256 research outputs found
Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras
Suppose that W is a finite, unitary, reflection group acting on the complex
vector space V and X is a subspace of V. Define N to be the setwise stabilizer
of X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction
defines a homomorphism from the algebra of W-invariant polynomial functions on
V to the algebra of C-invariant functions on X. In this note we consider the
special case when W is a Coxeter group, V is the complexified reflection
representation of W, and X is in the lattice of the arrangement of W, and give
a simple, combinatorial characterization of when the restriction mapping is
surjective in terms of the exponents of W and C. As an application of our
result, in the case when W is the Weyl group of a semisimple, complex, Lie
algebra, we complete a calculation begun by Richardson in 1987 and obtain a
simple combinatorial characterization of regular decomposition classes whose
closure is a normal variety.Comment: 11 pages revised 2/2012; to appear in Compos. Mat
Equivariant K-theory of generalized Steinberg varieties
We describe the equivariant K-groups of a family of generalized Steinberg
varieties that interpolates between the Steinberg variety of a reductive,
complex algebraic group and its nilpotent cone in terms of the extended affine
Hecke algebra and double cosets in the extended affine Weyl group. As an
application, we use this description to define Kazhdan-Lusztig "bar"
involutions and Kazhdan-Lusztig bases for these equivariant K-groups.Comment: 29 pages; final versio
Modules for Yokonuma-type Hecke algebras
This paper describes the module categories for a family of generic Hecke
algebras that specialize to the complex reflection groups G(r,1,n) and to the
certain endomorphism rings of permutation characters of finite general linear
groups. In particular, complete sets of inequivalent, irreducible modules for
semisimple specializations of these algebras are constructed.Comment: 24 pages; Introduction reworded and section 4 revised in response to
referee comment
Schur-Weyl duality and the free Lie algebra
We prove an analogue of Schur-Weyl duality for the space of homogeneous Lie
polynomials of degree r in n variables.Comment: 15 pages; revisions suggested by a refere
On the Invariants of the Cohomology of Complements of Coxeter Arrangements
We refine Brieskorn's study of the cohomology of the complement of the
reflection arrangement of a finite Coxeter group . As a result we complete
the verification of a conjecture by Felder and Veselov that gives an explicit
basis of the space of -invariants in this cohomology ring.Comment: 12 pages, 3 figures; final versio
Restricting invariants of unitary reflection groups
Suppose that G is a finite, unitary reflection group acting on a complex
vector space V and X is the fixed point subspace of an element of G. Define N
to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and
C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant
polynomial functions on V to the algebra of C-invariant functions on X.
Extending earlier work by Douglass and Roehrle for Coxeter groups, we
characterize when the restriction mapping is surjective for arbitrary unitary
reflection groups G in terms of the exponents of G and C, and their reflection
arrangements. A consequence of our main result is that the variety of G-orbits
in the G-saturation of X is smooth if and only if it is normal.Comment: 28 pages, includes tables; revisions suggested by a refere
Computations for Coxeter arrangements and Solomon's descent algebra III: Groups of rank seven and eight
In this paper we extend the computations in parts I and II of this series of
papers and complete the proof of a conjecture of Lehrer and Solomon expressing
the character of a finite Coxeter group W acting on the pth graded component of
its Orlik-Solomon algebra as a sum of characters induced from linear characters
of centralizers of elements of W for groups of rank seven and eight. For
classical Coxeter groups, these characters are given using a formula that is
expected to hold in all ranks.Comment: Minor changes; final versio
The homology of the Steinberg variety and Weyl group coinvariants
Let G be a complex, connected, reductive algebraic group with Weyl group W
and Steinberg variety Z. We show that the graded Borel-Moore homology of Z is
isomorphic to the smash product of the coinvariant algebra of W and the group
algebra of W.Comment: 17 pages, to appear in Documenta Mat
The Leray-Hirsch Theorem for equivariant oriented cohomology of flag varieties
We use the formal affine Demazure algebra to construct an explicit
Leray-Hirsch Theorem for torus equivariant oriented cohomology of flag
varieties. We then generalize the Borel model of such theory to partial flag
varieties.Comment: 17 pages. Comments are welcom
The Steinberg Variety and Representations of Reductive Groups
We give an overview of some of the main results in geometric representation
theory that have been proved by means of the Steinberg variety. Steinberg's
insight was to use such a variety of triples in order to prove a conjectured
formula by Grothendieck.
The Steinberg variety was later used to give an alternative approach to
Springer's representations and played a central role in the proof of the
Deligne-Langlands conjecture for Hecke algebras by Kazhdan and Lusztig.Comment: 37 pages; significant revision and extension; to appear in J. Algebr
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