arXiv subjects

Ana Balibanu

Publications and source records attributed to Ana Balibanu.

8 recordsLinked to original sources

arXiv2024

An analogue of Whittaker reduction for group-valued moment maps

We construct an analogue of Whittaker reduction for Poisson actions of a semisimple complex Poisson-Lie group G. The reduction takes place along a class of transversal slices to unipotent orbits in G, which are generalizations of the Steinberg cross-section and are indexed by conjugacy classes in the Weyl group. We give an interpretation of these reductions in the framework of Dirac geometry, and we use this to describe their symplectic leaves.

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arXiv2023

The partial compactification of the universal centralizer

The universal centralizer of a semisimple algebraic group is the family of centralizers of regular elements, parametrized by their conjugacy classes. When the group is of adjoint type, we construct a smooth, log-symplectic fiberwise compactification of the universal centralizer by taking the closure of each fiber in the wonderful compactification. We use the geometry of the wonderful compactification to give an explicit description of its symplectic leaves. We also show that the compactified centralizer fibers are isomorphic to certain Hessenberg varieties -- we apply this connection to compute the singular cohomology of the partial compactification, and to study the geometry of the corresponding universal Hessenberg family.

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arXiv2023

A quasi-Poisson structure on the multiplicative Grothendieck-Springer resolution

In this note we show that the multiplicative Grothendieck-Springer space has a natural quasi-Poisson structure. The associated group-valued moment map is the resolution morphism, and the quasi-Hamiltonian leaves are the connected components of the preimages of Steinberg fibers. This is a multiplicative analogue of the standard Poisson structure on the additive Grothendieck-Springer resolution, and an explicit illustration of a more general procedure of reduction along Dirac realizations which is developed in recent work of the author and Mayrand.

math.SG↗

arXiv2023

Wonderful varieties with a view towards Poisson geometry

These are expanded lecture notes from the author's minicourse at the 2022 Poisson Geometry Summer School, which took place at the Centre de Recerca Matematica in Barcelona, Spain. After giving a general introduction to wonderful varieties, and an explicit construction of the wonderful compactification of a semisimple adjoint group, we outline several connections to Poisson geometry and to varieties of Lagrangian subalgebras. This survey is intended to be accessible to readers without an extensive background in algebraic geometry.

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arXiv2022

Reduction along strong Dirac maps

We develop a general procedure for reduction along strong Dirac maps, which are a broad generalization of Poisson momentum maps. We recover a large number of familiar constructions in Poisson and quasi-Poisson geometry, and we introduce new examples of Poisson, quasi-Poisson, and Dirac reduced structures. In particular, we obtain quasi-Poisson analogues of several classes of spaces that are studied in geometric representation theory.

math.SG↗

arXiv2022

Perverse sheaves and the cohomology of regular Hessenberg varieties

We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot-action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the "Kahler package."

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arXiv2022

Steinberg slices and group-valued moment maps

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.

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arXiv2017

The Peterson Variety and the Wonderful Compactification

We look at the centralizer in a semisimple algebraic group $G$ of a regular nilpotent element, and show that its closure in the wonderful compactification is isomorphic to the Peterson variety. It follows that the closure in the wonderful compactification of the centralizer $G^x$ of any regular element $x$ is isomorphic to the closure of a general $G^x$-orbit in the flag variety. We also give a description of the $G^e$-orbit structure of the Peterson variety.

math.RT↗