Annales scientifiques de l'École normale supérieure

Convex cocompact actions in real projective geometry

Fanny Kassel — 2024-02-05

We study a notion of convex cocompactness for (not necessarily irreducible) discrete subgroups of the projective general linear group acting on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good properties of classical convex cocompact subgroups in rank-one Lie groups. Extending our earlier work [DGK3] from the context of projective orthogonal groups, we show that for word hyperbolic groups preserving a properly convex open set in projective space, the above general notion of convex cocompactness is equivalent to a stronger convex cocompactness condition studied by Crampon--Marquis, and also to the condition that the natural inclusion be a projective Anosov representation. We investigate examples.

Weight conjectures for l-compact groups and spetses

Günter Malle — 2024-02-05

Fundamental conjectures in modular representation theory of finite groups,
more precisely, Alperin's Weight Conjecture and Robinson's Ordinary Weight
Conjecture, can be expressed in terms of fusion systems. We use fusion systems
to connect the modular representation theory of finite groups of Lie type to the
theory of $\ell$-compact groups. Under some mild conditions we prove that the
fusion systems associated to homotopy fixed points of $\ell$-compact groups
satisfy an equation which for finite groups of Lie type is equivalent to
Alperin's Weight Conjecture.
\par
For finite reductive groups, Robinson's Ordinary Weight Conjecture is closely
related to Lusztig's Jordan decomposition of characters and the corresponding
results for Brauer $\ell$-blocks. Motivated by this, we define the principal
block of a spets attached to a spetsial $\ZZ_\ell$-reflection group, using the
fusion system related to it via $\ell$-compact groups, and formulate an analogue
of Robinson's conjecture for this block. We prove this formulation for an
infinite family of cases as well as for some groups of exceptional type.

Our results not only provide further strong evidence for the validity of
the weight conjectures, but also point toward some yet unknown structural
explanation for them purely in the framework of fusion systems.

A smooth complex rational affine surface with uncountably many real forms

Anna Bot — 2024-02-05

We exhibit a smooth complex rational affine surface with uncountably many nonisomorphic real forms.

Counting cusp forms by analytic conductor

Farrell Brumley — 2024-02-05

Let F be a number field and n ≥ 1 an integer. The universal family is the set F of all unitary cuspidal automorphic representations on GL_n over F, ordered by their analytic conductor. We prove an asymptotic for the size of the truncated universal family F(Q) as Q → ∞, under a spherical assumption at the archimedean places when n ≥ 3. We interpret the leading term constant geometrically and conjecturally determine the underlying Sato–Tate measure. Our methods naturally provide uniform Weyl laws with logarithmic savings in the level and strong quantitative bounds on the non-tempered discrete spectrum for GL_n.

Local index theory for Lorentzian manifolds

Christian Bär — 2024-02-05

Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a non-trivial index is caused by topologically non-trivial dynamics rather than non-trivial topology of the base manifold.
In this paper we prove a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes. This local formula implies an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions on Cauchy hypersurfaces. This is significantly more general than the previously known theorems that require the compatibility of the connection with Clifford multiplication and the spatial Dirac operator on the Cauchy hypersurface to be self-adjoint with respect to a positive definite inner product. The new Lorentzian index theorem now covers important geometrically natural operators such as the odd signature operator.

Sharp isoperimetric comparison on noncollapsed spaces with lower Ricci bounds

Gioacchino Antonelli — 2024-02-05

This paper studies sharp isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called N-dimensional RCD(K,N) spaces (X,d,H^N).
The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences.
Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies. Many other consequences of these results are explored in a companion paper by the authors.

On the space of ergodic measures for the horocycle flow on strata of Abelian differentials

Osama Khalil — 2024-02-05

We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2),$ showing that there is a sequence of horocycle ergodic measures, supported on a sequence of periodic horocycle orbits, which weak-$*$ converges to an $\mathrm{SL}_2(\mathbb{R})$-invariant but not ergodic measure. As a consequence we show that there are points in $\mathcal{H}(2)$ that do not equidistribute under the horocycle flow for any measure.

Classical solutions of the Boltzmann equation with irregular initial data

Christopher Henderson — 2024-02-05

This article considers the spatially inhomogeneous, non-cutoff Boltzmann equation. We construct a large-data classical solution given bounded, measurable initial data with uniform polynomial decay of mild order in the velocity variable. Our result requires no assumption of strict positivity for the initial data, except locally in some small ball in phase space. We also obtain existence results for weak solutions when our decay and positivity assumptions for the initial data are relaxed.

Because the regularity of our solutions may degenerate as $t\to 0$, uniqueness is a challenging issue. We establish weak-strong uniqueness under the additional assumption that the initial data possesses no vacuum regions and is Hölder continuous.

As an application of our short-time existence theorem, we prove global existence near equilibrium for bounded, measurable initial data that decays at a finite polynomial rate in velocity.