p-Adic Lifting Problems and Derived Equivalences

Generalized pseudo-coefficients of discrete series of $p$-adic groups

Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $σ$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $σ.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.

等变裂扩张与奇异等价

Equivariant Cleft Extensions and Singular Equivalences

研究阿贝尔范畴的裂扩张的等变提升及其对奇异范畴的影响，证明裂扩张的限制函子诱导奇异等价当且仅当其等变对应物也诱导奇异等价，并应用于斜群环。

Relation of Orbital Integrals on SO(5) and PGL(2)

We relate the "Fourier" orbital integrals of corresponding spherical functions on the p-adic groups SO(5) and PGL(2). The correspondence is defined by a "lifting" of representations of these groups. This is a local "fundamental lemma" needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.

p进群的一般对偶及其应用

The generic dual of p-adic groups and applications

该研究给出准分裂典型群、其相似群及一般旋量群的一般对偶的统一分类，证明相关函子提升映射的满射性，还分析局部Langlands参数结构并为每个局部L-packet构造特异元。

Types and unitary representations of reductive p-adic groups

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.

On Tannaka Duality

The purpose of this work is twofold: to expose the existing similarities between the generalizations of the Tannaka and Galois theories, and on the other hand, to develop in detail our own treatment of part of the content of Joyal and Street [1] paper, generalizing from vector spaces to an abstract tensor category. We also develop in detail the proof of the Tannaka equivalence of categories in the case of vector spaces. Saavedra Rivano [2], Deligne and Milne [3] generalize classical Tannaka theory to the context of K-linear tensor (or monoidal) categories. They obtain a lifting-equivalence int…

混合特征下Gaitsgory中心函子与Arkhipov-Bezrukavnikov等价

Gaitsgory's central functor and the Arkhipov-Bezrukavnikov equivalence in mixed characteristic

本文通过发展Iwahori水平的Wakimoto滤过理论证明抛物水平p-adic Hecke栈邻近循环函子的反常t-正合性及ℰ₁-中心提升性，构造对应Arkhipov-Bezrukavnikov函子并证明其对经典群及E₆、E₇型例外群的等价性。

Cyclic odd degree base change lifting for unitary groups in three variables

Let E/F be a quadratic number (resp. p-adic) field extension, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (resp. admissible) representations from the unitary group U(3,E/F) associated with E/F to the unitary group U(3,F'E/F'). As a consequence, we classify the invariant packets of U(3,F'E/F'), namely those which contain (irreducible) automorphic (resp. admissible) representations which are invariant under the action of the Galois group Gal(F'E/E). To do this we use the trace formula technique, and well-known results on the base…

Automorphismes, graduations et categories triangulees

We give a moduli interpretation of the outer automorphism group Out of a finite dimensional algebra similar to that of the Picard group of a scheme. We deduce that Out^0 is invariant under derived and stable equivalences. This allows us to transfer gradings between algebras and gives rise to conjectural homological constructions of interesting gradings on block of finite groups with abelian defect. We give applications to the lifting of stable equivalences to derived equivalences. We give a counterpart of the invariance result for smooth projective varieties: the product Pic^0xAut^0 is invaria…

关于 $p$-进域 $F$ 上 $\mathrm{GL}_n(F)$ ($n>2$) 的 Hecke 代数的一个问题

A problem on Hecke algebras for $\mathrm{GL}_n(F)$ for $n>2$ over $p$-adic field $F$

研究 $p$-进域上 $\mathrm{GL}_n$ ($n>2$) 的 Hecke 代数，通过 Bernstein 分解和交织代数证明不同域上的 Hecke 代数 Morita 等价。

On semisimple l-modular Bernstein-blocks of a p-adic general linear group

Let $G_n=\operatorname{GL}_n(F)$, where $F$ is a non-archimedean local field with residue characteristic $p$. Our starting point is the Bernstein-decomposition of the representation category of $G_n$ over an algebraically closed field of characteristic $\ell \neq p$ into blocks. In level zero, we associate to each block a replacement for the Iwahori-Hecke algebra which provides a Morita-equivalence just as in the complex case. Additionally, we will explain how this gives rise to a description of an arbitrary $G_n$-block in terms of simple $G_m$-blocks (for $m\leq n$), paralleling the approach …

Blocks with quaternion defect group over a 2-adic ring: the case \tilde{A}_4

Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to $k\tilde A_4$. The main ingredients are Erdmann's classification of tame blocks and work of Cabanes and Picaronny on perfect isometries between tame blocks.

典型群的表层表示的内镜提升

Endoscopic liftings of epipelagic representations for classical groups

本文针对p进典型群的Reeder-Yu表层表示，结合Mœglin广尖点支集理论与Bushnell-Kutzko覆盖类型理论，用诱导类型显式刻画其到一般线性群的内镜提升，引入表层层并重证了M. Oi的相关结果。

Cuspidal irreducible complex or l-modular representations of quaternionic forms of p-adic classical groups for odd p

Given a quaternionic form G of a p-adic classical group (p odd) we classify all cuspidal irreducible representations of G with coefficients in an algebraically closed field of characteristic different from p. We prove two theorems: At first: Every irreducible cuspidal representation of G is induced from a cuspidal type, i.e. from a certain irreducible representation of a compact open subgroup of G, constructed from a beta-extension and a cuspidal representation of a finite group. Secondly we show that two intertwining cuspidal types of G are up to equivalence conjugate under some element of G.

Monodromy of Galois representations and equal-rank subalgebra equivalence

We study l-independence of monodromy groups G_l of any compatible system of l-adic representations (in the sense of Serre) of number field K assuming semisimplicity. We prove that the formal character of the derived group of the identity component of G_l is independent of l and the (complexified) Lie algebra g_l of G_l satisfies an equal-rank subalgebra equivalence for all l. This equivalence is equivalent to the l-independence of the number of A_n factors for all n belonging to {6,9,10,11,...} and the parity of A_4 factors in g_l.