Stephen Griffeth

Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces

We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term--the top one--with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.

ORTHOGONAL FUNCTIONS GENERALIZING JACK POLYNOMIALS

The rational Cherednik algebra ℍ is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for ℍ. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible W-modules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on ℂ n with values in the representation S λ . For G(1,1, n) with λ = (n) these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M(λ) in the case in which the orthogonal functions are all well-defined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of ℍ so that the rational Cherednik algebra for G(r,p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G(r, p, n) by Clifford theory.

Towards a combinatorial representation theory for the rational Cherednik algebra of type G ( r, p, n )

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a self-contained and elementary proof of the analog for the groups G(r,p,n), with r>1, of Gordons theorem (previously Haimans conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p<r has been proved by Vale using a technique analogous to Gordons. Because of the combinatorial application to Haimans conjecture, the paper is logically self-contained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordons proof of his theorem.

Catalan numbers for complex reflection groups

We construct

Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the (k + 1)-Equals Ideal

(q,t)

Parabolic degeneration of rational Cherednik algebras

-Catalan polynomials and

Systems of parameters and holonomicity of A-hypergeometric systems

q

Affine Hecke algebras and the Schubert calculus

-Fuss-Catalan polynomials for any irreducible complex reflection group

Generalized Jack polynomials and the representation theory of rational Cherednik algebras

W

Cell modules and canonical basic sets for Hecke algebras from Cherednik algebras

. The two main ingredients in this construction are Rouquiers formulation of shift functors for the rational Cherednik algebras of