Papers

Let $X$ be a smooth projective curve and let $G$ be a reductive group over an algebraically closed field of characteristic zero. We compute the image of arbitrary finite-rank local systems on $\mathrm{Bun}_G(X)$ under the geometric Langlands equivalence, confirming a conjecture of V. Lafforgue in the case of the constant sheaf.

Let $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}$ be a Borel subalgebra of a basic classical Lie superalgebra over $\mathbb{C}$ with $\mathfrak{h}$ a Cartan subalgebra. We give an upper bound for the index $\mathrm{ind}(\mathfrak{b})$ of $\mathfrak{b}$; in some instances this bound is $0$, in which case $\mathrm{ind}(\mathfrak{b})=0$. Additionally we prove that $\mathrm{ind}(\mathfrak{b},\mathfrak{n})=0$, which implies $\mathrm{ind}(\mathfrak{b},\mathfrak{i})=0$ for all ideals $\mathfrak{i} \subseteq \mathfrak{n}$ of $\mathfrak{b}$. We also show that $\mathrm{ind}(\mathfrak{b},\mathfrak{a}^*)=0$ for all abelian ideals $\mathfrak{a} \subseteq \mathfrak{n}$ of $\mathfrak{b}$. These results are achieved by extending the theory of strongly orthogonal roots and the Kostant cascade to the theory of generalized root systems developed by Dimitrov and Fioresi.

Let $G$ be a quasi-split reductive group over a finite field and let $\^G$ be the Langlands dual group. Assuming the derived subgroup of $G$ is almost simple, we use techniques from the geometric Langlands program to relate special values of Bessel functions for generic principal series representations of $G$ to the trace of Frobenius acting on Kloosterman sheaves of Heinloth-Ngô-Yun for $\^G$. We also give explicit examples in essentially all possible cases, including exceptional groups.

We study degenerate Eisenstein series on symplectic groups and construct a new automorphic descent. We show that a certain Fourier coefficient, after restriction to a smaller symplectic group, is itself an Eisenstein series with the same inducing parameter \(s\). We describe the resulting holomorphic section explicitly in terms of the original section and local \(L\)-factors at unramified places. At every regular point of the local descent, we prove surjectivity at non-Archimedean places and dense image at Archimedean places. We finally indicate an application of the iterated descent to maximal Fourier coefficients of Eisenstein series.

The Isaacs–Navarro–Wolf conjecture states that if $G$ is a finite solvable group and $x$ is an element of $G$ such that $\chi(x)$ does not vanish for all irreducible characters $\chi$ of $G$, then $x$ must be contained in some nilpotent normal subgroup. In this paper we present a proof of the Isaacs–Navarro–Wolf conjecture that was discovered with the use of artificial intelligence systems.

We present a conjectural construction of Arthur packets within Fargues-Scholze's framework for the categorical local Langlands correspondence (CLLC). This construction consists of three parts. We first provide an overview the main statement of the CLLC, the underlying moduli stacks – $\mathrm{Par}_G$ of parameters and $\mathrm{Bun}_G$ of $G$-bundles – on the two sides of the correspondence, and its relation to representations of reductive p-adic groups. We then review the geometric Satake correspondence in order to define Hecke operators and the spectral action of sheaves on $\mathrm{Par}_G$ on sheaves on $\mathrm{Bun}_G$, and to construct semisimple parameters using excursion data. Finally, we generalize the geometric construction of Arthur packets from pushing-forward skyscraper sheaves on the regular conormal bundle of $\mathrm{Par}_G$ over $\mathbb C$ to a conjectural analogous operation on the stack of singularities on $\mathrm{Par}_G$ over $\overline{\mathbb Q}_\ell$.

Let $\mathbb{k}$ be a finite field and $\mathcal{A}, \mathcal{B}$ be $\mathbb{k}$-linear $\mathsf{Ext}$-finite hereditary abelian categories. A theorem of Cramer asserts that, under suitable assumptions, a derived equivalence $\mathcal{D}^b(\mathcal{A}) \!\longrightarrow\! \mathcal{D}^b(\mathcal{B})$ between two such categories induces an algebra isomorphism of the corresponding double Hall algebras $\mathsf{DH}_\mathcal{A} \!\longrightarrow\! \mathsf{DH}_\mathcal{B}$. It turns out that a counting formula for certain distinguished triangles in $\mathcal{D}^b(\mathcal{A})$, on which Cramer's proof relies, is incorrect in general. We give a corrected proof of Cramer's theorem which preserves the overall strategy of his approach.

For $X$ a conical affine symplectic singularity with $\mathbb{T}=T \times \mathbb{G}_m$-action, the fixed scheme $X^T$ and the map $X^T \rightarrow X$ carry much information about the geometry of $X$. In general, $X^T \rightarrow X$ fails to be a complete intersection. Thus, we study a derived intersection whose classical locus is the $T$-fixed subscheme $X^T$. We show that the structure of the symplectic singularity on $X$ produces a duality theorem for the structure sheaf of the derived intersection. The duality theorem allows us to study the structure of such derived intersections; in particular we describe their cohomological amplitude. An important source of symplectic singularities with $\mathbb{T}$-action are affine Grassmannian slices $\overline{W}^{\lambda}_{\mu}$. We pay particular attention to these slices when $G=\mathrm{SL}_{n+1}$, and we use the previously developed theory to characterize when $(\overline{W}^{\lambda}_{\mu})^T \rightarrow \overline{W}^{\lambda}_{\mu}$ is a complete intersection.

We show that various partitions of the canonical basis of quantum groups constructed by Lusztig and by Kashiwara coincide. Using this partition, we show that the subset corresponding to open Richardson varieties equals to the intersection of the subsets corresponding to the Schubert cells. We also show that in type $A$, the weights from open Richardson varieties are saturated in the corresponding Bruhat interval polytope.

Let $G$ be a quasi-reductive supergroup (so its underlying algebraic group $G_{\bar 0}$ is reductive). We consider two trivially intersecting classes of odd elements: neat elements and balanced elements. Neat elements are always $ad$-nilpotent and may be embedded into subalgebras that are isomorphic to $\mathfrak{osp}(1|2)$, a simple Lie superalgebra whose underlying Lie algebra is $\mathfrak{sl}_2$. Balanced odd elements, on the other hand, are a natural generalization of the notion of a self-commuting element (an element $x\in Lie(G)_{\bar 1}$ for which $[x,x]=0$). Balanced elements are used to define homology-type functors on the category of representations of $G$. We show that any element $x\in Lie(G)_{\bar 1}$ may be written as a sum of a neat and a balanced odd element which commute with each other. This theorem has a categorical application. Let $\mathfrak{g}^{(1|1)}$ be the $(1|1)$-dimensional Lie superalgebra generated by $x \in Lie(G)_{\bar 1}$. The semisimplification of the category of finite-dimensional super-representations of $\mathfrak{g}^{(1|1)}$ is a functor $S: Rep(\mathfrak{g}^{(1|1)}) \to Rep(SOSp(1|2))$. Any $x\in Lie(G)_{\bar 1}$ induces a homomorphism $ i_x:\mathfrak{g}^{(1|1)}\to Lie(G)$. Let $$\Phi_x=S\circ (-)\downarrow_{i_x}:Rep(G)\to Rep(SOSp(1|2))$$ be the composition of the restriction functor $(-)\downarrow_{i_x}$ and the functor $S $. We show that the functor $\Phi_x$ may be described explicitly using the homology-type functor $\Phi_{x_{bal}}$ corresponding to the balanced part of $x$ in the above decomposition. These homology-type functors are known as Duflo-Serganova functors. Finally, we provide a full classification of distinguished odd elements in simple quasi-reductive Lie superalgebras and show that in all cases except $\mathfrak{spe}(n)$, such elements are either balanced or neat.

This comment gives dimension-independent analytic proofs of the local inequalities arising in the odd- and even-dimensional constructions of higher-dimensional partition charge functions. The prior works established these inequalities by exhaustive computation in low dimensions and tested them numerically in selected higher dimensions.

We prove the wall-crossing formula conjectured by Gross–Joyce–Tanaka for equivariant enumerative invariants of CY4 categories equipped with framing functors. We also establish a version for stable pairs with fixed-determinant obstruction theories, as used in earlier applications by the first-named author. The main technical ingredient, which we develop in this work, is a construction of well-behaved CY4 pullback virtual classes using Jouanolou devices.

We show that Makhlin's polytopes in type $B_n$ are unimodularly equivalent to string polytopes for Demazure modules in type $B_{2n-1}$. The proof passes through type $C$, combining Makhlin's diagonal comparison with symplectic FFLV polytopes and the $B/C$ similarity for string cones.

Let $D(G)$ denote the derived category of smooth $G$-representations on $k$-vector spaces, where $G$ is a locally pro-$p$ group and $k$ is a field of characteristic $p$. In this paper we are primarily interested in the graded center of the subcategory of compact objects $Z^*(D(G)^c)$ and variants thereof. When $G$ is a $p$-adic Lie group, without proper open centralizers, we completely determine this center modulo locally nilpotent elements and give various applications.

Let $A$ be a finite-dimensional representation-finite algebra over an algebraically closed field $k$, and let $M$ be a finite-dimensional $A$-module such that $\operatorname{End}_A(M)$ has finite global dimension. We prove that $\operatorname{End}_A(M)$ has Cartan determinant one if $A$ has finite global dimension or $\operatorname{char}k\neq2$; in particular, the Cartan determinant conjecture holds for representation-finite algebras over an algebraically closed field.

We give examples of two derived equivalent gentle algebras which are not tilting-cotilting equivalent.

In this paper we prove that the Eaton–Moretó conjecture holds true for all $p$-solvable groups.

Associated with an even nilpotent element $e$ in a basic classical Lie superalgebra $\mathfrak{g}$, we study, in full generality, two constructions of finite $W$-superalgebras, defined via Whittaker models and isotropic subspaces, respectively. We prove that both formulations are independent of the various choices made in their constructions, thereby yielding, for a fixed good grading for $e$, at most two isomorphism classes of $W$-superalgebras. In the case when there are two non-isomorphic versions, we establish that they differ precisely by a Clifford extension. Consequently, when the two $W$-superalgebras are non-isomorphic, their module categories are equivalent up to a Clifford twist. Building on this equivalence and utilizing the Skryabin equivalence, we classify their irreducible representations in terms of generalized Whittaker modules over $\mathfrak{g}$

The classification of Brauer graph algebras under derived equivalence was recently given by Opper and Zvonareva. In this paper, we prove that two Brauer graph algebras are derived equivalent if and only if they are stably equivalent of Morita type. As an application, we show that every stable Picard group orbit of simple-images of Morita type contains a liftable representative. Moreover, we give a new proof of the fact that Brauer graph algebras are closed up to semisimple summands under stable equivalence of Morita type.

We present a derivation of the integral kernel of a metaplectic representation operator in the Bargmann-Fock model based solely on its defining intertwining property with the Heisenberg representation. Expressing a metaplectic operator as an integral operator transforms the corresponding infinitesimal intertwining identities into a system of first-order partial differential equations satisfied by its kernel. We show that this system determines the kernel (up to a unit scalar) which is precisely the classical Gaussian associated with the symplectic transformation. A similar result is obtained for the Schrödinger model.

We give a local newform theory for the metaplectic group of rank 1 over a non-archimedean local field of characteristic 0 with odd residual characteristic. It was suggested by some results of Kohnen and Ueda on the classical theory of modular forms of half-integral weight.

We classify Satake diagrams for general linear and orthosymplectic non-twisted affine Lie superalgebras and prove that each of them generates a family of proper spherical subalgebras. Furthermore, we demonstrate that every such family contains subalgebras with matrix invariants, which are viewed as classical analogs of K-matrices solving supersymmetric Reflection equation.

In this paper, for various pairs of topological spaces $X$ and $Y$, we introduce reflection functors between the categories $\mathfrak{KK}(X)$ and $\mathfrak{KK}(Y)$ of Kirchberg's ideal-related KK-theory for separable C*-algebras over $X$ and $Y$. We prove that these functors induce an equivalence $\mathfrak{KK}(X)_{\mathrm{loc}} \simeq \mathfrak{KK}(Y)_{\mathrm{loc}}$ between the localizing subcategories introduced by Meyer and Nest, and that this equivalence restricts to an equivalence $\mathcal{B}(X) \simeq \mathcal{B}(Y)$ between the bootstrap categories. Combining these equivalences with a combinatorial argument due to Bernstein, Gelfand, and Ponomarev from the representation theory of quivers, we show that, for a finite $T_0$-space $X$ whose Hasse diagram is an orientation of a tree, the categories $\mathfrak{KK}(X)_{\mathrm{loc}}$ and $\mathcal{B}(X)$ depend only on the underlying tree and not on its orientation. We also prove the analogous results for Dadarlat and Meyer's ideal-related E-theory. Moreover, we prove a rearrangement property of reflection functors, which yields an analogue of Coxeter functors. Finally, we apply reflection functors to prove that filtrated K-theory satisfies the universal coefficient theorem for C*-algebras over finite $T_0$-spaces whose Hasse diagrams are orientations of Dynkin diagrams of type A.

We construct Khovanov–Lauda–Rouquier-like generators for certain modified forms of non-degenerate affine Hecke algebras associated with arbitrary Weyl groups after localization. These generators yield non-graded KLR-like presentations of the modified algebras. As an application, we establish isomorphisms between direct sums of blocks of cyclotomic Hecke algebras of arbitrary Weyl groups and cyclotomic quotients of the resulting non-graded KLR-like algebras. We also explain how these isomorphisms identify the generalized weight-space decomposition of cyclotomic Hecke modules with the idempotent decomposition of modules over the KLR-like presentation.

Let $A$ be a finite-dimensional basic hereditary algebra and let $T(A)=A\ltimes D(A)$ be its trivial extension. Building on the classification of indecomposable $\tau$-rigid $T(A)$-modules, we give explicit Hom-vanishing conditions characterizing arbitrary basic $\tau$-rigid $T(A)$-modules. For such a module $M$, we also determine its maximal projective complement in terms of the support of the underlying $A$-module $U(M)$, and hence obtain an explicit criterion for $M$ to be support $\tau$-tilting. As an application, for the linearly oriented quiver of type $A_n$, we classify all basic rank-two $\tau$-rigid modules over $T(\Bbbk A_n)$ and prove that their number is \[ \binom{n}{2}\binom{n+1}{2}. \] We also show that every basic $\tau$-tilting $T(\Bbbk A_n)$-module contains an indecomposable projective direct summand. Finally, for two orientations of a quiver of type $D_4$, we determine the corresponding support $\tau$-tilting compatibility graphs and their face distributions, from which we obtain and compare the associated F-triangles.

We provide a criterion for determining when elements of the Bernstein center of a totally disconnected locally compact group are admissible invariant distributions in the sense of Harish-Chandra \cite{HC99}. As a consequence, we deduce the local integrability results for elements of bounded depth or Bernstein supports in the Bernstein centers of reductive $p$-adic groups. Our methods apply uniformly to both complex and mod-$\ell$ coefficients, generalizing results of Moy and Tadić \cite{MT02} in the complex case.

Let $A$ be the centralizer algebra of a matrix over an arbitrary field. We solve the fixed-source realization problem for matrix centralizers by proving that the matrix centralizer algebras derived equivalent to $A$ are precisely the opposite endomorphism algebras of tilting modules over $A$. We classify the basic tilting modules and determine their opposite endomorphism algebras. The tilting poset is a product of right weak orders on symmetric groups, with one factor for each primary block and degree equal to the number of distinct exponents in that block. Together with the center, this poset recovers the multiset of these numbers across all primary blocks, although it does not canonically match them with the local center factors. For each primary block, the target algebras are obtained by permuting the successive gaps between exponents, and their isomorphism classes are determined by the stabilizer of the gap word. Consequently, the quotient of the labeled mutation graph by target-algebra isomorphism is a Schreier multigraph. We also characterize when the weak-order orientation descends to its nonloop edges.

We study semi-orthogonal decompositions of perfect derived categories of gentle algebras via marked ribbon surfaces. We characterize such decompositions in terms of suitable disjoint union decompositions of full formal arc systems, and relate this description to good cuts of the corresponding surfaces. For gentle algebras, rotations of curves induce fully faithful functors from extension-closed subcategories of module categories to the components of the associated semi-orthogonal decompositions. Under additional Abelian and extension-comparison conditions, these constructions give derived decompositions of the module categories.

Let $G$ be a finite group. In this note, we construct an element ${\sigma}^G$ in the Burnside ring of $G$ over $\mathbb{Q}$, which gives a rational number $m(G)$ that is the sum of the inverses of the element orders in $G$, by using the deflation map ${\mathrm{Def}}^G_{G/N}$ induced by the $(G/N,G)$-biset $G/N$ for a normal subgroup $N$ of $G$. As a corollary to Theorem, we obtain another expression for the rank of the crossed Burnside ring $B^{\rm c}(G)$ of $G$ and the determinant of the Cartan matrix of $p$-local Mackey algebra of $G$ over a field $k$ of characteristic $p>0$ with big enough.

We show that for any real, connected, semi-simple matrix group, its Lie algebra of algebraic vector fields can be generated by three complete vector fields.