arXiv subjects

Sam Evens

Publications and source records attributed to Sam Evens.

24 records · Page 2Linked to original sources

arXiv2006

On the variety of Lagrangian subalgebras, II

When ${\frak g}$ is a complex semisimple Lie algebra, we study the variety ${\mathcal L}$ of subalgebras of ${\frak g}\oplus{\frak g}$ that are maximally isotropic with respect to $K_1 - K_2$, where $K_i$ is the Killing form on the ith factor. We show the irreducible components of ${\mathcal L}$ are smooth, classify them in terms of the generalized Belavin-Drinfeld triples introduced by Schiffmann, and relate them to orbits of the adjoint group $G\times G$. Building on ideas of Yakimov, we give a new proof of Karolinsky's classification of the diagonal $G$-orbits in ${\mathcal L}$. Our proof enables us to compute of the normalizer in ${\frak g}$ of a subalgebra in ${\mathcal L}$ under the diagonal action. As a consequence, we recover the classification of Belavin-Drinfeld triples. By results of math.DG/9909005, ${\mathcal L}$ is a Poisson variety and we determine the rank of the symplectic leaf at each point of ${\mathcal L}$ in terms of combinatorial data and relate the symplectic leaves to intersections of orbits of subgroups of $G\times G$. As a consequence, an intrinsically defined Poisson structure on each conjugacy class on $G$ has an open symplectic leaf and we determine the rank at each point of the conjugacy class.

math.QA↗

arXiv2003

Thompson's conjecture for real semi-simple Lie groups

A proof of Thompson's conjecture for real semi-simple Lie groups has been given by Kapovich, Millson, and Leeb. In this note, we give another proof of the conjecture by using a theorem of Alekseev, Meinrenken, and Woodward from symplectic geometry.

math.SG↗

arXiv2000

Representations of quantum tori and double-affine Hecke algebras

We study a BGG-type category of infinite dimensional representations of H[W], a semi-direct product of the quantum torus with parameter `q' built on the root lattice of a semisimple group G, and the Weyl group of G. Irreducible objects of our category turn out to be parameterized by semistable G-bundles on the elliptic curve C^*/q^Z. In the second part of the paper we construct a family of algebras depending on a parameter `v' that specializes to H[W] at v=0, and specializes to the double-affine Hecke algebra introduced by Cherednik, at v=1. We propose a Deligne-Langlands-Lusztig type conjecture relating irreducible modules over the double-affine Hecke algebra to Higgs G-bundles on the elliptic curve. The conjecture may be seen as a natural `v-deformation' of the classification of simple H[W]-modules obtained in the first part of the paper. Also, an `operator realization' of the double-affine Hecke algebra, as well as of its Spherical subalgebra, in terms of certain `zero-residue' conditions is given.

math.RT↗

arXiv1999

On the variety of Lagrangian subalgebras

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $Π$. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the real points of a De Concini-Procesi compactification and a compact homogeneous space. We study some properties of the Poisson structure $Π$ and show that it contains many interesting Poisson submanifolds.

math.DG↗

arXiv1997

Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$

We characterize the harmonic forms on a flag manifold $K/T$ defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson harmonic" with respect to the so-called Bruhat Poisson structure on $K/T$. This enables us to give Poisson geometrical proofs of many of the special properties of these harmonic forms. In particular, we construct explicit representatives for the Schubert basis of the $S^1$-equivariant cohomology of $K/T$, where the $S^1$-action is defined by $ρ$. Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmonic forms and the Schubert calculus on $K/T$ that was found by Kostant and Kumar in 1986. We also show that the Kostant harmonic forms are limits of the more familiar Hodge harmonic forms with respect to a family of Hermitian metrics.

dg-ga↗

arXiv1996

Transverse measures, the modular class, and a cohomology pairing for Lie algebroids

We show that every Lie algebroid $A$ over a manifold $P$ has a natural representation on the line bundle $Q_A = \wedge^{top}A \otimes \wedge^{top} T^*P$. The line bundle $Q_A$ may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of $Q_A$ may be viewed as transverse measures to $A$. As a consequence, there is a well-defined class in the first Lie algebroid cohomology $H^1(A)$ called the modular class of the Lie algebroid $A$. This is the same as the one introduced earlier by Weinstein using the Poisson structure on $A^*$. We show that there is a natural pairing between the Lie algebroid cohomology spaces of $A$ with trivial coefficients and with coefficients in $Q_A$. This generalizes the pairing used in the Poincare duality of finite-dimensional Lie algebra cohomology. The case of holomorphic Lie algebroids is also discussed, where the existence of the modular class is connected with the Chern class of the line bundle $Q_A$.

dg-ga↗