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Let M be a geometrically finite acylindrical hyperbolic $3$-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math.209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J., to appear, Preprint, 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$-manifold $M_0$, the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$. We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.
Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$. For the special case of the orbital measures, $\nu _{a_{i}}$, supported on the double cosets $Ka_{i}K$, where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).
Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda [24] conjectured that the formal degree of a square-integrable G-representation
$\pi $ can be expressed in terms of the adjoint
$\gamma $-factor of the enhanced L-parameter of
$\pi $. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations.
We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint
$\gamma $-factors.
For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.
We study harmonic analysis on the symmetric space $\text{GL}_n \times \text{GL}_n \backslash \text{GL}_{2n}$. We prove several standard results, e.g. Shalika germ expansion of orbital integrals, representability of the Fourier transform of orbital integrals and representability of spherical characters. These properties are not expected to hold for symmetric spaces in general.
La formule des traces relative de Jacquet–Rallis (pour les groupes unitaires ou les groupes linéaires généraux) est une identité entre des périodes des représentations automorphes et des distributions géométriques. Selon Jacquet et Rallis, une comparaison de ces deux formules des traces relatives devrait aboutir à une démonstration des conjectures de Gan–Gross–Prasad et Ichino–Ikeda pour les groupes unitaires. Les termes géométriques des groupes unitaires ou des groupes linéaires sont indexés par les points rationnels d'un espace quotient commun. Nous établissons que ces termes géométriques peuvent être vus comme des fonctionnelles sur des espaces d'intégrales orbitales semi-simples régulières locales. En outre, nous montrons que point par point ces distributions sont en fait égales, via l'identification des espaces d'intégrales orbitales locales donnée par le transfert et le lemme fondamental (essentiellement connus dans cette situation). Cela donne leur comparaison et cela clôt la partie géométrique du programme de Jacquet–Rallis. Notre résultat principal est donc un analogue de la stabilisation de la partie géométrique de la formule des traces due à Langlands, Kottwitz et Arthur.
Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.
Let
${\mathbf {G}}$
be a semisimple algebraic group over a number field K,
$\mathcal {S}$
a finite set of places of K,
$K_{\mathcal {S}}$
the direct product of the completions
$K_{v}, v \in \mathcal {S}$
, and
${\mathcal O}$
the ring of
$\mathcal {S}$
-integers of K. Let
$G = {\mathbf {G}}(K_{\mathcal {S}})$
,
$\Gamma = {\mathbf {G}}({\mathcal O})$
and
$\pi :G \rightarrow G/\Gamma $
the quotient map. We describe the closures of the locally divergent orbits
${T\pi (g)}$
where T is a maximal
$K_{\mathcal {S}}$
-split torus in G. If
$\# S = 2$
then the closure
$\overline {T\pi (g)}$
is a finite union of T-orbits stratified in terms of parabolic subgroups of
${\mathbf {G}} \times {\mathbf {G}}$
and, consequently,
$\overline {T\pi (g)}$
is homogeneous (i.e.
$\overline {T\pi (g)}= H\pi (g)$
for a subgroup H of G) if and only if
${T\pi (g)}$
is closed. On the other hand, if
$\# \mathcal {S}> 2$
and K is not a
$\mathrm {CM}$
-field then
$\overline {T\pi (g)}$
is homogeneous for
${\mathbf {G}} = \mathbf {SL}_{n}$
and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for
${\mathbf {G}} \neq \mathbf {SL}_{n}$
. As an application, we prove that
$\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$
for the class of non-rational locally K-decomposable homogeneous forms
$f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$
.
Let G be a semisimple real algebraic group defined over
${\mathbb {Q}}$
,
$\Gamma $
be an arithmetic subgroup of G, and T be a maximal
${\mathbb {R}}$
-split torus. A trajectory in
$G/\Gamma $
is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in T, we study the existence of non-obvious divergent trajectories under its action in
$G\kern-1pt{/}\kern-1pt\Gamma $
. We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that
$\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$
.
Let
$M\stackrel {\rho _0}{\curvearrowleft }S$
be a
$C^\infty $
locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking,
$\rho _0$
is parameter rigid if any
$C^\infty $
locally free action of S on M having the same orbits as
$\rho _0$
is
$C^\infty $
conjugate to
$\rho _0$
. In this paper we prove two types of result on parameter rigidity.
First let G be a connected semisimple Lie group with finite center of real rank at least
$2$
without compact factors nor simple factors locally isomorphic to
$\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$
or
$\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$
, and let
$\Gamma $
be an irreducible cocompact lattice in G. Let
$G=KAN$
be an Iwasawa decomposition. We prove that the action
$\Gamma \backslash G\curvearrowleft AN$
by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type.
Secondly we show that if
$M\stackrel {\rho _0}{\curvearrowleft }S$
is parameter rigid, then the zeroth and first cohomology of the orbit foliation of
$\rho _0$
with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.
We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.
This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.
We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$. We give explicit calculations when $\Gamma$ is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration $B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq 2$, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.
The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group
$\mathrm {SL}(d,{\mathbb {R}})$
and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.
In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let
$\pi $
be a minimal or next-to-minimal automorphic representation of G. We prove that any
$\eta \in \pi $
is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on
$\operatorname {GL}_n$
. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type
$D_5$
and
$E_8$
with a view toward applications to scattering amplitudes in string theory.
We show local rigidity of hyperbolic triangle groups generated by reflections in pairs of n-dimensional subspaces of
$\mathbb {R}^{2n}$
obtained by composition of the geometric representation in
$\mathsf {PGL}(2,\mathbb {R})$
with the diagonal embeddings into
$\mathsf {PGL}(2n,\mathbb {R})$
and
$\mathsf {PSp}^\pm (2n,\mathbb {R})$
.
We generalize Uhlenbeck’s generator theorem of
${\mathcal{L}}^{-}\operatorname{U}_{n}$
to the full rational loop group
${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{C}$
and its subgroups
${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{R}$
,
${\mathcal{L}}^{-}\operatorname{U}_{p,q}$
: they are all generated by just simple projective loops. Recall that Terng–Uhlenbeck studied the dressing actions of such projective loops as generalized Bäcklund transformations for integrable systems. Our result makes a nice supplement: any rational dressing is the composition of these Bäcklund transformations. This conclusion is surprising in the sense that Lie theory suggests the indispensable role of nilpotent loops in the case of noncompact reality conditions, and nilpotent dressings appear quite complicated and mysterious. The sacrifice is to introduce some extra fake singularities. So we also propose a set of generators if fake singularities are forbidden. A very geometric and physical construction of
$\operatorname{U}_{p,q}$
is obtained as a by-product, generalizing the classical construction of unitary groups.
Let F be a non-Archimedean local field of characteristic zero. Let G = GL(2, F) and
$3\widetildeG = \widetilde{GL}(2,F)$
be the metaplectic group. Let τ be the standard involution on G. A well-known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we make some modifications and adapt a topological argument of Tupan to the metaplectic group
$\widetildeG$
and give an elementary proof that any lift of the standard involution to
$\widetildeG$
; is also a dualizing involution.
For a locally compact group G, we study the distality of the action of automorphisms T of G on SubG, the compact space of closed subgroups of G endowed with the Chabauty topology. For a certain class of discrete groups G, we show that T acts distally on SubG if and only if Tn is the identity map for some
$n\in\mathbb N$
. As an application, we get that for a T-invariant lattice Γ in a simply connected nilpotent Lie group G, T acts distally on SubG if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on SubG if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on SubG. We also show that any compactly generated distal group G is Lie projective.
We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of
$\text{PGSp}_{6}$
discovered by Pollack, we prove that a cuspidal representation of
$\text{PGSp}_{6}$
is a (weak) functorial lift from the exceptional group
$G_{2}$
if its (partial) Spin L-function has a pole at
$s=1$
.