Character Sheaves Research Articles

Almost Commuting Scheme of Symplectic Matrices and Quantum Hamiltonian Reduction

Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of g=sp(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}=\\mathfrak {sp}(V)$$\\end{document} for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme Xnil\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{nil}$$\\end{document} of X and show that it is a complete intersection of dimension dim(g)+12dim(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {dim}(\\mathfrak {g})+\\frac{1}{2}\ ext {dim}(V)$$\\end{document} and compute its irreducible onents. We also study the quantum Hamiltonian reduction of the algebra D(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}(\\mathfrak {g})$$\\end{document} of differential operators on the Lie algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}$$\\end{document} tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type C. We contruct a category Cc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_c$$\\end{document} of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {D}$$\\end{document}-modules whose characteristic variety is contained in Xnil\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{nil}$$\\end{document} and construct an exact functor from this category to the category O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}$$\\end{document} of the above rational Cherednik algebra. Simple objects of the category Cc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_c$$\\end{document} are mirabolic analogs of Lusztig’s character sheaves.

Unipotent character sheaves and strata of a reductive group

Let H H be a connected reductive group over an algebraically closed field. We define a surjective map from the set C S ( H ) CS(H) of unipotent character sheaves on H H (up to isomorphism) to the set of strata of H H . To do this we use the generalized Springer correspondence. We also give a new parametrization of C S ( H ) CS(H) in terms of data coming from bad characteristic.

Character Sheaves for Classical Graded Lie Algebras

Character Sheaves for Classical Graded Lie Algebras

Character sheaves for graded Lie algebras: Stable gradings

In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [8].

Character sheaves for symmetric pairs: Special linear groups

We give an explicit description of character sheaves for the symmetric pairs associated to inner involutions of the special linear groups. We make use of the general strategy given in Vilonen and Xue [Character sheaves for classical symmetric pairs, Represent. Theory 26 (2022), 1097–1144] and central character consideration. We also determine the cuspidal character sheaves.

Character sheaves for classical symmetric pairs

We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.

The character table of [formula omitted

We compute the character table of GLn(Fq)⋊<σ>, σ being an order 2 exterior automorphism. We begin by giving the parametrisation of σ-stable irreducible characters of GLn(q) and that of the conjugacy classes contained in GLn(q)σ. Our main theorem is a formula expressing the extension of a σ-stable character of GLn(q) to GLn(q)σ as a linear combination of induced cuspidal functions. This formula is built upon a key result of J-L. Waldspurger concerning character sheaves. Using the main theorem, the determination of the character table is then reduced to computations in various Weyl groups and the generalised Green functions of classical groups. Finally, we explicitly determine the table for n=2 and 3.

Monodromic model for Khovanov–Rozansky homology

Abstract We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the Hecke category of type A we obtain a geometric description for Khovanov–Rozansky knot homology, essentially different from the one considered earlier by Webster and Williamson.

On Parabolic Restriction of Perverse Sheaves

We prove exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well-known result of Lusztig who established this property for character sheaves. We propose a conjectural (but known for character sheaves) t -exactness property of the Harish-Chandra transform and provide evidence for that conjecture. We also present two applications generalizing some results of Gabber and Loeser on perverse sheaves on an algebraic torus to an arbitrary reductive group.

Commutative character sheaves and geometric types for supercuspidal representations

We show that some types for supercuspidal representations of tamely ramified p-adic groups that appear in Jiu-Kang Yu’s work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu, assuming that the character vanishes on a certain derived subgroup. To define geometric types, we combine commutative character sheaves with Gurevich and Hadani’s geometrization of the Weil representation and Lusztig’s character sheaves.

Corrigendum to ‘Endoscopy for Hecke categories, character sheaves and representations’

Abstract We fix an error on a $3$ -cocycle in the original version of the paper ‘Endoscopy for Hecke categories, character sheaves and representations’. We give the corrected statements of the main results.

Correction to: On the Deligne–Lusztig involution for character sheaves

Correction to: On the Deligne–Lusztig involution for character sheaves

On the values of unipotent characters of finite Chevalley groups of type E6 in characteristic 3

Let G be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of G by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on G. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where G is one of the groups E6(q), E62(q), and q is a power of the bad prime p=3 for E6, by exploiting known facts about the representation theory of the Hecke algebra associated with G.

On the Deligne–Lusztig involution for character sheaves

For a reductive group G, we study the Drinfeld–Gaitsgory functor of the category of conjugation-equivariant D-modules on G. We show that this functor is an equivalence of categories, and that it has a filtration with layers expressed via parabolic induction of parabolic restriction. We use this to provide a conceptual definition of the Deligne–Lusztig involution on the set of isomorphism classes of irreducible character D-modules, which was defined previously in Lusztig (Adv Math 57:266–315, 1985, §15).

On the values of unipotent characters in bad characteristic

Let G(q) be a Chevalley group over a finite field \mathbb F_q . By Lusztig’s and Shoji’s work, the problem of computing the values of the unipotent characters of G(q) is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on G(q) . We show that this issue can be reduced to the case where q is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach.

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants int _S{mathcal {A}} of a surface S, determined by the choice of a braided tensor category {mathcal {A}}, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for {mathcal {A}}, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided {mathcal {A}}-modules are objects of the torus category int _{T^2}{mathcal {A}}. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of {mathcal {A}}={text {Rep}}_{q}G with the category {mathcal {D}}_q(G/G)-mod of equivariant quantum {mathcal {D}}-modules. When G=GL_n, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra {mathbb {SH}}_{q,t}.

Character sheaves on neutrally solvable groups

Let G G be an algebraic group over an algebraically closed field k \mathtt {k} of characteristic p &gt; 0 p&gt;0 . In this paper we develop the theory of character sheaves on groups G G such that their neutral connected components G ∘ G^\circ are solvable algebraic groups. For such algebraic groups G G (which we call neutrally solvable) we will define the set CS ⁡ ( G ) \operatorname {CS}(G) of character sheaves on G G as certain special (isomorphism classes of) objects in the category D G ( G ) \mathscr {D}_G(G) of G G -equivariant Q ¯ ℓ \overline {\mathbb {Q}}_{\ell } -complexes (where we fix a prime ℓ ≠ p \ell \neq p ) on G G . We will describe a partition of the set CS ⁡ ( G ) \operatorname {CS}(G) into finite sets known as L \mathbb {L} -packets and we will associate a modular category M L \mathscr {M}_L with each L \mathbb {L} -packet L L of character sheaves using a truncated version of convolution of character sheaves. In the case where k = F ¯ q \mathtt {k}=\overline {\mathbb {F}}_q and G G is equipped with an F q \mathbb {F}_q -Frobenius F F we will study the relationship between F F -stable character sheaves on G G and the irreducible characters of (all pure inner forms of) G F G^F . In particular, we will prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for neutrally solvable groups and that these almost characters coincide with the “trace of Frobenius” functions associated with F F -stable character sheaves. We will also prove that the matrix relating the irreducible characters and almost characters is block diagonal where the blocks on the diagonal are parametrized by F F -stable L \mathbb {L} -packets. Moreover, we will prove that the block in this transition matrix corresponding to any F F -stable L \mathbb {L} -packet L L can be described as the crossed S-matrix associated with the auto-equivalence of the modular category M L \mathscr {M}_L induced by F F .

A spectral incarnation of affine character sheaves

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$-modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al. [Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].

On Centers of Bimodule Categories and Induction–Restriction Functors

In this paper we study a toy categorical version of Lusztig's induction and restriction functors for character sheaves, but in the abstract setting of multifusion categories. Let $\mathscr{C}$ be an indecomposable multifusion category and let $\mathscr{M}$ be an invertible $\mathscr{C}$-bimodule category. Then the center $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ of $\mathscr{M}$ with respect to $\mathscr{C}$ is an invertible module category over the Drinfeld center $\mathscr{Z}(\mathscr{C})$ which is a braided fusion category. Let $\zeta_{\mathscr{M}}:\mathscr{Z}_{\mathscr{C}}(\mathscr{M})\longrightarrow\mathscr{M}$ denote the forgetful functor and let $\chi_{\mathscr{M}}:\mathscr{M}\longrightarrow\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ be its right adjoint functor. These functors can be considered as toy analogues of the restriction and induction functors used by Lusztig to define character sheaves on (possibly disconnected) reductive groups. In this paper we look at the relationship between the decomposition of the images of the simple objects under the above functors and the character tables of certain Grothendieck rings. In case $\mathscr{C}$ is equipped with a spherical structure and $\mathscr{M}$ is equipped with a $\mathscr{C}$-bimodule trace, we relate this to the notion of the crossed S-matrix associated with the $\mathscr{Z}(\mathscr{C})$-module category $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$.

Crossed S-matrices and character sheaves on unipotent groups

Let k be the algebraic closure of a finite field Fq of characteristic p. Let G be a connected unipotent group over k equipped with an Fq-structure given by a Frobenius map F:G⟶G. We will denote the corresponding algebraic group defined over Fq by G0. Character sheaves on G are certain special objects in the triangulated braided monoidal category DG(G) of bounded conjugation equivariant Q‾l-complexes (where l≠p is a prime number) on G. Boyarchenko has proved that the “trace of Frobenius” functions associated with F-stable character sheaves on G form an orthonormal basis of the space of class functions on G0(Fq) and that the matrix relating this basis to the basis formed by the irreducible characters of G0(Fq) is block diagonal with “small” blocks. In particular, there is a partition of the set of character sheaves as well as a partition of the set of irreducible characters of G0(Fq) into “small” families known as L-packets. In this paper we describe these block matrices relating character sheaves and irreducible characters corresponding to each L-packet. We prove that these matrices can be described as certain “crossed S-matrices” associated with each L-packet. We will also derive a formula for the dimensions of the irreducible representations of G0(Fq) in terms of certain modular categorical data associated with the corresponding L-packet. In fact we will formulate and prove more general results which hold for possibly disconnected groups G such that G∘ is unipotent. To prove our results, we will establish a formula (which holds for any algebraic group G) which expresses the inner product of the “trace of Frobenius” function of any F-stable object of DG(G) with any character of G0(Fq) (or of any of its pure inner forms) in terms of certain categorical operations.