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Affine Hecke algebras and the Schubert calculus
Using a combinatorial approach which avoids geometry, this paper studies the
ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized)
flag variety G/B. Here the data is a complex reductive algebraic group (or
symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T,
and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on
G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class
by a homogeneous line bundle (dominant or anti-dominant) and for products of a
Schubert class by a codimension 1 Schubert class. All of these Pieri-Chevalley
formulas are given in terms of the combinatorics of the Littelmann path model.
We give explicit computations of products of Schubert classes for the rank two
cases and this data allows us to make a "positivity conjecture" generalizing
the theorems of Brion and Graham, which treat the cases K(G/B) and H_T^*(G/B),
respectively
Cosets of affine vertex algebras inside larger structures
Given a finite-dimensional reductive Lie algebra equipped with
a nondegenerate, invariant, symmetric bilinear form , let
denote the universal affine vertex algebra associated to
and at level . Let be a vertex
(super)algebra admitting a homomorphism . Under some technical conditions on , we
characterize the coset for
generic values of . We establish the strong finite generation of this coset
in full generality in the following cases: , , and . Here and are
finite-dimensional Lie (super)algebras containing , equipped with
nondegenerate, invariant, (super)symmetric bilinear forms and which
extend , is fixed, and is a free field
algebra admitting a homomorphism .
Our approach is essentially constructive and leads to minimal strong finite
generating sets for many interesting examples. As an application, we give a new
proof of the rationality of the simple superconformal algebra with
for all positive integers .Comment: Some errors corrected, final versio
Eigencones and the PRV conjecture
Let be a complex semisimple simply connected algebraic group. Given two
irreducible representations and of , we are interested in some
components of . Consider two geometric realizations of
and using the Borel-Weil-Bott theorem. Namely, for , let \Li_i
be a -linearized line bundle on such that {\rm H}^{q_i}(G/B,\Li_i)
is isomorphic to . Assume that the cup product {\rm
H}^{q_1}(G/B,\Li_1)\otimes {\rm H}^{q_2}(G/B,\Li_2)\longto {\rm
H}^{q_1+q_2}(G/B,\Li_1\otimes\Li_2) is non zero. Then, {\rm
H}^{q_1+q_2}(G/B,\Li_1\otimes\Li_2) is an irreducible component of ; such a component is said to be {\it cohomological}. Solving a
Dimitrov-Roth conjecture, we prove here that the cohomological components of
are exactly the PRV components of stable multiplicity one.
Note that Dimitrov-Roth already obtained some particular cases. We also
characterize these components in terms of the geometry of the Eigencone of .
Along the way, we prove that the structure coefficients of the Belkale-Kumar
product on {\rm H}^*(G/B,\ZZ) in the Schubert basis are zero or one
Invariant subalgebras of affine vertex algebras
Given a finite-dimensional complex Lie algebra g equipped with a
nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the
universal affine vertex algebra associated to g and B at level k. For any
reductive group G of automorphisms of V_k(g,B), we show that the invariant
subalgebra V_k(g,B)^G is strongly finitely generated for generic values of k.
This implies the existence of a new family of deformable W-algebras W(g,B,G)_k
which exist for all but finitely many values of k.Comment: Final version, proof of main result simplified. arXiv admin note:
substantial text overlap with arXiv:1006.562
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