7,819 research outputs found

    Combinatorics and invariant differential operators on multiplicity free spaces

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    We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.Comment: 36 pages, some typos correcte

    Invariant functions on symplectic representations

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    Let G be a connected reductive group. In this paper we are studying the invariant theory of symplectic G-modules. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration over an affine space with rational generic fibers. Of particular interest are those modules for which the generic orbit is coisotropic. We prove that they are cofree. This result has been used in another paper (math.SG/0505268) to classify all these modules. Our main tool is a symplectic version of the local structure theorem.Comment: v1: 24 pages; v2: 31 pages, expanded exposition, new introduction, some facts (esp. Thm. 7.2+Corollaries, Thm. 8.4) which were only implicit in v1 are now spelled ou

    Spherical roots of spherical varieties

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    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

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    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

    Functoriality properties of the dual group

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    Let GG be a connected reductive group. In a previous paper, arxiv:1702.08264, is was shown that the dual group GX∨G^\vee_X attached to a GG-variety XX admits a natural homomorphism with finite kernel to the Langlands dual group G∨G^\vee of GG. Here, we prove that the dual group is functorial in the following sense: if there is a dominant GG-morphism X→YX\to Y or an injective GG-morphism Y→XY\to X then there is a canonical homomorphism GY∨→GX∨G^\vee_Y\to G^\vee_X which is compatible with the homomorphisms to G∨G^\vee.Comment: v1:14 pages; v2: 16 pages, changed Rem. 2.3, Rem. 2.9, proof of Thm. 3.2; v3: 2 typos correcte

    Graded cofinite rings of differential operators

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    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
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