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Emmanuel Letellier

Publications and source records attributed to Emmanuel Letellier.

17 recordsLinked to original sources
arXiv2024

$\mathrm{PGL}_n(\mathbb{C})$-character stacks and Langlands duality over finite fields

In this paper we study the mixed Poincar\'e polynomial of generic $\mathrm{PGL}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of $\mathrm{PGL}_n(\mathbb{C})$ which have non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that this conjectured formula interpolates the structure coefficients of the two based rings$ \left(\mathcal{C}(\mathrm{PGL}_n(\mathbb{F}_q)),Loc(\mathrm{PGL}_n),*\right)$ and $\left(\mathcal{C}(\mathrm{SL}_n(\mathbb{F}_q)), CS(\mathrm{SL}_n),\cdot\right) $ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc(\mathrm{PGL}_n)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of $\mathrm{PGL}_n$ and $CS(\mathrm{SL}_n)$ the basis of characteristic functions of Lusztig's character-sheaves on $\mathrm{SL}_n$. Our result reminds us of a non-abelian Fourier transform.

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arXiv2024

Ennola duality for decomposition of tensor products

The aim of this paper is to investigate Ennola duality for decomposition of tensor products of irreducible characters of finite general linear groups and finite unitary groups. We prove that Ennola duality holds generically and give a geometric interpretation using the cohomology of quiver varieties. For non-generic characters (like unipotent characters), Ennola duality does not work just by replacing q by -q. We construct two-variable polynomials that interpolate multiplicities for finite general linear groups and finite unitary groups in the unipotent case (which can be considered as Ennola duality).

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arXiv2023

Saxl conjecture and the tensor square of unipotent characters of GL(n,q)

We know from Letellier that if for some triple of partitions the corresponding Kronecker coefficient is non-zero then the corresponding multiplicities for unipotent characters of GL(n,q) is also non-zero. A conjecture of Saxl says that the tensor square of an irreducible character of the symmetric group corresponding to a staircase partition contains all the irreducible characters. Therefore Saxl conjecture implies its analogue for unipotent characters. In this paper we prove the analogue of Saxl conjecture for unipotent characters and we describe conjecturally the set of all partitions for which the tensor square of the associated unipotent character contains all the unipotent characters.

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arXiv2023

Geometrization of the two orthogonality formulas for Green functions

The Green functions were first introduced by Green to compute the character table of GLn(q) in 1955. They were later generalized by Deligne and Lusztig for an arbitrary finite group of Lie type G(q) using l-adic cohomological methods (1976). They proved that these Green functions satisfy an orthogonality relation (we call the first orthogonality relation). Ten years later Kawanaka proved that they satisfy an other orthogonality relation (we call the second orthogonality relation). In this notes, we provide a geometrical understanding of these two orthogonality relations and explain how we can see geometrically that the two orthogonality relations are in fact equivalent.

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arXiv2023

Fourier transform from the symmetric square representation of $PGL_2$ and $SL_2$

Let $G$ be a connected reductive group over $\overline{\mathbb{F}}_q$ and let $\rho^\vee:G^\vee\rightarrow GL_n$ be an algebraic representation of the dual group $G^\vee$. Assuming that $G$ and $\rho^\vee$ are defined over $\mathbb{F}_q$, Braverman and Kazhdan defined an operator on the space $\mathcal{C}(G(\mathbb{F}_q))$ of complex valued functions on $G(\mathbb{F}_q)$. In this paper we are interested in the case where $G$ is either $SL_2$ or $PGL_2$ and $\rho^\vee$ is the symmetric square representation of $G^\vee$. We construct a natural $G\times G$-equivariant embedding $G\hookrightarrow\mathcal{G}=\mathcal{G}_\rho$ and an involutive operator (Fourier transform) $\mathcal{F}^{\mathcal{G}}$ on the space of functions $\mathcal{C}(\mathcal{G}(\mathbb{F}_q))$ that extends Braverman-Kazhdan's operator.

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arXiv2022

E-series of character varieties of non-orientable surfaces

In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincaré series). In case (B), we discuss the mixed Poincaré series when the surface is the real projective plane and k=1.

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arXiv2020

On the derived Lusztig correspondence

Let $G$ be a connected reductive group, $T$ a maximal torus of $G$, and $N$ the normalizer of $T$ in $G$. In this paper we study the connection between the derived category of l-adic sheaves on the stack $[Lie(T)/N]$ and the derived category of $l$-adic sheaves on $[Lie(G)/G]$.

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arXiv2019

Notes on a conjecture of Braverman-Kazhdan

Given a connected reductive algebraic group G over a finite field together with a representation of the dual group of G in GL(n), Braverman and Kazhdan defined an exotic Fourier operator on the space of complex valued functions on the finite group of rational points of G. In these notes we give an explicit formula for the Fourier kernel and a geometrical interpretation of this formula (as conjectured by Braverman and Kazhdan under some assumption).

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arXiv2018

Locally free representations of quivers over commutative Frobenius algebras

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t^2). Using these results together with results of Geiss, Leclerc and Schroer we give, when k is algebraically closed, a classification of pairs (Q,R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally, when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.

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arXiv2016

Higgs bundles and indecomposable parabolic bundles over the projective line

In this paper we count the number of isomorphism classes of geometrically indecomposable quasi-parabolic structures of a given type on a given vector bundle on the projective line over a finite field. We give a conjectural cohomological interpretation for this counting using the moduli space of Higgs fields on the given vector bundle over the complex projective line with prescribed residues. We prove a certain number of results which bring evidences to the main conjecture. We detail the case of rank 2 vector bundles.

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arXiv2014

DT-invariants of quivers and the Steinberg character of GL_n

In this paper we give a simple description of DT-invariants of double quivers without potential as the multiplicity of the Steinberg character in some representation associated with the quiver. When the dimension vector is indivisible we use this description to express these DT-invariants as the Poincaré polynomial of some singular quiver varieties. Finally we explain the connections with previous work of Hausel-Letellier-Villegas where DT-invariants are expressed as the graded multiplicities of the trivial representation of some Weyl group in the cohomology of some non-singular quiver varieties attached to an extended quiver.

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arXiv2014

Character varieties with Zariski closures of GL_n-conjugacy classes at punctures

In a previous paper (joint with Hausel and Rodriguez-Villegas) we gave a conjectural formula for the mixed Hodge polynomials of character varieties with generic semisimple conjugacy classes at punctures and we prove a formula for the E-polynomial. We also proved that these character varieties are irreducible. In this paper we extend the above results to character varieties with Zariski closures of arbitrary generic conjugacy classes at punctures working with intersection cohomology. We also study Weyl group action on the intersection cohomology of the partial resolutions of these character varieties and give a conjectural formula for the two-variables polynomials that encode the trace of the elements of the Weyl group on the subquotients of the weight filtration. Finally, we compute the generating function of the stack count of character varieties with Zariski closure of unipotent regular conjugacy class at punctures.

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arXiv2013

Torus orbits on homogeneous varieties and Kac polynomials of quivers

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.

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arXiv2012

Tensor products of unipotent characters of general linear groups over finite fields

We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials and give a necessary and sufficient condition in terms of the representation theory of symmetric groups for these polynomials to be non-zero.

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arXiv2012

Positivity of Kac polynomials and DT-invariants for quivers

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomas- invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich-Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isoytpical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincaré polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties.

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arXiv2011

Arithmetic harmonic analysis on character and quiver varieties II

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.

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