Related papers: $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \ma…

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $\mathrm{Sch}/\Bbbk$ denote the category of schemes of finite type over $\Bbbk$. Let $B$ be a connected projective scheme over $\Bbbk$ and let $\mathcal L$ be an ample…

In this article we continue our investigation of the Derived Equivalences over noetherian quasi-projective schemes $X$, over affine schemes $\spec{A}$. For integers $k\geq 0$, let $C{\mathbb M}^k(X)$ denote the category of coherent…

Let $\Gamma$ denote a distance-regular graph with vertex set $X$ and diameter $D \geq 3$. Fix a vertex $x \in X$. Let the field $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $\operatorname{Mat}_X(\mathbb{F})$ denote the…

Let X be a proper scheme and Z a prestack over X equipped with a flat connection. We give a local-to-global description of D-modules on the prestack S(Z) of flat sections of Z. Examples of S(Z) include the moduli stacks of principal…

We consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, Moeller and Zorich showing that the sum of the first k…

In Commutative Algebra structure results on minimal free resolutions of Gorenstein modules are of classical interest. We define Gorenstein modules of finite length over the weighted polynomial ring via symmetric matrices in divided powers.…

The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $\pi_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give…

Let $k$ be a field, $Q$ a finite directed graph, and $kQ$ its path algebra. Make $kQ$ an $\NN$-graded algebra by assigning each arrow a positive degree. Let $I$ be a homogeneous ideal in $kQ$ and write $A=kQ/I$. Let $\QGr A$ denote the…

We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent…

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-module gives rise to a family of…

We study GL-equivariant modules over the infinite variable polynomial ring $S = k[x_1, x_2, ..., x_n, ...]$ with $k$ an infinite field of characteristic $p > 0$. We extend many of Sam--Snowden's far-reaching results from characteristic zero…

Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which the categories QGr(A) and QGr(kQ) are equivalent:…

We apply the machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different…

In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on…

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. If $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of…

Let $(R, \mf, k_R)$ be regular local $k$-algebra satisfying the weak Jacobian criterion, such that $k_R/k$ is an algebraic field extension. Let $D_R$ be the ring of $k$-linear differential operators of $R$. We give an explicit decomposition…

Let G be a connected reductive group over an algebraic closure of a finite field Fq. In this paper it is proved that the infinite dimensional Steinberg module of kG defined by N. Xi in 2014 is irreducible when k is a field of positive…

This is the second in a series of two papers developing a moduli-theoretic framework for differential ideal sheaves associated with formally integrable, involutive systems of algebraic partial differential equations (PDEs). Building on…

We show that the Hall algebra of the category of coherent sheaves on an elliptic curve (or, equivalently, the algebra of unramified automorphic forms for GL(n) for all n) is equal to the stable limit of spherical double affine Hecke…

We give a general construction for finite dimensional representations of $U_q(\hat{\G})$ where $\hat{\G}$ is a non-twisted affine Kac-Moody algebra with no derivation and zero central charge. At $q=1$ this is trivial because…