The spectral gluing theorem revisited

Beraldo, Dario

doi:10.46298/epiga.2020.volume4.5940

Cited by 9 publications

(14 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof of Theorem F. The proof rests on a contractibility statement proven in [2], to which we reduce via a "microlocal" argument as in [7]. Hence, we expect this functor to be given by the action of an object F DL ∈ D(N) ⇒ : indeed, by a conjecture of [4] and [7], we expect to have…”

Section: 2mentioning

confidence: 90%

“…In this case, we do need the definitions of Eis enh,spec P and CT enh,spec P : they are recalled in Section 3.1. Using the techniques of [2] and [7], we will be able to simplify the functor DL spec Ǧ to obtain:…”

Section: Theorem E the Functormentioning

confidence: 99%

“…We assume familiarity with the theory of singular support for coherent sheaves on quasi-smooth stacks, see [1] and [2]. We also assume some familiarity with the theory of H, as developed in [5] and in [7]. The two latter references are not strictly necessary for the proof, but they help streamline the argument.…”

Section: Proof Of Theorem Fmentioning

confidence: 99%

See 2 more Smart Citations

Deligne-Lusztig duality on the stack of local systems

Beraldo

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun G (that is, the difference between !-extensions and * -extensions) is controlled, Langlands-dually, by the locus of semisimple Ǧ-local systems. To see this, we first rephrase the question in terms of Deligne-Lusztig duality and then study the Deligne-Lusztig functor DL spec G acting on the spectral Langlands DG category IndCoh N (LS G ).We prove that DL spec G is the projection IndCoh N (LS G ) ։ QCoh(LS G ), followed by the action of a coherent D-module St G ∈ D(LS G ), which we call the Steinberg D-module. We argue that St G might be regarded as the dualizing sheaf of the locus of semisimple G-local systems. We also show that DL spec G , while far from being conservative, is fully faithful on the subcategory of compact objects.

show abstract

Section: 2mentioning

confidence: 90%

Section: Theorem E the Functormentioning

confidence: 99%

See 1 more Smart Citation

Deligne-Lusztig duality on the stack of local systems

Beraldo

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…) brings several new tools to the Betti geometric Langlands program: the notion of singular support over LS Ǧ for objects of D Betti (Bun G (X)), the notion of categorified Eisenstein series [11], the strong spectral gluing theorem [12], and so on. We hope to exploit these tools to give an explicit construction of the conjectural functor L Betti G .…”

Section: The Action Of H(lsmentioning

confidence: 99%

The topological chiral homology of the spherical category

Beraldo

2019

Journal of Topology

Self Cite

View full text Add to dashboard Cite

We consider the spherical DG category Sph G attached to an affine algebraic group G. By definition, Sph G := IndCoh(LSG(S 2 )) consists of ind-coherent sheaves on the (derived) stack of G-local systems on the 2-sphere S 2 . The three-dimensional version of the pair of pants endows Sph G with an E3-monoidal structure. More generally, for an algebraic stack Y and n −1, wewhere IndCoh0 is the sheaf theory introduced by Arinkin and Gaitsgory. The case of Sph G is recovered by setting Y = BG and n = 2.The cobordism hypothesis associates to Sph(Y, n) an (n + 1)-dimensional TFT, whose value on a manifold M d of dimension d n + 1 (possibly with boundary) is given by the topological chiral homology M d Sph(Y, n). In this paper, we compute such chiral homology, obtaining the Stokes style formulawhere the formal completion is constructed using the obvious projectionThe most interesting instance of this formula is for Sph G Sph(BG, 2), the original spherical category, and X a Riemann surface. In this case, we obtain a monoidal equivalenceis the stack of G-local systems on the topological space underlying X and H is a sheaf theory related to Hochschild cochains.

show abstract

“…1.6.1. Let us recall the statement, which was briefly discussed in [8] and [10]. Roughly, the theorem says that any F ∈ D(Bun SL2 ) can be reconstructed from its tempered part and its constant term.…”

mentioning

confidence: 99%

On the geometric Ramanujan conjecture

Beraldo¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we prove two results pertaining to the (unramified and global) geometric Langlands program. The first result is an analogue of the Ramanujan conjecture: any cuspidal D-module on Bun G is tempered. We actually prove a more general statement: any D-module that is * -extended from a quasicompact open substack of Bun G is tempered. Then the assertion about cuspidal objects is an immediate consequence of a theorem of Drinfeld-Gaitsgory. Building up on this, we prove our second main result, the automorphic gluing theorem for the group SL 2 : it states that any D-module on Bun SL 2 is determined by its tempered part and its constant term. This theorem (vaguely speaking, an analogue of Langlands' classification for the group SL 2 (R)) corresponds under geometric Langlands to the spectral gluing theorem of Arinkin-Gaitsgory and the author.

show abstract

The spectral gluing theorem revisited

Cited by 9 publications

References 11 publications

Deligne-Lusztig duality on the stack of local systems

Deligne-Lusztig duality on the stack of local systems

The topological chiral homology of the spherical category

On the geometric Ramanujan conjecture

Contact Info

Product

Resources

About