Sommers, Eric

Job Title

Associate Professor, Department of Mathematics and Statistics

Last Name

Sommers

First Name

Eric

Discipline

Mathematics

Expertise

Algebraic Groups
Representation Theory

Introduction

My work involves studying the properties of reductive algebraic groups. Algebraic groups are groups equipped with the Zariski topology such that the multiplication and inverse maps are maps of varieties. They behave like Lie groups, except that there is the freedom to work over any field.
Specifically, I study the objects that arise when trying to understand the representation theory of algebraic groups, especially nilpotent orbits and affine Weyl groups. Recently I have been thinking about the connection between nilpotent orbits, Borel-stable ideals in the nilradical, Kazhdan-Lusztig cells, and certain duality maps.

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Now showing 1 - 10 of 12

Open Access
Exponents for B-stable ideals
(2006-01) Sommers, E; Tymoczko, J
Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types. When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.
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COHOMOLOGY OF LINE BUNDLES ON THE COTANGENT BUNDLE OF A GRASSMANNIAN
(2009-01) Sommers, E
e show that certain line bundles on the cotangent bundle of a Grassmannian arising from an anti-dominant character have cohomology groups isomorphic to those of a line bundle on the cotangent bundle of the dual Grassmannian arising from the dominant character , where is the longest element of the Weyl group of
Open Access
A characterization of Dynkin elements
(2003-01) Gunnells, PE; Sommers, E
We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element is the unique point of minimal length in its $N$-region. In type $A_n$ this translates into a statement about the regions determined by the canonical left Kazhdan-Lusztig cells, which leads to some conjectures in representation theory.
Open Access
B-stable ideals in the nilradical of a Borel subalgebra
(2005-01) Sommers, EN
Let $G$G be a connected simple algebraic group over the complex numbers and $B$B a Borel subgroup. Let $\germ g$g be the Lie algebra of $G$G and $\germ b$b the Lie algebra of $B$B . A subspace of the nilradical of $\germ b$b which is stable under the action of $B$B is called a $B$B -stable ideal of the nilradical. It is called strictly positive if it intersects the simple root spaces trivially. The author counts the number of strictly positive $B$B -stable ideals in the nilradical of a Borel subalgebra and proves that the set of minimal roots of any $B$B -stable ideal is conjugate by an element of the Weyl group to a subset of simple roots. He also counts the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.
Open Access
Normality of very even nilpotent varieties in D-2l
(2005-01) Sommers, E
For the classical groups, Kraft and Procesi have resolved the question of which nilpotent orbits have closures that are normal and which do not, with the exception of the very even orbits in D2l that have partitions of the form (a2k, b2) for a > b even natural numbers satisfying ak + b = 2l.
Open Access
Normality of nilpotent varieties in E-6
(2003-01) Sommers, E
We determine which nilpotent orbits in E6 have closures which are normal varieties and which do not. At the same time we are able to verify a conjecture in [E. Sommers, Comm. Math. Univ. Sancti Pauli 49 (1) (2000) 101–104] concerning functions on non-special nilpotent orbits for E6.
Open Access
EXTERIOR POWERS OF THE REFLECTION REPRESENTATION IN SPRINGER THEORY
(2010-01) Sommers, E
We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original conjecture to all nilpotent orbits and also takes into account the action of the component group. The method is to use Shoji's approach to the orthogonality formulas for Green functions to relate the symmetric algebra to a sum over Green functions. In the second part of the paper we give an explanation of the appearance of the Orlik-Solomon exponents using a result from rational Cherednik algebras.
Open Access
PIECES OF NILPOTENT CONES FOR CLASSICAL GROUPS
(2010-01) Achar, Promad; Henderson, Anthony; Sommers, E
We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato's exotic nilpotent cone. We prove that the number of $\F_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\F_q$-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.
Open Access
Component groups of unipotent centralizers in good characteristic
(2003-01) McNinch, GJ; Sommers, E
Let G be a connected, reductive group over an algebraically closed field of good characteristic. For uG unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checking, that the structure of the component group A(u) is independent of good characteristic.
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Lusztig's canonical quotient and generalized duality
(2001-01) Sommers, E
We give a new characterization of Lusztig's canonical quotient, a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra. This group plays an important role in the classification of unipotent representations of finite groups of Lie type. We also define a duality map. To each pair of a nilpotent orbit and a conjugacy class in its fundamental group, the map assigns a nilpotent orbit in the Langlands dual Lie algebra. This map is surjective and is related to a map introduced by Lusztig (and studied by Spaltenstein). When the conjugacy class is trivial, our duality map is just the one studied by Spaltenstein and by Barbasch and Vogan which has an image of the set of special nilpotent orbits.