On Arthur's unitarity conjecture for split real groups
Joseph Hundley, Stephen D. Miller
备注：
34 pages, 1 figure, 3 tables. Post-publication version 3 includes comments (in blue) included to navigate literature statements about normalized intertwining operators
Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces of automorphic forms. We prove the unitarity of the "Langlands element" (i.e., the one specified by Arthur) of all unipotent Arthur packets for split real groups. The proof uses Eisenstein series, Langlands' constant term formula and square integrability criterion, analytic properties of intertwining operators, and some mild arithmetic input from the theory of Dirichlet L-functions, to reduce to a more combinatorial problem about intertwining operators. This updated arXiv posting also includes some comments (in blue) concerning statements about normalized intertwining operators we quoted from the literature in Section 9.
In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O_{2n} established by Arthur. We give two applications: Prasad's conjecture and Gross--Prasad conjecture for O_{n}. Also, we discuss the Arthur conjecture for O_{2n}, and establish the Arthur multiplicity formula for O_{2n}.
We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the coefficients of unipotent orbital integrals are expressed by the Dedekind zeta function, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.
Let G be a special orthogonal group or an inner form of a symplectic group over a number field F such that there exists a non-empty set S of real places of F at which G has discrete series and outside of which G is quasi-split. We prove Arthur's multiplicity formula for automorphic representations of G having algebraic regular infinitesimal character at all places in S.
A Langlands parameter, in the Langlands dual group, can be decomposed into a product of a tempered parameter and a positive quasi-character. Fixing a tempered parameter, Arthur conjectured that positive quasi-characters corresponding to certain weighted Dynkin diagrams for the centralizer of the tempered parameter will yield unitary representations. In this paper, we treat one basic case in Arthur's conjecture for the special orthogonal groups. We establish the unitarity for a class of Langlands-Vogan parameters in Arthur's packet.
Let $G=\mathbf{G}(\mathbb{R})$ be the group of real points of a quasi-split connected reductive algebraic group defined over $\mathbb{R}$. Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on …
On the complementary Arthur representations and unitary dual for p-adic classical groups
University of Michigan（密歇根大学）；University of Minnesota（明尼苏达大学）；Purdue University（普渡大学）；National University of Singapore（新加坡国立大学）；Huazhong University of Science and Technology（华中科技大学）
本文针对特征零的非阿基米德局部域上连通约化群的酉对偶结构猜想，给出辛群和分裂奇特殊正交群互补 Arthur 表示的刻画，得到\(\mathrm{GL}_N\)（\(N = 2,3\)）不可约自对偶尖点自守表示局部分量的限制。
Arthur has conjectured that the unitarity of a number of representations can be shown by finding appropriate automorphic realizations. This has been verified for classical groups by Moeglin and for the exceptional Chevalley group G_2 by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups E_6, E_7, E_8, and F_4. In particular we prove Arthur's conjecture that the spherical constituent of an unramified principal series of a Chevalley group over any local field of characteristic zero is unitarizable if its Langlands parameter coincides with …
Let $k$ be a $p$-adic field and let $\mathbf{G}(k)$ be the $k$-points of a connected reductive group, inner to split. The set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form an Arthur packet for $\mathbf{G}(k)$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets which we call \emph{weak} Arthur packets and describe their constituents in terms of thei…
We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms. This work is applied to transferring known results in the second-named author's earlier work for quasi-split special unitary groups to their non quasi-split inner forms.
In spirit of Gan-Ichino's work on the Arthur's multiplicity formula for metaplectic groups, we have established the Arthur's multiplicity formula for even orthogonal or unitary groups with Witt index less than or equal to one. In that multiplicity formula, some local packets defined using the stable range theta lifts are involved. In this paper, we prove that at non-Archimedean places, the definition of the local packets involved in that multiplicity formula is independent of the choice of the dual-pairs used in their construction. Moreover, at those places where the groups are quasi-split, we…
In this paper we establish the endoscopic classification of tempered representations of quasi-split unitary groups over local fields, and the endoscopic classification of the discrete automorphic spectrum of quasi-split unitary groups over global number fields. The method is analogous to the work of Arthur on orthogonal and symplectic groups, based on the theory of endoscopy and the comparison of trace formulas on unitary groups and general linear groups.
We study the reducibility of parabolically induced representations of non-split inner forms of quasi-split classical groups. The isomorphism of Arthur R-groups, endoscopic R-groups and Knapp-Stein R-groups is established, as well as showing these R-groups are isomorphic to the corresponding ones for the quasi-split form. This shows R-groups are an invariant of the L-packets. The results are applied to classify the elliptic spectrum.
Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover,we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe-Gan for quasi-split even orthogonal groups.
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 e…
Let $π$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $π$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(\frac{1}{2},π\timesχ)$, with a certain automorphic character $χ$ of ${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general ${\mathfrak n}\geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results…
In this paper we gives the Langlands parameters of Langlands' packets of discrete series using the twisted endoscopy as explained by Arthur; this holds for orthogonal, symplectic, unitary and G-Spin groups and gives the most simple proof available. We have assume that the groups are quasi-split but this is just for simplicity. The proof explaines first what is the classification from the representation's theory point of view; this gives the Langlands' packets purely in terms of representation theory. And then using the theory of L-function of Shahidi and the doubling method of Rallis and Piate…
We prove Ibukiyama's conjectures on Siegel modular forms of half-integral weight and of degree 2 by using Arthur's multiplicity formula on the split odd special orthogonal group $\SO_5$ and Gan-Ichino's multiplicity formula on the metaplectic group $\Mp_4$. In the proof, the representation theory of the Jacobi groups also plays an important role.
We prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. The proof holds for tempered representations of quasi-split real K-groups and quasi-split p-adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.
In this paper, we reformulate Moeglin's explicit construction of local A-packets of split odd special orthogonal groups and symplectic groups. By this reformulation together with results of the previous paper with Minguez, we can compute the A-packets explicitly. Also, we give a non-vanishing criterion of our parametrization, and an algorithm to compute certain derivatives. Finally, we prove a formula for the Aubert duals of irreducible representations of Arthur type.
We study the Knapp-Stein $R$--groups for inner forms of the split group $SL_n(F),$ with $F$ a $p$--adic field of characteristic zero. Thus, we consider the groups $SL_m(D),$ with $D$ a central division algebra over $F$ of dimension $d^2,$ and $m=n/d.$ We use the generalized Jacquet-Langlands correspondence and results of the first named author to describe the zeros of Plancherel measures. Combined with a study of the behavior of the stabilizer of representations by elements of the Weyl group we are able to determine the Knapp-Stein $R$--groups in terms of those for $SL_n(F).$ We show the $R$--…
This paper is concerned with the structure of packets of representations and some refinements that are helpful in endoscopic transfer for real groups. It includes results on the structure and transfer of packets of limits of discrete series representations. It also reinterprets the Adams-Johnson transfer of certain nontempered representations via spectral analogues of the Langlands-Shelstad factors, thereby providing structure and transfer compatible with the associated transfer of orbital integrals. The results come from two simple tools introduced here. The first concerns a family of splitti…
Let $\mathbf{G}$ be a split algebrac group of type $E_n$ defined over a $p$-adic field. This group contains a dual pair $G \times G'$ where one of the groups is of type $G_2$. The minimal representation of $\mathbf{G}$, when restricted to the dual pair, gives a correspondence of representations of the two groups in the dual pair. We prove a matching of spherical Hecke algebras of $G$ and $G'$, when acting on the minimal representation. This implies that the correspondence is functorial, in the sense of Arthur and Langlands, for spherical representations.
Let $G$ be a connected quasi-split reductive group over $\mathbb{R}$, and more generally, a quasi-split $K$-group over $\mathbb{R}$. Arthur had obtained the formal formula for the spectral side of the stable local trace formula, by using formal substitute of Langlands parameters. In this paper, we construct the spectral side of the stable trace formula and endoscopy trace formula directly for quasi-split $K$-groups over $\mathbb{R}$, by incorporating the works of Shelstad. In particular we give the explicit expression for the spectral side of the stable local trace formula, in terms of Langlan…
Let $\mathbf{G}$ be an unramified quasi-split unitary group over a p-adic field of odd residual characteristic. The goal of this paper is to describe the supercuspidal representations within certain L-packets of $\mathbf{G}$, which are classified by Arthur and Mok using the theory of endoscopy. The description is given in terms of the cuspidal types constructed by Bushnell-Kutzko and Stevens. As a starting example, we require the parameters of our packets to satisfy certain regularity conditions, such that these packets consist of very cuspidal representations in the sense of Adler and Reeder.…
Local Intertwining Relations and Co-tempered $A$-packets of Classical Groups
Kyoto University（京都大学）；National University of Singapore（新加坡国立大学）；University of Bonn（波恩大学）；Universidad de Sevilla（塞维利亚大学）；University of Vienna（维也纳大学）；UC Berkeley（加州大学伯克利分校）；Korea Institute for Advanced Study（韩国高等科学研究院）
Using the results of J. Arthur on the representation theory of classical groups with additional work by Colette Moeglin and its relation with representations of affine Hecke algebras established by the author, we show that the category of smooth complex representations of a split $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). A statement of this kind had been conjecture by G. Lusztig. All classical groups (general li…
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 sphe…
