arXiv subjects
Publications and source records attributed to Yakov Varshavsky.
This paper extends the affine Springer theory developed by Bouthier, Kazhdan, and the second author (see [BKV]) to the mixed characteristic case. In particular, we introduce a theory of perfectly placid perfect infinity stacks and establish their dimension theory. Furthermore, we prove that, in the Witt vector setting, the Chevalley morphism between arc spaces is flat.
In this paper we record the formalism of algebro-geometric DG categories (in short AGCat) following a suggestion of V. Drinfeld. This formalism will be applied to ``real-world" problems in papers sequel to this one, [GRV2] and [GRV3].
The main goal of this note is to show that the local L-packet of Fargues-Scholze [FS], corresponding to an elliptic L-parameter, has an endoscopic decomposition. Our argument is strongly motivated by a beautiful paper of Chenji Fu [Fu], where the stable case is proven. To put our results in a more general context, we also construct a general endoscopic decomposition over complex numbers based on results of Arthur, and a generalization of this decomposition over an arbitrary algebraically closed field of characteristic zero based on a recent work [KSV].
For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). We show that an elliptic G-representation (in the sense of Arthur) can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. We also study the action of the automorphism group of C/Q on complex G-representations. We prove that the sets of essentially square-integrable representations and of elliptic representations are stable under Gal(C/Q).
Let $G$ be a connected reductive group over a field $F=\mathbb{F}_q((t))$ splitting over $\overline{\mathbb{F}}_q((t))$. Following [KV,DR], a tamely unramified Langlands parameter $λ:W_F\to{}^L G(\overline{\mathbb{Q}}_{\ell})$ in general position gives rise to a finite set $Π_λ$ of irreducible admissible representations of $G(F)$, called the $L$-packet. The main goal of this work is to provide a geometric description of characters $χ_π$ of $π\inΠ_λ$ and of their endoscopic linear combinations $χ_λ^κ$ in terms of homology of affine Springer fibers, thus establishing an analog of Lusztig conjectures in this case. Furthermore, each $χ_λ^κ$ can be described as the trace of Frobenius function of a conjugation equivariant perverse sheaf on the loop group by the sheaf-function correspondence. As another application, we prove that the sum $χ_λ^{st}:=\sum_{π\inΠ_λ}χ_π$ is stable and show that the $χ_λ^{st}$'s are compatible with inner twistings. More generally, we prove that each $χ_λ^κ$ is $\mathcal{E}_{λ,κ}$-stable.
Using techniques of [BKV], we construct a perverse t-structure on the infinity-category of l-adic LG-equivariant sheaves on the regular-semisimple bounded locus of the loop group LG and prove that the derived $τ$-coinvariants of affine Grothendieck--Springer sheaves are perverse. Our main new ingredient is a theorem of Yun on compatibility of actions.
We prove that the adjoint equivariant derived category of a reductive group $G$ is equivalent to the appropriately defined monoidal center of the torus-equivariant version of the Hecke category. We use this to give new proofs, independent of sheaf-theoretic set up, of the fact that the Drinfeld center of the abelian Hecke category is equivalent to the abelian category of unipotent character sheaves; and of a characterization of strongly-central sheaves on the torus.
Let $G$ be a connected reductive group over an algebraically closed field $k$, and let $Fl$ be the affine flag variety of $G$. For every regular semisimple element $γ$ of $G(k((t)))$, the affine Springer fiber $Fl_γ$ can be presented as a union of closed subvarieties $Fl^{\leq w}_γ$, defined as the intersection of $Fl_γ$ with an affine Schubert variety $Fl^{\leq w}$. The main result of this paper asserts that if elements $w_1,\ldots,w_n$ are sufficiently regular, then the natural map $H_i(\bigcup_{j=1}^n Fl^{\leq w_j}_γ)\to H_i(Fl_γ)$ is injective for every $i\in{\mathbb Z}$. It plays an important role in our work [BV]. One can view this statement as providing a categorification of the notion of a weighted orbital integral. Along the way we also show that every affine Schubert variety can be written as an intersection of closures of semi-infinite orbits.
In this paper we introduce the categorical "true local terms" maps for Artin stacks and show that they are additive and commute with proper pushforwards, smooth pullbacks and specializations. In particular, we generalizing results of [Va2] to this setting. As an application, we supply proofs of two theorems stated in [AGKRRV]. Namely, we show that the "true local terms" of the Frobenius endomorphism coincide with the "naive local terms" and that the "naive local terms" commute with !-pushforwards. The latter result is a categorical version of the classical Grothendieck--Lefschetz trace formula.
The goal of this work is to construct a perverse t-structure on the infinity-category of l-adic LG-equivariant sheaves on the loop Lie algebra Lg and to show that the affine Grothendieck-Springer sheaf S is perverse. Moreover, S is an intermediate extension of its restriction to the locus of ``compact" elements with regular semi-simple reduction. Note that classical methods do not apply in our situation because LG and Lg are infinite-dimensional ind-schemes.
The goal of this note is to show that in the case of transversal intersections the "true local terms" appearing in the Lefschetz trace formula equal to the "naive local terms". To prove the result we extend the method of [Va], where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of any etale sheaf to the normal cone is monodromic and the assertion that in some cases local terms are "constant in families". As an application, we get a generalization of the Deligne-Lusztig trace formula.
In this work we give a geometric proof of Hrushovski's generalization of the Lang-Weil estimates on the number of points in the intersection of a correspondence with the graph of Frobenius.
We apply the dimension theory developed in [BKV] to establish some of Lusztig's conjectures [Lu].
The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.
In this paper we prove an explicit formula for the Bernstein projector to representations of depth at most r. As a consequence, we show that the depth zero Bernstein projector is supported on topologically unipotent elements and it is equal to the restriction of the character of the Steinberg representation. As another application, we deduce that the depth r Bernstein projector is stable. Moreover, for integral depths our proof is purely local.
In this note we construct a "restriction" map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on a parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig-Spaltenstein on induced unipotent classes to all infinite fields.
The goal of this note is to give a short geometric proof of a theorem of Hrushovski asserting that an intersection of a correspondence with a graph of a sufficiently large power of Frobenius is non-empty.
In this note we formulate and give a self-contained proof of the Yoneda lemma for infinity categories in the language of complete Segal spaces.