Paving Springer fibers for E7
The purpose of this paper is to show that, for each unipotent element in type E7, the corresponding Springer fiber can be paved by affine spaces.
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arXiv subjects
Corrado De Concini
Publications and source records attributed to Corrado De Concini.
A differential algebra and the homotopy type of the complement of a toric arrangement
We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.
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Cohomology rings of compactifications of toric arrangements
Some projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T were constructed in [3]. In this paper we describe their integer cohomology rings by generators and relations.
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On some modules of covariants for a reflection group
Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak g)^\mathfrak g$ of $(\bigwedge \mathfrak g)^\mathfrak g\cong S(\mathfrak h)^W$-modules, where $\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation. New version with different title. Various improvements. New section 7.
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Projective Wonderful Models for Toric Arrangements
In this paper we illustrate an algorithmic procedure which allows to build projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T. The main step of the construction is a combinatorial algorithm that produces a toric variety by subdividing in a suitable way a given smooth fan.
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Topics in Hyperplane arrangements - Errata
We have received an e-mail from Bryan Gillespie pointing out that a proposition, that is Proposition 8.5, of our book, [1] is incorrect as stated. The given formula (8.5) is valid only in the generic case that is assuming that for any point of the arrangement $p, X_p$ is formed by a basis. The correct proposition is slightly weaker, in general one must replace Formula 8.5 of the book with the next Formula (3). Accordingly one has to change Proposition 9.2 in the obvious way. The remaining parts of the book are not affected but one should remove the first line of 11.3.3 which quotes the incorrect formula. Here we discuss the correct proposition, replacing Proposition 8.5.
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The adjoint representation inside the exterior algebra of a simple Lie algebra
For a simple complex Lie algebra $\mathfrak g$ we study the space of invariants $A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$, (which describes the isotypic component of type $\mathfrak g$ in $ \bigwedge \mathfrak g^*$) as a module over the algebra of invariants $\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$. As main result we prove that $A$ is a free module, of rank twice the rank of $\mathfrak g$, over the exterior algebra generated by all primitive invariants in $(\bigwedge \mathfrak g^*)^{\mathfrak g}$, with the exception of the one of highest degree.
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On special covariants in the exterior algebra of a simple Lie algebra
We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n-th symmetric power of the defining representation. As main result we prove that this subspace is a free module over the subalgebra of the exterior algebra generated by all primitive invariants except the one of highest degree.
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Geometry of the analytic loop group
We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of the quantum affine algebra (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters, are parameterized by certain G-bundles on an elliptic curve.
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A generalized Steinberg section and branching rules for quantum groups at roots of 1
In this paper we construct a generalization of the classical Steinberg section for the quotient map of a semisimple group with respect to the conjugation action. We then give various applications of our construction including the construction of a sort of Gelfand Zetlin basis for a generic irreducible representation of quantum GL(n) at odd roots of unity.
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Infinitesimal index: cohomology computations
In this note several computations of equivariant cohomology groups are performed. For the compactly supported equivariant cohomology, the notion of infinitesimal index developed in arXiv:1003.3525, allows to describe these groups in terms of certain spaces of distributions arising in the theory of splines. The new version contains a large number of improvements.
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Vector partition functions and index of transversally elliptic operators
Let G be a torus acting linearly on a complex vector space M, and let X be the list of weights of G in M. We determine the equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit difference equations associated to X, introduced by Dahmen--Micchelli in the context of the theory of splines in order to study vector partition functions. This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah-Singer which is in a sense universal for index computations.
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The quotient of a complete symmetric variety
We study the quotient of a completion of a symmetric variety G/H under the action of H. We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points.
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Nested sets and Jeffrey Kirwan cycles
For the complement of a hyperplane arrangement we construct a dual homology basis to the no broken circuit basis of cohomology. This is based on the theory of wonderful embeddings and nested sets.
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