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We develop a theory of modular forms on the groups SO ( 3 , n + 1 ) , n ≥ 3 . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO ( 3 , n + 1 ) , there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO ( 2 , n ) . As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E 8 has rational Fourier expansion.

In this paper, we give a precise definition of an analytic γ-factor of an irreducible representation of a classical group over a local function field of odd characteristic so that it satisfies some notable properties which are enough to define it uniquely. We use the doubling method to define the γ-factor, and the main theorem extends works of Lapid-Rallis, Gan, Yamana, and the author to a classical group over a local function field of odd characteristic.

We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig in Lfromto, (2011) and that the Kazhdan–Lusztig map in Kazhdan and Lusztig, (1998) is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the “skeleta” of affine Springer fibers.

We explain how to develop the twisted doubling integrals for Brylinski–Deligne extensions of connected classical groups. This gives a family of global integrals which represent Euler products for this class of nonlinear extensions.

This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.

The article is about the representation theory of an inner form G of a general linear group over a non-Archimedean local field. We introduce semisimple characters for G whose intertwining classes describe conjecturally via the Local Langlands correspondence the behaviour on wild inertia. These characters also play a potential role to understand the classification of irreducible smooth representations of inner forms of classical groups. We prove the intertwining formula for semisimple characters and an intertwining implies conjugacy like theorem. Further we show that endo-parameters for G, i.e. invariants consisting of simple endo-classes and a numerical part, classify the intertwining classes of semisimple characters for G. They should be the counter part for restrictions of Langlands-parameters to wild inertia under the Local Langlands correspondence.

Let [Formula: see text] be a classical group over an arbitrary field [Formula: see text], acting on an [Formula: see text]-dimensional [Formula: see text]-space [Formula: see text]. All those maximal subgroups of [Formula: see text] are classified each of which normalizes a solvable subgroup [Formula: see text] of [Formula: see text] not lying in [Formula: see text].

Let G be a classical group over an arbitrary field F, acting on an n-dimensional vector space V = V(n, F) over a field F. In this paper, we classify the maximal subgroups of G, which normalizes a solvable subgroup N of GL(n, F) not lying in F*1V.

We consider the subgroups $ H $ in a symplectic or orthogonal group over a finite field of odd characteristic such that $ O_{r}(H)\neq 1 $ for some odd prime $ r $ . We obtain a refinement of the well-known Aschbacher Theorem on subgroups of classical groups for this case.

Isospectral are the groups with coinciding sets of element orders. We prove that no finite group isospectral to a finite simple classical group has the exceptional groups of types $ E_{7} $ and $ E_{8} $ among its nonabelian composition factors.

Recently, Raimund Preusser displayed very short polynomial expressions of elementary generators in classical groups over an arbitrary commutative ring as products of conjugates of an arbitrary matrix and its inverse by absolute elementary matrices. In particular, this provides very short proofs for description of normal subgroups. In 2018, the author discussed various generalizations of these results to exceptional groups, specifically those of types E6 and E7. Here, a further variation of Preusser’s wonderful idea is presented. Namely, in the case of GL(n, R), n ≥ 4, similar expressions of elementary transvections as conjugates of g ∈ GL(n, R) and g−1 by relative elementary matrices x ∈ E(n, J) and then x ∈ E(n, R, J), for an ideal J ⊴ R, are obtained. Again, in particular, this allows to give very short proofs for the description of subgroups normalized by E(n, J) or E(n, R, J), and thus also of subnormal subgroups in GL(n, R).

Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow $p$-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic $p$-group acting on an elementary abelian $p$-group., Comment: Example 5.12 has been corrected and expanded

The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $\mathbb{R}^4/\Gamma$ by a finite subgroup $\Gamma\subset SU(2).$ We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm's equations [Che09] via the Nahm correspondence. Recently in [Nak18a] and [NT17], based on [Nak03], Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory [COS11]. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group., Comment: To appear in memorial volume dedicated to Michael Atiyah. 53 pages, 1 figure

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of [Formula: see text]-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

In Classical group testing, one is given a population of n items N which contains some defective d items inside. A group test (pool) is a test on a subset of N. Under the circumstance of no errors, a test is negative if the testing pool contains no defective items and the test is positive if the testing pool contains at least one defective item but we don’t know which one. The goal is to find all defectives by using as less tests as possible, mainly to minimize the number of tests (in the worst case situation). Let M(d, n) denote the minimum number of tests in the worst case situation where $$|N|=n$$ and d is the number of defectives. In this paper, we focus on estimating M(d, n) and obtain a better result than known ones in various cases of d and n.

This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of A-parameters which governs the branching laws for $GL_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in our earlier work including Bessel models and Fourier-Jacobi models., 70 pages, to appear in Compositio Math

We give a short combinatorial proof of Asai's decomposition formula for Lusztig induction of unipotent characters in groups of classical type, relying solely on the Mackey formula.

A GLd-pseudocharacter is a function from a group Γ to a ring k satisfying polynomial relations that make it “look like” the character of a representation. When k is an algebraically closed field of characteristic 0, Taylor proved that GLd-pseudocharacters of Γ are the same as degree-d characters of Γ with values in k, hence are in bijection with equivalence classes of semisimple representations Γ → GLd(k). Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group H over an algebraically closed field k of characteristic 0 and for any group Γ, there exists an infinite collection of functions and relations which are naturally in bijection with H(k)-conjugacy classes of semisimple homomorphisms Γ→ H(k). In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG algebra. We then define generating sets and generating relations for these objects and show that, for all H as above, the corresponding FFG-algebra is finitely presented up to radical. Hence one can always define H-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations up to radical of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of homomorphisms, following Larsen.

In this article we consider the set G G of rational points of a quaternionic form of a symplectic or an orthogonal group defined over a non-Archimedean local field of odd residue characteristic. We construct all full self-dual semisimple characters for G G and we classify their intertwining classes using endo-parameters. We compute the set of intertwiners between self-dual semisimple characters, and prove an intertwining and conjugacy theorem. Finally we count all G G -intertwining classes of full self-dual semisimple characters which lift to the same G ~ \tilde {G} -intertwining class of a full semisimple character for the ambient general linear group G ~ \tilde {G} for G G .

By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $\pi$ of a classical group (which may be not-quasi-split) over a nonarchimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing $\pi$. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to arbitrary tame Langlands parameters., Comment: 36 pages

We prove a slope 1 stability range for the homology of the symplectic, orthogonal and unitary groups with respect to the hyperbolic form, over any fields other than $F_2$, improving the known range by a factor 2 in the case of finite fields. Our result more generally applies to the automorphism groups of vector spaces equipped with a possibly degenerate form (in the sense of Bak, Tits and Wall). For finite fields of odd characteristic, and more generally fields in which -1 is a sum of two squares, we deduce a stability range for the orthogonal groups with respect to the Euclidean form, and a corresponding result for the unitary groups. In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$, and use it to recover also the improved range of Galatius-Kupers-Randal-Williams in the case of finite fields, at the characteristic., v2: Revision. Now recovers the Galatius-Kupers-Randal-Williams improved stability range for general linear groups over finite fields

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group $G$ over $\mathbf {Z}$ , and provide numerical applications in absolute rank $\leq 8$ . Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of $\mathrm{GL}_{n}$ over $\mathbf {Q}$ ( $n$ arbitrary) whose motivic weight is $\leq 24$ . In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on $\mathrm{GL}_{n}$ , and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of $G$ , for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for $\mathrm{Sp}_{2g}(\mathbf {Z})$ : we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.

We analyse the complexity of constructing involution centralisers in unitary groups over fields of odd order. In particular, we prove logarithmic bounds on the number of random elements required to generate a subgroup of the centraliser of a strong involution that contains the last term of its derived series. We use this to strengthen previous bounds on the complexity of recognition algorithms for unitary groups in odd characteristic. Our approach generalises and extends two previous papers by the second author and collaborators on strong involutions and regular semisimple elements of linear groups., Comment: 48 pages. Revised after suggestions from Eamonn O'Brien and an anonymous referee. To appear J. Algebra

For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma .

This article is part of a project which consists of investigating Arthur packets for real classical groups. Our goal is to give an explicit description of these packets and to establish the multiplicity one property (which is known to hold for $p$-adic and complex groups). The main result in this paper is a construction of packets from unipotent packets on $c$-Levi factors using cohomological induction. An important tool used in the argument is a statement of commutativity between cohomological induction and spectral endoscopic transfer.

We study the top Fourier coefficients of isobaric automorphic representations of $${\mathrm {GL}}_n({\mathbb {A}})$$ of the form $$\begin{aligned} \Pi _{\underline{s}} = {\mathrm {Ind}}^{{\mathrm {GL}}_n({\mathbb {A}})}_{P({\mathbb {A}})} \Delta (\tau _1,b_1) |\cdot |^{s_1} \otimes \cdots \otimes \Delta (\tau _r,b_r) |\cdot |^{s_r}, \end{aligned}$$ where $$s_i\in {\mathbb {C}}$$ , $$\Delta (\tau _i,b_i)$$ ’s are Speh representations in the discrete spectrum of $${\mathrm {GL}}_{a_ib_i}({\mathbb {A}})$$ with $$\tau _i$$ ’s being unitary cuspidal representations of $${\mathrm {GL}}_{a_i}({\mathbb {A}})$$ , and $$n = \sum _{i=1}^r a_ib_i$$ . In particular, we prove a part of a conjecture of Ginzburg, and also a conjecture of Jiang under certain assumptions. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.

Given a (complex, smooth) irreducible representation $\pi$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $\sigma$ of a classical group, we show that the (normalized) parabolic induction $\pi\rtimes\sigma$ is reducible if there exists $\rho$ in the supercuspidal support of $\pi$ such that $\rho\rtimes\sigma$ is reducible. In special cases we also give irreducibility criteria for $\pi\rtimes\sigma$ when the above condition is not satisfied., Oberwolfach Preprints;2017,09

In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$ a reductive group, to geometric $\ell$-adic representations. In this note we take $G$ to be a classical group and construct many examples of $G$-irreducible representations to which these new lifting methods apply, but to which the lifting methods provided by potential automorphy theorems do not.

We introduceadmissible collectionsfor a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy theQuillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of|G|\lvert G\rvert. Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex,Ann. of Math. (2)137(1993), 3, 473–529], our techniques are simpler.