Hecke Modules Research Articles

Regular Schur labeled skew shape posets and their 0-Hecke modules

Abstract Assuming Stanley’s P-partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets P with underlying set $\{1, 2, \ldots , |P|\}$ such that the P-partition generating function is symmetric and the set of linear extensions of P, denoted $\Sigma _L(P)$ , is a left weak Bruhat interval in the symmetric group $\mathfrak {S}_{|P|}$ . We describe the permutations in $\Sigma _L(P)$ in terms of reading words of standard Young tableaux when P is a regular Schur labeled skew shape poset, and classify $\Sigma _L(P)$ ’s up to descent-preserving isomorphism as P ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$ -Hecke modules $\mathsf {M}_P$ associated with regular Schur labeled skew shape posets P up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the finite posets P whose linear extensions form a dual plactic-closed subset of $\mathfrak {S}_{|P|}$ . Using this characterization, we construct distinguished filtrations of $\mathsf {M}_P$ with respect to the Schur basis when P is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$ -Hecke modules $\mathsf {M}_P$ are also discussed.

Freeness of Hecke modules at non-minimal levels

Freeness of Hecke modules at non-minimal levels

ıSchur duality and Kazhdan-Lusztig basis expanded

Expanding the classical works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an ıSchur duality between an ıquantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wang's quasi-split ıSchur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the ıcanonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the ıSchur duality.

Overconvergent Hilbert modular forms via perfectoid modular varieties

We give a new construction of p-adic overconvergent Hilbert modular forms by using Scholze’s perfectoid Shimura varieties at infinite level and the Hodge–Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary our definitions in p-adic families. We show that the resulting spaces are isomorphic as Hecke modules to earlier constructions of Andreatta, Iovita and Pilloni. Finally, we give a new direct construction of sheaves of arithmetic Hilbert modular forms, and compare this to the construction via descent from the geometric case.

The Derived Hecke Algebra for Dihedral Weight One Forms

We study the action of the derived Hecke algebra in the setting of dihedral weight one forms and prove a conjecture of the second- and fourth- named authors relating this action to certain Stark units associated to the symmetric square L-function. The proof exploits the theta correspondence between various Hecke modules as well as the ideas of Merel and Lecouturier on higher Eisenstein elements.

Weak Bruhat interval modules of the 0-Hecke algebra

The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restriction, and (anti-)involution twists of weak Bruhat interval modules.

Functorial properties of pro‐p‐Iwahori cohomology

Suppose $F$ is a finite extension of $\mathbb{Q}_p$, $G$ is the group of $F$-points of a connected reductive $F$-group, and $I_1$ is a pro-$p$-Iwahori subgroup of $G$. We construct two spectral sequences relating derived functors on mod-$p$ representations of $G$ to the analogous functors on Hecke modules coming from pro-$p$-Iwahori cohomology. More specifically: (1) using results of Ollivier--Vign\'eras, we provide a link between the right adjoint of parabolic induction on pro-$p$-Iwahori cohomology and Emerton's functors of derived ordinary parts; and (2) we establish a "Poincar\'e duality spectral sequence" relating duality on pro-$p$-Iwahori cohomology to Kohlhaase's functors of higher smooth duals. As applications, we calculate various examples of the Hecke modules $\textrm{H}^i(I_1,\pi)$.

Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

Let N and p be primes such that p divides the numerator of $$\frac{N-1}{12}$$ . In this paper, we study the rank $$g_p$$ of the completion of the Hecke algebra acting on cuspidal modular forms of weight 2 and level $$\Gamma _0(N)$$ at the p-maximal Eisenstein ideal. We give in particular an explicit criterion to know if $$g_p \ge 3$$ , thus answering partially a question of Mazur. In order to study $$g_p$$ , we develop the theory of higher Eisenstein elements, and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler mass formula in characteristic p.

Supersingular Hecke modules as Galois representations

Let $F$ be a local field of mixed characteristic, let $k$ be a finite extension of its residue field, let ${\mathcal H}$ be the pro-$p$-Iwahori Hecke $k$-algebra attached to ${\rm GL}_{d+1}(F)$ for some $d\ge1$. We construct an exact and fully faithful functor from the category of supersingular ${\mathcal H}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over $k$. More generally, for a certain $k$-algebra ${\mathcal H}^{\sharp}$ surjecting onto ${\mathcal H}$ we define the notion of $\sharp$-supersingular modules and construct an exact and fully faithful functor from the category of $\sharp$-supersingular ${\mathcal H}^{\sharp}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over $k$.

On a rigidity of some modular Galois deformations

Let $F$ be a totally real field and $\mathbf{\rho} = (\rho_{\lambda})_{\lambda}$ be a compatible system of two dimensional $\lambda$-adic representations of the Galois group of $F$. We assume that $\mathbf{\rho}$ has a residually modular $\lambda$-adic realization for some $\lambda$. In this paper, we consider local behaviors of modular deformations of $\lambda$-adic realizations of $\mathbf{\rho}$ at unramified primes. In order to control local deformations at specified unramified primes, we construct certain Hecke modules. Applying Kisin's Taylor-Wiles system, we obtain an $R = T$ type result supplemented with local conditions at specified unramified primes. As a consequence, we shall show a potential rigidity of some modular deformations of infinitely many $\lambda$-adic realizations of $\mathbf{\rho}$.

On the Cohomology of Congruence Subgroups of GL3 over the Eisenstein Integers

Let F be the imaginary quadratic field of discriminant −3 and its ring of integers. Let Γ be the arithmetic group and for any ideal let be the congruence subgroup of level consisting of matrices with bottom row In this paper we compute the cohomology spaces as a Hecke module for various levels where ν is the virtual cohomological dimension of Γ. This represents the first attempt at such computations for GL3 over an imaginary quadratic field, and complements work of Grunewald–Helling–Mennicke and Cremona, who computed the cohomology of over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on

Newforms of half-integral weight: the minus space of Sk+1/2(Γ0(8M))

We compute generators and relations for a certain 2-adic Hecke algebra of level 8 associated with the double cover of SL2 and a 2-adic Hecke algebra of level 4 associated with PGL2. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the 2-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight k + 1/2 and level 8M where M is odd and square-free. Using these operators and our previous results on half-integral weight forms of level 4M we define a subspace of the space of half-integral weight forms as a common -1 eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace, which we call the minus space, is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight 2k and level 4M. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.

Hecke modules based on involutions in extended Weyl groups

Let X X be the group of weights of a maximal torus of a simply connected semisimple group over C \mathbf {C} and let W W be the Weyl group. The semidirect product W ( ( Q ⊗ X ) / X ) W((\mathbf {Q}\otimes X)/X) is called an extended Weyl group. There is a natural C ( v ) \mathbf {C}(v) -algebra H \mathbf {H} called the extended Hecke algebra with basis indexed by the extended Weyl group which contains the usual Hecke algebra as a subalgebra. We construct an H \mathbf {H} -module with basis indexed by the involutions in the extended Weyl group. This generalizes a construction of the author and Vogan.

Hecke modules for arithmetic groups via bivariant K-theory

Let $\Gamma$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $\Gamma$-$C^{*}$-algebra on which the commensurator of $\Gamma$ acts with Hecke operators. When $\Gamma$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $\Gamma$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $\Gamma$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the $KK$-groups associated to an arithmetic group $\Gamma$ become true Hecke modules. We conclude by discussing Hecke equivariant maps in $KK$-theory in great generality and apply this to the Borel-Serre compactification as well as various noncommutative compactifications associated with $\Gamma$. Along the way we discuss the relation between the $K$-theory and the integral cohomology of low-dimensional manifolds as Hecke modules.

Freeness of spherical Hecke modules of unramified U(2,1) in characteristic p

Let F be a non-archimedean local field of odd residue characteristic p. Let G be the unramified unitary group U(2,1)(E/F) in three variables, and K be a maximal compact open subgroup of G. For an irreducible smooth representation σ of K over F‾p, we prove that the compactly induced representation indKGσ is free of infinite rank over the spherical Hecke algebra H(K,σ).

On Hecke modules generated by eta-products and imaginary quadratic fields

On Hecke modules generated by eta-products and imaginary quadratic fields

Modular elliptic curves over the field of twelfth roots of unity

In this paper we perform an extensive study of the spaces of automorphic forms for $\text{GL}_{2}$ of weight $2$ and level $\mathfrak{n}$, for $\mathfrak{n}$ an ideal in the ring of integers of the quartic CM field $\mathbb{Q}({\it\zeta}_{12})$ of twelfth roots of unity. This study is conducted through the computation of the Hecke module $H^{\ast }({\rm\Gamma}_{0}(\mathfrak{n}),\mathbb{C})$, and the corresponding Hecke action. Combining this Hecke data with the Faltings–Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curves over this field.Supplementary materials are available with this article.

Hecke stability and weight $$1$$ 1 modular forms

The Galois representations associated to weight 1 eigenforms over $$\bar{\mathbb {F}}_{p}$$ are remarkable in that they are unramified at $$p$$ , but the effective computation of these modular forms presents challenges. One complication in this setting is that a weight 1 cusp form over $$\bar{\mathbb {F}}_{p}$$ need not arise from reducing a weight 1 cusp form over $$\bar{\mathbb {Q}}$$ . In this article we propose a unified Hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a finite-dimensional Hecke module of ratios of modular forms must consist of genuine modular forms. We conclude with some applications of the Hecke stability method that are motivated by the refined inverse Galois problem.

Hecke modules and supersingular representations of U(2,1)

Let F F be a nonarchimedean local field of odd residual characteristic p p . We classify finite-dimensional simple right modules for the pro- p p -Iwahori-Hecke algebra H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) , where G G is the unramified unitary group U ( 2 , 1 ) ( E / F ) \textrm {U}(2,1)(E/F) in three variables. Using this description when C = F ¯ p C = \overline {\mathbb {F}}_p , we define supersingular Hecke modules and show that the functor of I ( 1 ) I(1) -invariants induces a bijection between irreducible nonsupersingular mod- p p representations of G G and nonsupersingular simple right H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) -modules. We then use an argument of Paškūnas to construct supersingular representations of G G .

Categorification of a parabolic Hecke module via sheaves on moment graphs

We investigate certain categories, associated by Fiebig with the geometric representation of a Coxeter system, via sheaves on Bruhat graphs. We modify Fiebig's definition of translation functors in order to extend it to the singular setting and use it to categorify a parabolic Hecke module. As an application we obtain a combinatorial description of indecomposable projective objects of (truncated) non-critical singular blocks of (a deformed version of) category $\mathcal{O}$, using indecomposable special modules over the structure algebra of the corresponding Bruhat graph.