Caroline Lassueur, Jacques Thévenaz
For the group of endo-permutation modules of a finite \(p\)-group, there is a surjective reduction modulo \(p\) homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic \(p\). We prove that this reduction map always has a section which is a group homomorphism.
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We introduce endo-permutation Mackey functors and the Mackey-Dade group of a $p$-group and study their basic properties. We exhibit relations between the Mackey-Dade group of a finite $p$-group $P$ and the Dade groups of the Weyl groups of all subgroups of $P$. As a result, we show that the Mackey-Dade group tensored with $\mathbb Q$ over $\mathbb Z$ is the kernel of the linearization map for Mackey functors, introduced by the author.
We verify Dade's invariant conjecture for the simple Ree groups 2F4(2^{2n+1}) for all n > 0 in the defining characteristic, i.e., in characteristic 2. This completes the proof of Dade's conjecture for the simple Ree groups 2F4(2^{2n+1}).
A Galois-equivariant Dade-Glauberman-Nagao correspondence with central defect
本文为中心亏式群上的广义 Dade-Glauberman-Nagao 对应构造了高度为零的普通不可约特征之间的伽罗瓦等变对应，并满足块关系，适用于 Alperin-McKay-Navarro 约化。
Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade $kG$-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group $D(G)$ defined by Lassueur. We also consider the subgroup $D^Ω (G)$ of $D(G)$ generated by relative syzygies $Ω_X$, where $X$ is a finite $G$-set. If $C(G,p)$ denotes the group of superclass functions…
Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace $p$ by a set of primes pi and prove a pi-version of Dade's conjecture for pi-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a pi-version of Alperin's weight conjecture previously established by the authors.
A reduction theorem for the Character Triple Conjecture
本文证明特征三元组猜想对所有有限群成立只需对所有拟单群成立，从而解决了Dade猜想自约化形式的存在性问题，并给出了Späth猜想的一般形式。
In this paper, we prove Späth's Character Triple Conjecture for $p$-solvable groups. This is a conjecture proposed by Späth during the reduction process of Dade's Projective Conjecture to quasisimple groups. In addition, as suggested by Isaacs and Navarro, we take into account the $p$-residue of characters.
Fusion-stable endosplit $p$-permutation resolutions
本文证明饱和融合系下满足特定 Dade 类条件的稳定内置换模具有内分裂 $p$-置换分解，并由此推出 Morita 等价的两个块是华丽 Rickard 等价的。
Let $K$ be a normal subgroup of the finite group $H$. To a block of a $K$-interior $H$-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to a block given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results "Block extensions" Section 12.
The Thompson group F has a natural unitary representation on $H=L^2[0,1]$. With some projections, we construct a family of projective unitary representations on a Fermionic Fock space associated with $H$. It comes from the representation of the associated CAR algebra. After $H^2(F;S^1)$ is obtained, we mainly study whether any of these projective unitary representations can be lifted to an ordinary one. We will discuss the lifting problem of these projective representations.
On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups
该研究针对有限群主块的Eaton-Moretó猜想，通过构造主p-块中满足次数条件的不可约特征标，结合Dade射影猜想证明了该猜想成立。
We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1. This is also related to various questions about Fried's modular tower program and properties of Hurwitz spaces of covers of curves.
We present a new criterion to predict if a character of a finite group extends. Let $G$ be a finite group and $p$ a prime. For $N\lhd G$, we consider $p$-blocks $b$ and $b'$ of $N$ and ${\rm N}_N(D)$, respectively, with $(b')^N=b$, where $D$ is a defect group of $b'$. Under the assumption that $G$ coincides with a normal subgroup $G[b]$ of $G$, which was introduced by Dade early 1970's, we give a character correspondence between the sets of all irreducible constituents of $ϕ^G$ and those of $(ϕ')^{{\rm N}_G(D)}$ where $ϕ$ and $ϕ'$ are irreducible Brauer characters in $b$ and $b'$, respectively…
We formulate the transfer factor of character lifting from orthogonal groups to symplectic groups by Adams in the framework of symplectic Dirac cohomology for the Lie superalgebras and the Rittenberg-Scheunert correspondence of representations of the Lie superalgebra $\fro\frsp(1|2n)$ and the Lie algebra $\fro(2n+1)$. This leads to formulation of a direct lifting of characters from the linear symplectic group $Sp(2n,\bbR)$ to its nonlinear covering metaplectic group $Mp(2n,\bbR)$.
In this paper we study the theta lifting for loop groups and extend the classical tower property established by S. Rallis to the loop setting. As an application we obtain cusp forms on loop groups, and we give the first example where the cusp forms constructed using this method are nonvanishing.
Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct summand, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from R_p to abelian groups are exactly the rational p-biset functors. This yields in particular very effi…
Let N be the normalizer of a maximal torus T in a split reductive group over F_q and let w be an involution in the Weyl group N/T. We construct explicitly a lifting n of w in N such that the image of n under the Frobenius map is equal to the inverse of n.
A local-global principle for unipotent characters
本文针对类型A、B、C的单连通有限约化群，将Dade猜想和Späth特征三元组猜想适配到单酉特征，给出计数公式，并构造与特征三元组同构相容的参数化，提供局部-全局原理的几何版本。
Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\varepsilon$ of $\tilde{G}$ of finite order, let $G$ denote the connected part of the group of $\varepsilon$-fixed points. Then there exists a lifting from packets of representations of $G(k)$ to packets for $\tilde{G}(k)$. In the case of Deligne-Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.
For a positive integer $g$, let $\mathrm{Sp}_{2g}(R)$ denote the group of $2g \times 2g$ symplectic matrices over a ring $R$. Assume $g \ge 2$. For a prime number $\ell$, we give a self-contained proof that any closed subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$ which surjects onto $\mathrm{Sp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ must in fact equal all of $\mathrm{Sp}_{2g}(\mathbb{Z}_\ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.
We show that the Humbert surfaces rationally generate the Picard groups of Siegel modular threefolds.
One aspect of the Langlands program for linear groups is lifting of characters, which relates virtual representations on a group $G$ with those on an endoscopic group for $G$. The goal of this paper is to extend this theory to nonlinear two-fold covers of real groups in the simply laced case. Suppose $\tG$ is a two-fold cover of a real reductive group $G$. The main result is that there is an operation, denoted $\Lift_G^{\tG}$, taking a stable virtual character of $G$ to 0 or a virtual genuine character of $\tG$, and $\Lift_G^{\tG}(Θ_π)$ may be explicitly computed if $π$ is a stable sum of stan…
We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture.
Let N be the normalizer of a maximal torus T in a split reductive group over F_q, and let w be an involution in the Weyl group N/T. We construct a section of W satisfying the braid relations, such that the image of the lift n of w under the Frobenius map is equal to the inverse of n.
We give a necessary and sufficient condition for a modular representation of a group $G=C_{p^h} \rtimes C_m$ in a field of characteristic zero to be lifted to a representation over local principal ideal domain of characteristic zero containing the $p^h$ roots of unity.
We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
Any sufficiently often differentiable curve in the orbit space of a compact Lie group representation can be lifted to a once differentiable curve into the representation space.
We characterize those regular, holomorphic or formal maps into the orbit space $V/G$ of a complex representation of a finite group $G$ which admit a regular, holomorphic or formal lift to the representation space $V$. In particular, the case of complex reflection groups is investigated.
We prove that all endo-$p$-permutation modules for a finite group are liftable from characteristic $p>0$ to characteristic $0$.
We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
Theta liftings for $(\mathrm{GL}_n,\mathrm{GL}_n)$ type dual pairs of loop groups
该研究针对圈群$(\n\thrm{GL}_n,\n\thrm{GL}_n)$型对偶对的相关问题，证明了由经典尖点形式诱导的圈群$\n\thrm{GL}_n$上尖点形式的theta提升是艾森斯坦级数，为圈群表示论相关研究提供了新的结论。
Given an abelian $p$-group $G$ of rank $n$, we construct an action of the torus $\mathbb{T}^n$ on the stable module $\infty$-category of $G$-representations over a field of characteristic $p$. The homotopy fixed points are given by the $\infty$-category of module spectra over the Tate construction of the torus. The relationship thus obtained arises from a Galois extension in the sense of Rognes, with Galois group given by the torus. As one application, we give a homotopy-theoretic proof of Dade's classification of endotrivial modules for abelian $p$-groups. As another application, we give a sl…
We improve the main results in the paper from the title using a recent refinement of Bronshtein's theorem due to Colombini, Orrú, and Pernazza. They are then in general best possible both in the hypothesis and in the outcome. As a consequence we obtain a result on lifting smooth mappings in several 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…
Suppose $G$ is a reductive algebraic group, $T$ is a Cartan subgroup, $N=\text{Norm}(T)$, and $W=N/T$ is the Weyl group. If $w\in W$ has order $d$, it is natural to ask about the orders lifts of $w$ to $N$. It is straightforward to see that the minimal order of a lift of $w$ has order $d$ or $2d$, but it can be a subtle question which holds. We first consider the question of when $W$ itself lifts to a subgroup of $N$ (in which case every element of $W$ lifts to an element of $N$ of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all l…
Any sufficiently often differentiable curve in the orbit space $V/G$ of a real finite-dimensional orthogonal representation $G \to O(V)$ of a finite group $G$ admits a differentiable lift into the representation space $V$ with locally bounded derivative. As a consequence any sufficiently often differentiable curve in the orbit space $V/G$ can be lifted twice differentiably. These results can be generalized to arbitrary polar representations. Finite reflection groups and finite rotation groups in dimensions two and three are discussed in detail.
We study splitting densities of primitive elements of a discrete subgroup of a connected non-compact semisimple Lie group of real rank one with finite center in another larger such discrete subgroup. When the corresponding cover of such a locally symmetric negatively curved Riemannian manifold is regular, the densities can be easily obtained from the results due to Sarnak or Sunada. Our main interest is a case where the covering is not necessarily regular. Specifically, for the cases of the modular group and its congruence subgroups, we determine the splitting densities explicitly. As an appli…
Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with Frobenius kernel $G_{(1)}$. It is known that when $p\ge 2h-2$, where $h$ is the Coxeter number of $G$, the projective indecomposable $G_{(1)}$-modules (PIMs) lift to $G$, and this has been conjectured to hold in all characteristics. In this paper, we explore the lifting problem via extensions of algebraic groups, following the work of Parshall and Scott who in turn build upon ideas due to Donkin. We prove various results which augment this approach, and as an application demo…
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.
We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor-Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GL_n.
Lifting representations of finite reductive groups II: Explicit conorms
本文针对有限约化群的表示提升问题，通过证明映射与同源、Weil限制相容及分解为简单映射复合，以及构造显式k-态射两种方式，让稳定半单共轭类映射更显式，且该提升在重要情形下与Shintani提升一致。
We restrict a Siegel modular cusp form of degree 2 and square free level that is a Yoshida lifting (a lifting from the orthogonal group of a definite quaternion algebra) to the embedded product of two half planes and compute the Petersson product against the product of two elliptic cuspidal Hecke eigenforms. The square of this integral can be explicitly expressed in terms of the central critical value of an L-function attached to the situation. The result is related to a conjecture of Gross and Prasad about restrictions of automorphic representations of special orthogonal groups.
Lifting banal representations of classical groups
本文研究了局部非阿基米德域上辛群或分裂正交群的平凡素数表示，证明了任何光滑不可约表示均可提升至Q_lbar，同时探讨了更一般的经典群的类似结果，并应用此证明了强平凡情况下辛-正交或单元ary对的Howe对偶性。
We study a new lifting of automorphic representations using the theta representation $Θ$ on the $4$-fold cover of the symplectic group, $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to $Θ$. We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. Assuming this, we compute the effect of our lift on Arthur parameters and show th…
Global liftings between inner forms of GSp(4)
本文研究数域上约化群到其拟分裂内形式的自守提升，证明许多情形下非平凡上同调弱整体提升的存在性，并针对 GSp(4) 内形式解决局部提升的精细细节，证实了 Ibukiyama-Kitayama 关于平方自由水平参数模新形式的猜想。
Liftable braids and the coloured braid groupoid
本文针对圆盘的分歧覆盖，将辫群的提升同态推广到着色辫群胚到映射类群胚的映射，并刻画了每个着色辫的提升，恢复了经典结果。
For two derived equivalent $k$-algebras $\barΛ$ and $\barΓ$, we introduce a correspondence between $\OO$-orders reducing to $\barΛ$ and $\OO$-orders reducing to $\barΓ$. We outline how this may be used to transfer properties like uniqueness (or non-existence) of a lift between $\barΛ$ and $\barΓ$. As an application, we look at tame algebras of dihedral type with two simple modules, where, most notably, we are able to show that among those algebras only the algebras $\mathcal D^{κ,0}(2A)$ and $\mathcal D^{κ,0}(2B)$ can actually occur as basic algebras of blocks of group rings of finite groups.
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。
Projective representations on operator algebras
该研究针对算子代数上的射影酉表示的2上同调提升障碍建立新限制，证明江-苏代数上的射影表示可提升为群表示，刻画有限群在两类代数上的射影表示的提升障碍值，还证明性质(T)群的部分2-上闭链无法作为超有限II₁因子上射影表示的提升不变量。
Given an orthogonal representation of a compact group, we show that any element of the connected component of the isometry group of the orbit space lifts to an equivariant isometry of the original Euclidean space. Corollaries include a simple formula for this connected component, which applies to ``most'' representations.
Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual group of F one can associate certain meromorphic function on the set of isomorphism classed of irreducible representations of G (which we call gamma-functions). In this paper we describe a general framework for an explicit construction of these gamma-functions. We make this idea precise in certain special cases. As a byproduct we give explicit conjectural for…
Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $Γ$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the connected part of the group of $Γ$-fixed points in $\tilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $(\tilde{G},Γ)$, and consider any group $G$, not just the $Γ$-fixed points of $\tilde{G}$, satisfying the axioms. (In fact, the axioms do not require $Γ$ to act on all of $\tilde{G}$.) If both $\tilde{G}$ and $G$ …
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