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Olivier Schiffmann

Publications and source records attributed to Olivier Schiffmann.

At least 37 records · Page 2Linked to original sources
arXiv2015

Two descriptions of the quantum affine algebra $U_v(\hat{\mathfrak{sl}}_2)$ via Hall algebra approach

We compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver $\overr{Q}$ and of the category of coherent sheaves on ${\mathbb P}^1$. Using this approach, we show that the Drinfeld--Beck isomorphism for the quantized enveloping algebra $U_v(\hat{\mathfrak{sl}}_2)$ is a corollary of an equivalence between the derived categories $D^b\bigl(\Rep(\overr{Q})\bigr)$ and $D^b\bigl(\Coh({\mathbb P}^1)\bigr)$. This technique also allows to reprove several technical results on the integral form of $U_v(\hat{\mathfrak{sl}}_2)$.

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arXiv2015

Composition algebra of a weighted projective line

In this article, we deal with properties of the reduced Drinfeld double of the composition subalgebra of the Hall algebra of the category of coherent sheaves on a weighted projective line. This study is motivated by applications in the theory of quantized enveloping algebras of some Lie algebras. We obtain a new realization of quantized enveloping algebras of affine Lie algebras of simply-laced type and get new embeddings between such algebras. Moreover, our approach allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$. In particular, our method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.

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arXiv2014

Counting Higgs bundles

We prove a closed formula counting semistable twisted Higgs bundles of fixed rank and degree over a smooth projective curve defined over a finite field. We also prove a formula for the Donaldson-Thomas invariants of the moduli spaces of twisted Higgs bundles over a curve.

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arXiv2014

Indecomposable vector bundles and stable Higgs bundles over smooth projective curves

We prove that the number of indecomposable vector bundles of fixed rank r and degree d over a smooth projective curve X defined over a finite field is given by a polynomial (depending only on the pair (r,d) and the genus g of X) in the Weil numbers of X. We provide a closed formula -expressed in terms of generating series- for this polynomial. We also show that the same polynomial computes the number of points of the moduli space of stable Higgs bundles of rank r and degree d over X (at least for large characteristics). This entails a closed formula for the Poincaré polynomial of the moduli spaces of stable Higgs bundles over a compact Riemann surface, and hence also for the Poincaré polynomials of the character varieties for the groups GL(r).

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arXiv2012

On the number of points of the Lusztig nilpotent variety over a finite field

In this note we give a closed expression for the number of points over finite fields of the Lusztig nilpotent variety associated to any quiver without edge loops, in terms of Kac's A-polynomial. We conjecture a similar result for quivers in which edge loops are allowed. Finally, we give a formula for the number of points over a finite field of the various stratas of the Lusztig nilpotent variety involved in the geometric realization of the crystal graph.

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arXiv2012

A presentation of the deformed W_{1+\infty} algebra

We provide a generators and relation description of the deformed W_{1+\infty}-algebra introduced in previous joint work of E. Vasserot and the second author. This gives a presentation of the (spherical) cohomological Hall algebra of the one-loop quiver, or alternatively of the spherical degenerate double affine Hecke algebra of GL(\infty).

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arXiv2012

Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A^2

We construct a representation of the affine W-algebra of gl_r on the equivariant homology space of the moduli space of U_r-instantons on A^2, and identify the corresponding module. As a corollary we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r). Another proof has been announced by Maulik and Okounkov. Our approach uses a suitable deformation of the universal enveloping algebra of the Witt algebra W_{1+\infty}, which is shown to act on the above homology spaces (for any r) and which specializes to all W(gl_r). This deformation is in turn constructed from a limit, as n tends to infinity, of the spherical degenerate double affine Hecke algebra of GL_n.

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arXiv2012

The Hall algebra of a curve

Let X be a smooth projective curve over a finite field. We describe H, the full Hall algebra of vector bundles X as a Feigin-Odesskii shuffle algebra. This shuffle algebra corresponds to the scheme S of all cusp eigenforms and to the rational function of two variables on S coming from the Rankin-Selberg L-functions. This means that the zeroes of these L-functions control all the relations in H. The scheme S is a disjoint union of countably many G_m-orbits. In the case when X has a theta-characteristic defined over the base field, we embed H into the space of regular functions on the symmetric powers of S.

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arXiv2012

The spherical Hall algebra of Spec(Z)

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. We concentrate on the subalgebra SH in H generated by functions on the Arakelov Picard group of Spec(Z). We identify H with a Feigin-Odesskii type shuffle algebra, with the function defining the shuffle algebra expressed through the Riemann zeta function. As an application we study relations in H. Quadratic relations express the functional equation for the Eisenstein-Maass series. We show that the space of additional cubic relations (lying an an appropriate completion of H and considered modulo rescaling), is identified with the space spanned by nontrivial zeroes of the zeta function.

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arXiv2012

The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A^2. We show that it is isomorphic to the elliptic Hall algebra, and hence to the spherical DAHA of GL_\infty. We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K-theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K-theory of the moduli space of framed torsion free sheaves over P^2, virtual fundamental classes, shuffle algebras,...).

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arXiv2010

Drinfeld realization of the elliptic Hall algebra

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.

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arXiv2010

Spherical Hall algebras of curves and Harder-Narasimhan stratas

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.

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arXiv2009

Lectures on canonical and crystal bases of Hall algebras

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. It also contains an exposition of Kashiwara and Saito's geometric construction of the crystal graph in terms of irreducible components of Lusztig's lagrangian in the cotangent bundle to the above moduli spaces. The last section deals with the Hall algebras of curves. It contains a few new results and conjectures. Apart from these, the text is purely expositional.

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arXiv2009

Lectures on Hall algebras

These are notes for a minicourse on Hall algebras given at the ICTP in Trieste in January 2006. After giving the definition and first properties of Hall algebras, we study in some details the classical Hall algebra, the Hall algebra of quivers, and the Hall algebra of coherent sheaves on smooth projective curves. The last section deals with the Hall algebras in the context of derived categories. (The figures only seem to come out nicely in the .ps file)

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arXiv2009

On the Hall algebra of an elliptic curve, I

In this article we describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X of H_X by algebra automorphisms. Next, we study a certain natural subalgebra U_X of DH_X for which we give a presentation by generators and relations. This algebra turns out to be a flat two-parameter deformation of the ring of diagonal invariants C[x_1^{\pm 1}, ..., y_1^{\pm 1},...]^{S_{\infty}}, i.e. the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.

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arXiv2009

On the Hall algebra of an elliptic curve, II

This paper is a sequel to math.AG/0505148, where the Hall algebra U^+_E of the category of coherent sheaves on an elliptic curve E defined over a finite field was explicitly described, and shown to be a two-parameter deformation of the ring of diagonal invariants R^+=C[x_1^{\pm 1}, ..., y_1,...]^{S_{\infty}} (in infinitely many variables). In the present work, we study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on E. This allows us to define a "canonical basis" B of U^+_E in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice and a certain PBW-type basis. Finally we provide some (loose) interpretation of the above construction in terms of Schubert varieties in (nonexistent!) "double affine grassmanian", and in terms of character sheaves of loop groups (both of type A).

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