Hecke group algebras as quotients of affine Hecke algebras at level 0

The Hecke group algebra of a finite Coxeter group , as introduced by the first and last authors, is obtained from by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when is the finite Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity of small order, is the natural quotient of the affine Hecke algebra H(W)(q) through its level 0 representation.The proof relies on the following core combinatorial result: at level 0 the 0-Hecke algebra H(W)(0) acts transitively on . Equivalently, in type A, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level 0 representation is a calibrated principal series representation M(t) for a suitable choice of character t, so that the quotient factors (non-trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the 0-Hecke algebra and that of the affine Hecke algebra H(W)(q) at this specialization.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Polynomial identities and noncommutative versal torsors

To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, is a cleft H-Galois extension of a “big” commutative algebra . Any “form” of can be obtained from by a specialization of and vice versa. If the algebra is simple, then is an Azumaya algebra with center . The algebra is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of are satisfied. We construct an embedding of into ; this embedding maps the center of into when the algebra is simple. In this case, under an additional assumption, , thus turning into a central localization of . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.     

The structure of analytic τ-sheaves

Let R be a complete discrete valuation -algebra whose residue field is algebraic over , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T_?(M) of M.For a given τ-sheaf M on , we prove the existence of a maximal model for M on , an R-model of , and, over a finite separable extension R′ of R, of nondegenerate models for M.We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂C, and a filtration

     

On twisted representations of vertex algebras

In this paper, we develop a formalism for working with representations of vertex and conformal algebras by generalized fields—formal power series involving non-integer powers of the variable. The main application of our technique is the construction of a large family of representations for the vertex superalgebra corresponding to an integer lattice Λ. For an automorphism coming from a finite-order automorphism we find the conditions for existence of twisted modules of . We show that the category of twisted representations of is semisimple with finitely many isomorphism classes of simple objects.     

Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let and be two atomic nests on complex Hilbert spaces H and K, respectively. Denote the nest algebra associated with and the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from onto and a thorough classification of numerical radius isometries from onto .     

Perturbations of subalgebras of type II1 factors

In this paper we consider two von Neumann subalgebras and of a type II₁ factor . For a map φ on , we define

     

A fixed point localization formula for the Fourier transform of regular semisimple coadjoint orbits

Let be a Lie group acting on an oriented manifold M, and let ω be an equivariantly closed form on M. If both and M are compact, then the integral is given by the fixed point integral localization formula (Theorem 7.11 in Berline et al. Heat Kernels and Dirac Operators, Springer, Berlin, 1992). Unfortunately, this formula fails when the acting Lie group is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of in such a way that all fixed points are accounted for.Let be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly closed forms is the symplectic volume form dβ of a coadjoint orbit Ω. Even if Ω is not compact, the integral exists as a distribution on the Lie algebra . This distribution is called the Fourier transform of the coadjoint orbit.In this article, we will apply the localization results described in [L1,L2] to get a geometric derivation of Harish-Chandra's formula (9) for the Fourier transforms of regular semisimple coadjoint orbits. Then, we will make an explicit computation for the coadjoint orbits of elements of which are dual to regular semisimple elements lying in a maximally split Cartan subalgebra of .     

Equivariant chain complexes, twisted homology and relative minimality of arrangements

We show that the π-equivariant chain complex (), , associated to a Morse-theoretic minimal CW-structure X on the complement of an arrangement , is independent of X. The same holds for all scalar extensions, , a field, where X is an arbitrary minimal CW-structure on a space M. When is a section of another arrangement , we show that the divisibility properties of the first Betti number of the Milnor fiber of  obstruct the homotopy realization of  as a subcomplex of a minimal structure on .If is aspherical and is a sufficiently generic section of , then may be described in terms of π, L and , for an arbitrary local system L; explicit computations may be done, when is fiber-type. In this case, explicit -presentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of , π_p(M), may also be obtained. For nonresonant abelian scalar extensions, the -rank of is combinatorially determined.     

A note on maximality of analytic crossed products

Let G be a compact abelian group with the totally ordered dual group which admits the positive semigroup . Let N be a von Neumann algebra and be an automorphism group of on N. We denote to the analytic crossed product determined by N and α. We show that if is a maximal σ-weakly closed subalgebra of , then induces an archimedean order in .     

Caractérisation des feuilletages holomorphes singuliers, contenus dans des feuilletages Levi flat, sur les surfaces complexes compactes

Let be a holomorphic foliation with reduced singularities on a complex surface M and a real analytic codimension one foliation on M whose leaves contain the ones of . We show that a Levi flat group of diffeomorphisms of is resoluble and holomorphically conjugate to his normal form. We deduce, in one hand, that each singularity of  is conjugate to his normal form. In the other hand at each singularity m of , where is not defined, up a conjugacy, by the one form ω=xdy+ydx, one of the local invariant curves of , with non obvious holonomy, is contained in the set of singularities of . Moreover if M is a compact Stein variety we show, under some generic conditions, that has a 1-Liouvillian first integrating factor.     

Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C^k manifold in , for any , and for any τ>0 let