11,064 research outputs found
Fuchsian polyhedra in Lorentzian space-forms
Let S be a compact surface of genus >1, and g be a metric on S of constant
curvature K\in\{-1,0,1\} with conical singularities of negative singular
curvature. When K=1 we add the condition that the lengths of the contractible
geodesics are >2\pi. We prove that there exists a convex polyhedral surface P
in the Lorentzian space-form of curvature K and a group G of isometries of this
space such that the induced metric on the quotient P/G is isometric to (S,g).
Moreover, the pair (P,G) is unique (up to global isometries) among a particular
class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of
A.D. Alexandrov and Rivin--Hodgson concerning the sphere to the higher genus
cases, and it is also the polyhedral version of a theorem of
Labourie--Schlenker
Faces of highest weight modules and the universal Weyl polyhedron
Let be a highest weight module over a Kac-Moody algebra ,
and let conv denote the convex hull of its weights. We determine the
combinatorial isomorphism type of conv , i.e. we completely classify the
faces and their inclusions. In the special case where is
semisimple, this brings closure to a question studied by Cellini-Marietti [IMRN
2015] for the adjoint representation, and by Khare [J. Algebra 2016; Trans.
Amer. Math. Soc. 2017] for most modules. The determination of faces of
finite-dimensional modules up to the Weyl group action and some of their
inclusions also appears in previous work of Satake [Ann. of Math. 1960],
Borel-Tits [IHES Publ. Math. 1965], Vinberg [Izv. Akad. Nauk 1990], and
Casselman [Austral. Math. Soc. 1997].
For any subset of the simple roots, we introduce a remarkable convex cone
which we call the universal Weyl polyhedron, which controls the convex hulls of
all modules parabolically induced from the corresponding Levi factor. Namely,
the combinatorial isomorphism type of the cone stores the classification of
faces for all such highest weight modules, as well as how faces degenerate as
the highest weight gets increasingly singular. To our knowledge, this cone is
new in finite and infinite type.
We further answer a question of Michel Brion, by showing that the
localization of conv along a face is always the convex hull of the weights
of a parabolically induced module. Finally, as we determine the inclusion
relations between faces representation-theoretically from the set of weights,
without recourse to convexity, we answer a similar question for highest weight
modules over symmetrizable quantum groups.Comment: Final version, to appear in Advances in Mathematics (42 pages, with
similar margins; essentially no change in content from v2). We recall
preliminaries and results from the companion paper arXiv:1606.0964
Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant
under the action of a Fuchsian group of isometries (i.e. a group of isometries
leaving globally invariant a totally geodesic surface, on which it acts
cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric
to a hyperbolic metric with conical singularities of positive singular
curvature on a compact surface of genus greater than one. We prove that these
metrics are actually realised by exactly one convex Fuchsian polyhedron (up to
global isometries). This extends a famous theorem of A.D. Alexandrov.Comment: Some little corrections from the preceding version. To appear in Les
Annales de l'Institut Fourie
- …