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Olivier Schiffmann

  1. The spherical Hall algebra of Spec(Z).

    Втр, 02/21/2012 - 15:01 — SomNambul
    Authors: Olivier Schiffmann, Eric Vasserot, Mikhail Kapranov
    Subjects: Algebraic Geometry
    Abstract

    We study an arithmetic analog of the Hall algebra of a curve, when the curve
    is replaced by the spectrum of the integers compactified at infinity. The role
    of vector bundles is played by lattices with quadratic forms. This algebra H
    consists of automorphic forms with respect to GL_n(Z), n>0, with multiplication
    given by the parabolic pseudo-Eisenstein series map.

  2. Drinfeld realization of the elliptic Hall algebra.

    Пт, 04/16/2010 - 15:01 — SomNambul
    Authors: Olivier Schiffmann
    Subjects: Quantum Algebra
    Abstract

    We give a new presentation of the Drinfeld double of the elliptic Hall
    algebra introduced in a previous work with I. Burban. This presentation is
    similar in spirit to Drinfeld's `new realization' of quantum affine algebras.
    This answers, in the case of elliptic curves, a question of Kapranov concerning
    functional relations satisfied by (principal, unramified) Eisenstein series for
    the groups GL(n) over a function field. It also provides proofs of some recent
    conjectures of Feigin, Feigin, Jimbo, Miwa and Mukhin.

  3. Spherical Hall algebras of curves and Harder-Narasimhan stratas.

    Пт, 02/05/2010 - 16:01 — SomNambul
    Authors: Olivier Schiffmann
    Subjects: Quantum Algebra
    Abstract

    Let X be any smooth projective curve defined over a finite field. We show
    that the characteristic functions of any Harder-Narasimhan strata S_a of
    Bun_{GL_n}X belongs to the spherical Hall algebra H_X^{sph} of X. We give a
    geometric analog of the above result: the intersection cohomology sheaf IC(S_a)
    belongs to the category of simple Eisenstein sheaves over Bun_{GL_n}X.

  4. Lectures on canonical and crystal bases of Hall algebras.

    Пнд, 10/26/2009 - 14:11 — SomNambul
    Authors: Olivier Schiffmann
    Subjects: Quantum Algebra
    Abstract

    These are the notes for a series of lectures given on the theory of canonical
    and crystal bases for Hall algebras (for a summer school in Grenoble in 2008).
    It may be viewed as a follow-up to arXiv:math/0611617. It covers the
    construction, due to Lusztig, of the canonical bases for the Hall algebra of a
    quiver Q in terms of a certain category of perverse sheaves over the moduli
    space of representations of Q.

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