Hecke algebras, new vectors and new forms on $Γ_0(m)$

3和5级的模2赫克代数

Mod-$2$ Hecke algebras of level $3$ and $5$

该研究利用形变理论分析素数级模2模形式的大赫克代数，证明3、5级时其极大约化商与泛形变环的对应商同构并确定其结构，还描述了模p赫克代数的自然分次。

Hecke algebras related to the unimodular and modular groups over hermitian fields and definite quaternion algebras

We investigate the structure of the Hecke algebras related to the unimodular and modular group over hermitian fields and definite quaternion algebras. In particular we show that in general there is no decomposition into primary components. We can give a set of generators and in some special cases we deduce the law of interchange of the Siegel $Φ$-operator.

The amplification method in the GL(3) Hecke algebra

This article contains all of the technical ingredients required to implement an effective, explicit and unconditional amplifier in the context of GL(3) automorphic forms. In particular, several coset decomposition computations in the GL(3) Hecke algebra are explicitly done.

派生Hecke作用在除代数平凡上同调上的推广

Derived Hecke action on the trivial cohomology of division algebras

本文将Venkatesh的结构定理推广到所有数域上的除代数，证明Hecke平凡上同调的稳定子模是严格派生Hecke代数作用下的自由模，并建立保持有理结构的理性形式。

Generic pro-$p$ Hecke algebras

This is an extended and corrected version of the author's Diplomarbeit. A class of algebras called generic pro-$p$ Hecke algebras is introduced, enlarging the class of generic Hecke algebras by considering certain extensions of (extended) Coxeter groups. Examples of generic pro-$p$ Hecke algebras are given by pro-$p$-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion of an orientation of a Coxeter group is introduced and used to define `Bernstein maps' intimately related to Bernstein's presentation and to Cherednik's cocycle. It is shown that certain relations in the Hecke algebra …

A tale of two Hecke algebras

We use Bernstein's presentation of the Iwahori-Matsumoto Hecke algebra to obtain a simple proof of the Satake isomorphism and, in the same stroke, compute the center of the Iwahori-Matsumoto Hecke algebra.

Parabolic inductions for pro-$p$-Iwahori Hecke algebras

We give some properties of parabolic inductions and their adjoint functors for pro-$p$-Iwahori Hecke algebras.

Extension between simple modules of pro-$p$-Iwahori Hecke algebras

We calculate the extension groups between simple modules of pro-$p$-Iwahori Hecke algebras.

The Spectral Hecke Algebra

We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the Langlands correspondence, i.e. on moduli spaces of Galois representations. We verify the simplest form of derived local-global compatibility between the action of the spectral Hecke algebra on the derived Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh's derived Hecke algebra on the cohomology of arithmetic groups.

On the AF-algebra of a Hecke eigenform

An AF-algebra is assigned to each cusp form f of weight two; we study properties of this operator algebra, when f is a Hecke eigenform.

Hecke algebras for tame supercuspidal types

Let $F$ be a non-archimedean local field of residue characteristic $p\neq 2$. Let $G$ be a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Yu constructed types which are called tame supercuspidal types and conjectured that Hecke algebras associated with these types are isomorphic to Hecke algebras associated with depth-zero types of some twisted Levi subgroups of $G$. In this paper, we prove this conjecture. We also prove that the Hecke algebra associated with a regular supercuspidal type is isomorphic to the group algebra of a certain abelian group.

Generic Hecke algebra and theta correspondence over finite fields

We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at $1$ can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we generalize the results of Aubert-Michel-Rouquier an…

Involutions on pro-$p$-Iwahori Hecke algebras

The pro-$p$-Iwahori Hecke algebra has an involution $ι$ defined in terms of Iwahori-Matsumoto basis. Then for a module $π$ of pro-$p$-Iwahori Hecke, $π^ι= π\circ ι$ is also a module. We calculate $π^ι$ for simple modules $π$. We also calculate the dual of $π$.

A Satake homomorphism for the $\bmod \, p$ derived Hecke algebra

We explore the structure of the derived spherical Hecke algebra of a $p$-adic group, a graded associative algebra whose degree $0$ subalgebra is the classical spherical Hecke algebra. Working with $\mathbb Z/ p^a$ coefficients, we establish a Satake homomorphism relating this graded algebra to the corresponding graded algebra for the torus. We investigate the image of this homomorphism in degree $1$, as well as other properties, such as transitivity with respect to inclusion of Levi subgroups.

Iwahori-Hecke algebras for p-adic loop groups

This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dime…