Split Classical Groups Research Articles

On the Weak Local Arthur Packets Conjecture for Split Classical Groups

Abstract Recently, motivated by the theory of real local Arthur packets, making use of the wavefront sets of representations over non-Archimedean local fields $F$, Ciubotaru, Mason-Brown, and Okada defined the weak local Arthur packets consisting of certain unipotent representations and conjectured that they are unions of local Arthur packets. In this paper, we prove this conjecture for $\textrm{Sp}_{2n}(F)$ and split $\textrm{SO}_{2n+1}(F)$ with the assumption of the residue field characteristic of $F$ being large. In particular, this implies the unitarity of these unipotent representations. We also discuss the generalization of the weak local Arthur packets beyond unipotent representations, which reveals the close connection with a conjecture of Jiang on the structure of wavefront sets for representations in local Arthur packets.

Bounded reduction of orthogonal matrices over polynomial rings

We prove that a matrix from the split orthogonal group over a polynomial ring with coefficients in a small-dimensional ring can be reduced to a smaller matrix by a bounded number of elementary orthogonal transformations. The bound is given explicitly. This result is an effective version of the early stabilisation of the orthogonal K1 functor proven by Suslin and Kopeiko in [39]. Since the similar effective results for special linear and symplectic groups are obtained by Vaserstein in [44], the present paper closes the problem for split classical groups.

Two identities relating Eisenstein series on classical groups

In this paper we introduce two general identities relating Eisenstein series on split classical groups, as well as double covers of symplectic groups. The first identity can be viewed as an extension of the doubling construction introduced in [CFGK19]. The second identity is a generalization of the descent construction studied in [GRS11].

On typical representations for depth-zero components of split classical groups

Let G \mathbf {G} be a split classical group over a non-Archimedean local field F F with the cardinality of the residue field q F > 5 q_F>5 . Let M M be the group of F F -points of a Levi factor of a proper F F -parabolic subgroup of G \mathbf {G} . Let [ M , σ M ] M [M, \sigma _M]_M be an inertial class such that σ M \sigma _M contains a depth-zero Moy–Prasad type of the form ( K M , τ M ) (K_M, \tau _M) , where K M K_M is a hyperspecial maximal compact subgroup of M M . Let K K be a hyperspecial maximal compact subgroup of G ( F ) \mathbf {G}(F) such that K K contains K M K_M . In this article, we classify s \mathfrak {s} -typical representations of K K . In particular, we show that the s \mathfrak {s} -typical representations of K K are precisely the irreducible subrepresentations of ind J K ⁡ λ \operatorname {ind}_J^K\lambda , where ( J , λ ) (J, \lambda ) is a level-zero G G -cover of ( K ∩ M , τ M ) (K\cap M, \tau _M) .

Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula

We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we explicitly compute the Euler-Poincare characteristic of the level one discrete automorphic spectrum of $\mathbf{G}$ with respect to a finite-dimensional representation of $\mathbf{G}(\mathbb{R})$. For such a group $\mathbf{G}$, Arthur's endoscopic classification of the discrete spectrum allows to analyse precisely this Euler-Poincare characteristic. For example one can deduce the number of everywhere unramified automorphic representations $\pi$ of $\mathbf{G}$ such that $\pi_{\infty}$ is isomorphic to a given discrete series representation of $\mathbf{G}(\mathbb{R})$. Dimension formulae for the spaces of vector-valued Siegel modular forms are easily derived.

ON THE LOCAL LANGLANDS CORRESPONDENCE FOR SPLIT CLASSICAL GROUPS OVER LOCAL FUNCTION FIELDS

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

The LS method for the classical groups in positive characteristic and the Riemann Hypothesis

We provide a definition for an extended system of $\gamma$-factors for products of generic representations $\tau$ and $\pi$ of split classical groups or general linear groups over a non-archimedean local field of characteristic $p$. We prove that our $\gamma$-factors satisfy a list of axioms (under the assumption $p\neq 2$ when both groups are classical groups) and show their uniqueness (in general). This allows us to define extended local $L$-functions and root numbers. We then obtain automorphic $L$-functions $L(s,\tau \times \pi)$, where $\tau$ and $\pi$ are globally generic cuspidal automorphic representations of split classical groups or general linear groups over a global function field. In addition to rationality and the functional equation, we prove that our automorphic $L$-functions satisfy the Riemann Hypothesis.

Residues of Eisenstein series and the automorphic cohomology of reductive groups

AbstractLet $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $. In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, $q\lt {q}_{ \mathsf{res} } $, for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $, ${C}_{n} $, ${D}_{n} $.

Parabolic factorizations of split classical groups

Parabolic factorizations of split classical groups

Constructing automorphic representations in split classical groups

In this paper we introduce a general construction for acorrespondence between certain Automorphic representations inclassical groups. This construction is based on the method of smallrepresentations, which we use to construct examples of CAPrepresentations.

Functors for unitary representations of classical real groups and affine Hecke algebras

We define exact functors from categories of Harish–Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with “real infinitesimal character”) into the corresponding p-adic spherical unitary dual.

Admissibility and permissibility for minuscule cocharacters in orthogonal groups

For a given cocharacter μ, μ-admissibility and μ-permissibility are combinatorial notions introduced by Kottwitz and Rapoport that arise in the theory of bad reduction of Shimura varieties. In this paper we prove that μ-admissibility is equivalent to μ-permissibility in all previously unknown cases of minuscule cocharacters μ in Iwahori–Weyl groups attached to split orthogonal groups. This, combined with other cases treated previously by Kottwitz–Rapoport and the author, establishes the equivalence of μ-admissibility and μ-permissibility for all minuscule cocharacters in split classical groups, as conjectured by Rapoport.

More variations on the decomposition of transvections

The method of decomposition of unipotents consists in writing elementary matrices as products of factors lying in proper parabolic subgroups whose images under inner automorphisms also fall into proper parabolic subgroups of certain types. For the general linear group, this method was first proposed by Stepanov in 1987 to simplify the proof of Suslin’s normality theorem. Soon thereafter, Vavilov and Plotkin generalized it to other classical groups and Chevalley groups. Subsequently, many further results of that type have been discovered. In the present paper, new versions of decomposition of unipotents, which allow one to expand its applicability well beyond the present scope, are described. Here these ideas are merely illustrated for split classical groups, in some simplest cases. Detailed calculations are relegated to subsequent publications. Bibliography: 34 titles.

On the residual spectrum of split classical groups supported in the Siegel maximal parabolic subgroup

For the split symplectic and special orthogonal groups over a number field, we decompose the part of the residual spectrum supported in the maximal parabolic subgroup with the Levi factor isomorphic to GL n . The decomposition depends on the analytic properties of the symmetric and exterior square automorphic L-functions, but seems sufficient for the computation of the corresponding part of the Eisenstein cohomology. We also prove that if one assumed Arthur’s conjectural description of the discrete spectrum for the considered groups, then one would be able to find the poles of the L-functions in question, and would make the decomposition more precise.

The unitary spherical spectrum for split classical groups

AbstractThis paper gives a complete description of the spherical unitary dual of split classical real andp-adic groups. The proof makes heavy use of the affine graded Hecke algebra.

On the cuspidal cohomology of arithmetic groups

It is the main objective of this paper to show a vanishing result for cuspidal cohomology of arithmetic groups in classical groups~$G$ defined over some number field~$k$. Our approach relies on the fact that cuspidal automorphic forms are nonsingular in the sense of Howe. This result puts a strong constraint on the (archimedean components of) irreducible cuspidal automorphic representations of $G$ that can possibly contribute to the cuspidal cohomology. Combining this with the classification of Vogan-Zuckerman of unitary representations with nonzero cohomology provides a constant $r_0(G/k)$, only depending on $G/k$, below which the cuspidal cohomology vanishes. We will give a formula for this constant for each classical group of type (I) in the classification scheme due to Weil. We conclude with making this result explicit for some split classical groups over a totally real algebraic number field.

Residual Spectra of Split Classical Groups and their Inner Forms

Abstract.This paper is concerned with the residual spectrum of the hermitian quaternionic classical groups G′n and H′n defined as algebraic groups for a quaternion algebra over an algebraic number field. Groups G′n and H′n are not quasi-split. They are inner forms of the split groups SO4n and Sp4n. Hence, the parts of the residual spectrum of G′n and H′n obtained in this paper are compared to the corresponding parts for the split groups SO4n and Sp4n.

Functoriality for the Classical Groups over Function Fields

Langlands' functoriality for generic representations from the split classical groups to an appropriate GL N is established. The functorial lift or transfer to GL N is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands-Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands-Shahidi method has only been developed over number fields. To overcome many difficulties, stability of γ-factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.

On Certain Classes of Unitary Representations for Split Classical Groups

AbstractIn this paper we prove the unitarity of duals of tempered representations supported onminimal parabolic subgroups for split classical p-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups.

On the non-unitary unramified dual for classical 𝑝–adic groups

In this paper we give a Zelevinsky type classification of unramified irreducible representations of split classical groups.