires unipotents
$\unicode[STIX]{x1D70E}_{n,k}$
, ou bien lorsque le caractère infinitésimal est régulier.
We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math. 737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group
${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$
satisfying certain conditions, where
$K$
is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that
${\mathcal{A}}$
possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in
$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$
containing a Zariski dense subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$
.
Soit
$F$
un corps local non archimédien de caractéristique
${\geqslant}0$
, et soit
$G=GL(N,F)$
,
$N\geqslant 1$
. Un élément
$\unicode[STIX]{x1D6FE}\in G$
est dit quasi régulier si le centralisateur de
$\unicode[STIX]{x1D6FE}$
dans
$M(N,F)$
est un produit d’extensions de
$F$
. Soit
$G_{\text{qr}}$
l’ensemble des éléments quasi réguliers de
$G$
. Pour
$\unicode[STIX]{x1D6FE}\in G_{\text{qr}}$
, on note
$O_{\unicode[STIX]{x1D6FE}}$
l’intégrale orbitale ordinaire sur
$G$
associée à
$\unicode[STIX]{x1D6FE}$
. On remplace ici le discriminant de Weyl
$|D_{G}|$
par un facteur de normalisation
$\unicode[STIX]{x1D702}_{G}:G_{\text{qr}}\rightarrow \mathbb{R}_{{>}0}$
permettant d’obtenir les mêmes résultats que ceux prouvés par Harish-Chandra en caractéristique nulle: pour
$f\in C_{\text{c}}^{\infty }(G)$
, l’intégrale orbitale normalisée
$I^{G}(\unicode[STIX]{x1D6FE},f)=\unicode[STIX]{x1D702}_{G}^{\frac{1}{2}}(\unicode[STIX]{x1D6FE})O_{\unicode[STIX]{x1D6FE}}(f)$
est bornée sur
$G$
, et pour
$\unicode[STIX]{x1D716}>0$
tel que
$N(N-1)\unicode[STIX]{x1D716}<1$
, la fonction
$\unicode[STIX]{x1D702}_{G}^{-\frac{1}{2}-\unicode[STIX]{x1D716}}$
est localement intégrable sur
$G$
.
A classical result due to Paley and Wiener characterizes the existence of a nonzero function in
$L^{2}(\mathbb{R})$
, supported on a half-line, in terms of the decay of its Fourier transform. In this paper, we prove an analogue of this result for Damek–Ricci spaces.
This paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.
We study
$\text{Sp}_{2n}(F)$
-distinction for representations of the quasi-split unitary group
$U_{2n}(E/F)$
in
$2n$
variables with respect to a quadratic extension
$E/F$
of
$p$
-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Mœglin–Tadić classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit
$L$
-packets with no distinguished members that transfer under base change to
$\text{Sp}_{2n}(E)$
-distinguished representations of
$\text{GL}_{2n}(E)$
.
We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of
$\text{GL}(n)$
over a non-archimedean local field by means of the local Jacquet–Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.
where
$\tau :{\open R}^n\to {\open R}^n$
is a general function. In particular, for the linear choices
$\tau (x)=0$
,
$\tau (x)=x$
and
$\tau (x)={x}/{2}$
this covers the well-known Kohn–Nirenberg, anti-Kohn–Nirenberg and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions τ and here we investigate the corresponding calculus in the model case of
${\open R}^n$
. We also give examples of nonlinear τ appearing on the polarized and non-polarized Heisenberg groups.
We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.
A space X is said to be Lipschitz 1-connected if every Lipschitz loop 𝛾 : S1 → X bounds a O (Lip(𝛾))-Lipschitz disk f : D2 → X. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.
In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.
A problem in representation theory of
$p$
-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix
$(m_{u,v})$
of the Casselman basis to another natural basis in terms of certain polynomials that are deformations of the Kazhdan–Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality
$q$
to
$q^{-1}$
. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix
$(m_{u,v})$
to its inverse.