134 research outputs found
Spherical roots of spherical varieties
Brion proved that the valuation cone of a complex spherical variety is a
fundamental domain for a finite reflection group, called the little Weyl group.
The principal goal of this paper is to generalize this fundamental theorem to
fields of characteristic unequal to 2. We also prove a weaker version which
holds in characteristic 2, as well. Our main tool is a generalization of
Akhiezer's classification of spherical rank-1-varieties.Comment: v1; 19 pages; v2: 19 pages, reformatted to LaTeX, slightly expanded,
a couple of minor errors corrected; v3: 19 pages, minor modifications, final
versio
Convexity of Hamiltonian manifolds
Let K be a connected Lie group and M a Hamiltonian K-manifold. In this paper,
we introduce the notion of convexity of M. It implies that the momentum image
is convex, the moment map has connected fibers, and the total moment map is
open onto its image. Conversely, the three properties above imply convexity. We
show that most Hamiltonian manifolds occuring "in nature" are convex (e.g., if
M is compact, complex algebraic, or a cotangent bundle). Moreover, every
Hamiltonian manifold is locally convex. This is an expanded version of section
2 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian manifolds.Comment: 12 pages, to appear in J. Lie Theor
Graded cofinite rings of differential operators
We classify subalgebras of a ring of differential operators which are big in
the sense that the extension of associated graded rings is finite. We show that
these subalgebras correspond, up to automorphisms, to uniformly ramified finite
morphisms. This generalizes a theorem of Levasseur-Stafford on the generators
of the invariants of a Weyl algebra under a finite group.Comment: v1: 9 pages; v2: 22 pages, completely rewritten, main theorem fixed,
applications added; v3: 23 pages, references added, minor correction
Functoriality properties of the dual group
Let be a connected reductive group. In a previous paper,
arxiv:1702.08264, is was shown that the dual group attached to a
-variety admits a natural homomorphism with finite kernel to the
Langlands dual group of . Here, we prove that the dual group is
functorial in the following sense: if there is a dominant -morphism
or an injective -morphism then there is a canonical homomorphism
which is compatible with the homomorphisms to .Comment: v1:14 pages; v2: 16 pages, changed Rem. 2.3, Rem. 2.9, proof of Thm.
3.2; v3: 2 typos correcte
A connectedness property of algebraic moment maps
Let a connected reductive group G act on the smooth connected variety X. The
cotangent bundle of X is a Hamiltonian G-variety. We show that its "total
moment map" has connected fibers.
This is an expanded version of section 6 of my paper dg-ga/9712010 on Weyl
groups of Hamiltonian manifolds.Comment: 15 page
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