16 pages. New version contains an appendix by Sean Cotner, which allows us to generalize Theorem 2.5 compared to the previous version. Comments welcome!
For a reductive group over a nonarchimedean local field, we define the stack of spherical Langlands parameters, using the inertia-invariants of the Langlands dual group. This generalizes the stack of unramified Langlands parameters in case the group is unramified. We then use this stack to deduce the Eichler--Shimura congruence relations for Hodge type Shimura varieties, without restrictions on the ramification.
We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze's basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results. Our key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the …
We describe the image, under the local Langlands correspondence for tori, of the characters of a torus which are trivial on its Iwahori subgroup. Let $k$ be a non-archimedian local field. Let $\boldsymbol{G}$ be a connected reductive group defined over $k$, which is quasi-split and split over a tamely ramified extension. Let $K$ be a special maximal parahoric subgroup of $\boldsymbol{G}(k)$. To the class of representations of $\boldsymbol{G}(k)$, having a non-zero vector fixed under $K$, we establish a bijection, in a natural way, with the twisted semi-simple conjugacy classes of the inertia f…
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 …
Braverman and Kahzdan have introduced an influential conjecture on local functional equations for general Langlands $L$-functions. It is related to L. Lafforgue's equally influential conjectural construction of kernels for functorial transfers. We formulate and prove a version of Braverman and Kazhdan's conjecture for spherical representations over an archimedean field that is suitable for application to the trace formula. We then give a global application related to Langlands' beyond endoscopy proposal. It is motivated by Ngô's suggestion that one combine nonabelian Fourier transforms with th…
We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical $G$-variety $X$, one associates a dual group ${^LG_X}$ and an $L$-value (encoded in a representation of ${^LG_X}$), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism ${^LG_X}\to {^LG_Y}$ of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approac…
This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition fu…
Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory (due to Vogan and Soergel) describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands (due to Fargues, Zhu and Hellmann) describe representations with varying central character of a large family of groups associated to isocrystals in term…
By applying the residue method for period integrals and Langlands-Shahidi's theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals and certain automorphic L-functions. In some cases, we also study the local multiplicity of the spherical varieties.
Let F be a global field, G an unramified quasi-split reductive group over F and chi an everywhere unramified automorphic character of a maximal maximally split torus of G. Using Langlands-Shahidi theory, we compute the meromorphic function defined by the action of a global standard intertwining operator associated to chi on a spherical vector and show that the ratio of its poles in the positive Weyl chamber is well behaved.
Langlands has described the irreducible admissible representations of $T$, when $T$ is the group of points of an algebraic torus over a local field. Also, Langlands described the automorphic representations of $T_{\mathbb A}$ when $T_{\mathbb A}$ is the group of adelic points of an algebraic torus over a global field $F$. We describe irreducible (in the local setting) and automorphic (in the global setting) $ε$-genuine representations for covers of tori, also known as metaplectic tori, which arise from a framework of Brylinski and Deligne. In particular, our results include a description of sp…
We state a conjecture that relates the derived category of smooth representations of a p-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of GL_n by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and Emerton-Helm.
Let $G$ be a split reductive group over a number field $F$. We consider the computation of the inner product of two $K$-spherical pseudo Eisenstein series of $G$ supported in $[T,\mathcal{O}(1)]$ by means of residues, following a classical approach initiated by Langlands. We show that only the singularities of the intertwining operators due to the poles of the completed Dedekind zeta function $Λ_F$ contribute to the spectrum, while the singularities caused by the zeroes of $Λ_F$ do not contribute to any of the iterated residues which arise as a result of the necessary contour shifts. In the co…
Kottwitz's conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze's more general spaces of local shtukas. Using a new Lefschetz-Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that for an irreducible smooth representation of an inner form of $\math…
For a split reductive group G over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of l in the Langlands correspondence for function fields.
On the unramified spectrum of spherical varieties over p-adic fields
群G的不可约表示的描述可以看作是调和分析中的一个问题;也就是说，通过左乘和右乘将G上的一个合适的函数空间分解成G x G的作用的不可约表示。对于局部非阿基米德域上的可裂p-adic约化群G，非分歧不可约光滑表示与"Langlands对偶“群中的半单共轭类成双射。我们将这种描述推广到G的任意球面簇X上，如下：空间$C_c^\infty(X)$的不可约非分歧子是自然的"几乎双射“，其中有许多$A_X^*/W_X$的副本，一个复环面与X的"小外尔群”的商。这导致了对非分歧向量的Hecke模的描述（Gaitsgory和Nadler的几何结果的弱模拟），并理解了某些子群的表示是函子提升的现象。在证明过程中，考察了球面簇的合理性性质，并对F. Knop，在Borel轨道集合上的外尔群。
Let H be a split reductive group over a local non-archimedean field, and let H^ denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight representation of the dual group, considered as a series of elements in the Hecke algebra of H. This offers an alternative approach to a solution of the same problem by Wen-Wei Li. Moreover, we generalize the notion of "Satake transform" and perform the analogous calculation for a large class of spherical varieties.
In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_κ$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of end…
Let X be an affine spherical variety, possibly singular, and $L^+X$ its arc space. The intersection complex of $L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L-functions. Such relationships were previously established in Braverman-Finkelberg-Gaitsgory-Mirkovic for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier-Ngo-Sakellaridis for toric varieties and L-monoids. In this paper, we compute this intersection complex for the large class of those spherical …
