Papers

These lecture notes are intended as an introduction to the theory of rational Dunkl operators, the associated special functions and related Markov processes with an emphasis on examples which are related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones of positive semidefinite matrices. We start with a comprehensive introduction into Dunkl theory: Dunkl operators, the intertwining operator and its positivity, the Dunkl kernel and the Dunkl transform, the Dunkl Laplacian and the associated heat semigroup. We further give an outline of the connection with Calogero-Moser-Sutherland models and generalized Hermite polynomials. Moreover, of central interest will be product formulas, generalized translations and associated commutative hypergroup structures on closed Weyl chambers. In particular, we explain how Dunkl theory for particular multiplicities is related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones, and how this leads to a bunch of multiplicities for which the Weyl-group invariant Dunkl theory admits a probability preserving translation and an associated commutative hypergroup structure on the closed Weyl chamber. We finally discuss Markov processes on R^N which are related with Dunkl theory with an emphasis on connections to random walk on groups and hypergroups. In particular associated martingales, martingale characterizations, moment functions and Appell characters are studied, i.e. diffusion-reflection processes with the Dunkl Laplacians as generators.

This paper gives a criterion for an endopermutation Scott module to be slash indecomposable. In particular, under suitable assumptions, we only need to prove certain endopermutation Scott modules over local subgroups are slash indecomposable.

Finite monodromy provides a bridge between group representations and algebraic solutions of differential equations. We study this connection for the Katz-Long-Moody construction, which transforms representations of the semidirect product of a free group and a braid group into new representations of the same group and is related to Knizhnik-Zamolodchikov (KZ)-type equations. For a fixed finite-image input, we classify the parameter values for which the resulting representations have finite image, both on the semidirect product and on its free-group and braid-group subgroups. In particular, finiteness of the braid-group image is independent of the admissible parameter. These results give necessary and sufficient conditions for all solutions of the corresponding regular-singular KZ-type equations to be algebraic. On restriction to the free group, they also characterize finite monodromy and algebraicity of all solutions of the associated Fuchsian systems, connecting the classification to classical questions about algebraic hypergeometric functions.

Nonlinear activations can create equivariant interactions between irreducible representations that linear maps cannot. We use the Gaussian degree decomposition to extend ordinary polynomial degree to such nonlinear maps, and prove that for a fixed coordinatewise equivariant layer each degree factors into a polynomial determined by the linear maps and a scalar determined by the activation. This separates three distinct obstructions, coming from symmetry, coordinates, and activation.

We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space $X$, we prove that the Gabriel spectrum of ${\mathbf{Sh}}(X)$ is homeomorphic to ${\mathrm{Sk}}({\mathrm{Sob}}(X))$, where ${\mathrm{Sob}}(X)$ denotes the sobrification of $X$ and ${\mathrm{Sk}}$ indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if $X$ is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ${\mathbb R}^n$ is homeomorphic to the space of ideals of ${\mathbb R}^n$ with a natural topology.

In this article, we consider a general version of the classical group ring isomorphism problem, called the twisted group ring isomorphism problem (TGRIP), which determines an isomorphism between the twisted complex group algebras of finite groups. Although for isoclinic groups $G$ and $H$ their complex group algebras are isomorphic, i.e, $\mathbb C G \cong \mathbb C H$, their twisted complex group algebras need not be isomorphic. Continuing this line of investigation, we establish sufficient conditions under which two isoclinic groups have isomorphic twisted complex group algebras, thereby providing solutions to (TGRIP) for isoclinic groups. As applications, we study (TGRIP) for special $p$-groups of rank $2$, unicentral groups, and nilpotent groups of class $2$ with elementary abelian Schur multipliers, all considered up to isoclinism. We also present several examples illustrating the main results, including a complete solution of (TGRIP) for special $p$-groups of rank $2$ of order $p^6(p \ge 3)$

The aim of this article is to leverage Enomoto's classification of exact structures to the dg level. Let $\mathscr{A}$ be a connective additive idempotent complete dg category. We establish an intrinsic Enomoto-type classification of exact dg structures on $\mathscr{A}$ in terms of bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Our proof is a direct dg argument based on dg totalizations of $3$-term complexes and dg duality, rather than a reduction to the greatest exact dg structure and the classification of its substructures. As a consequence, we establish a bijection between exact dg structures on $\mathscr{A}$ and bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Moreover, this bijection is an isomorphism of posets and yields a complete lattice structure on the class of exact dg structures on $\mathscr{A}$. In particular, we provide another construction of Rump–Chen's greatest exact dg structure on $\mathscr{A}$.

We study $n$-torsionfree objects in abelian categories with enough projectives. Frobenius functors preserve $n$-torsionfreeness, and faithful ones reflect it. We prove that stabilization of the torsionfree filtration implies weak Gorensteinness. For Frobenius extensions satisfying a generator condition, we compare the terms of minimal injective resolutions and obtain transfer of Auslander-type conditions and of the Auslander–Gorenstein conjecture. We also compute a family of non-Gorenstein algebras whose torsionfree filtrations stabilize at level two and contain explicit nonprojective Gorenstein projective modules.

In this paper we obtain a complete description of all indecomposable characters (central positive-definite functions) of "block-diagonal" inductive limits of semidirect products $\mathbb{T}^N\rtimes\mathfrak{S}_{N}$ of the compact $N$-dimensional torus $\mathbb{T}^{N}$ with the symmetric group $\mathfrak{S}_{N}$. The classification is proved by means of a variant of the ergodic method of Vershik and Kerov, namely by studying the weak limits of characters of finite-dimensional subgroups. As an application of our results, we give another proof of the classification of indecomposable characters of the infinite unitary group with block diagonal embeddings.

We introduce posets of type $\mathbb{D}$, a family of posets described by an admissible Dynkin quiver of type $D_n$ together with a compatible set of extra arrows (alien arrows), and show that their socle-projective representation category is of finite representation type. Continuing, in the same spirit, a program initiated for posets of type $\mathbb{A}$ [R. Schiffler, R-J. Serna, A geometric realization of socle-projective categories for posets of type $\mathbb{A}$, J. Pure Appl. Algebra 224 (2020), no. 12, 106436], we build on an existing geometric model for cluster-tilted algebras of type $D_n$, whose geometric combinatorics is more intricate than that of type $\mathbb{A}$. Our main result establishes a $\Bbbk$-linear categorical equivalence $\Theta$ between a full subcategory $(\mathcal C/T)_F$ of the arc category of a punctured $(n+3)$-gon, whose objects are certain $sp$-arcs, and the category of finitely generated socle-projective $\Bbbk\mathscr{P}$-modules, for $\mathscr{P}$ a poset of type $\mathbb{D}$. As a consequence, when the set of alien arrows is empty, we conclude that the cluster subalgebra generated by the $sp$-arcs coincides with the full cluster algebra.

We introduce an analogue of Gröbner bases, or equivalently, Bergman's diamond lemma, for linear (super)monoidal categories. It gives a systematic way to study monoidal ideals and to prove basis theorems. The central notion is that of a monoidal Gröbner system and is based on higher linear rewriting theory. We then apply the theory to categories of affine Brauer type: linear (super)monoidal categories that have the same hom-basis as the affine Brauer category, but possibly distinct composition and tensor product. Our main result is a criterion for a category to be of affine Brauer type, reducing the question to an explicit list of local computations, which we partially implement in the computer algebra system FORM. This gives new combinatorial proofs of the basis theorems for the affine Brauer category, the nil-Brauer category and the affine VW supercategory, and yields new examples. In particular, we construct the odd nil-Brauer category, a conjectural supercategorification of the split $\imath$quantum group of rank one, and the quantized affine VW supercategory, whose non-affine part is the quantized periplectic Brauer category and which we expect to satisfy a higher quantum Schur–Weyl duality. Finally, we classify the categories of affine Brauer type admitting a sufficiently simple presentation.

In this paper we present a formalism of categorical representation theory for affine group schemes of finite type over fields (both in the weak and strong settings) which applies to arbitrary base fields. This is based on the development of an appropriate $\infty$-categorical setting for the study of representations and Harish-Chandra (bi)modules of such group schemes, which constitute the building blocks and basic examples of such structures. We also study categorical traces in this context, and show in particular that the categorical trace of the identity morphism, resp.~Frobenius morphism, on the appropriate $\infty$-category of representations of a connected reductive algebraic group $G$ over an algebraically closed field $k$ of positive characteristic identifies (under suitable assumptions) with the $\infty$-category of Ind-coherent sheaves on the adjoint quotient $G/G$, resp.~of the fixed points $G^{\mathrm{F}}$ of the Frobenius.

We construct free immersions of $P^3 \times S^1$ and $S^3 \times S^1$ into $14$-space. These are the first examples of free immersions in critical dimension of closed manifolds that are not spheres, projective spaces, tori, or surfaces. The idea is to mimic De Leo's construction for $m$-tori with $m \leq 5$. De Leo constructs a map that is symmetric with respect to a representation of the Lie group formed by the $(m - 1)$-torus; we replace this group with $\mathrm{SO}(3)$. In our examples, the osculating determinant is constant.

The purpose of this paper is to prove that every finite simple group of type $\mathsf B$ or $\mathsf C$ satisfies the inductive blockwise Alperin weight condition at every prime $\ell$ dividing its order. For ${\rm PSp}_{2n}(q)$, where $q$ is odd and $n\geq3$, a permutation lattice argument based on Conlon's induction theorem removes the unitriangularity assumption from earlier results on the inductive condition at odd nondefining primes. At $\ell=2$ in odd defining characteristic, errors and omissions in the underlying classification of radical $2$-subgroups affect an earlier parametrisation of weights for arbitrary blocks. We verify the required weight parametrisation for principal blocks and use Jordan reduction to establish the inductive condition for all blocks. At odd $\ell$ in even defining characteristic and rank at least four, we prove the inductive condition using generic weights and Jordan reduction, without assuming unitriangularity. For $\Omega_{2n+1}(q)$, where $q$ is odd and $n\geq3$, we remove the unitriangularity assumption at odd nondefining primes using Conlon's induction theorem. At $\ell=2$, we prove the stabiliser and extension property for Brauer characters of $\operatorname{Spin}_{2n+1}(q)$ that was assumed in earlier work on the inductive condition. We also give proofs of the inductive condition for the remaining sporadic groups $J_4$, $Fi'_{24}$, the Baby Monster and the Monster, without making a priority claim. Together with the previously established cases, these results imply that the blockwise Alperin weight conjecture holds at $\ell$ for every finite group each of whose nonabelian simple sections of order divisible by $\ell$ is of type $\mathsf B$, of type $\mathsf C$, or sporadic.

Transcendence questions of periods and quasi-periods of Anderson modules, as well as the hyperderivatives thereof, are of major interest in number theory over function fields. In the case of Drinfeld modules, this question was answered by the second author. In this paper, we show that these results hold for general Anderson $t$-modules under certain assumptions on their Galois representations. The proofs use the explicit description of the $\mathfrak{p}$-adic Galois representation of the first author by means of a rigid analytic trivialization of the associated $t$-motive, as well as prolongations of $t$-modules and $t$-motives. The general result is applied to determine algebraic relations between periods and quasi-periods, and their hyperderivatives, of finitely many Drinfeld modules. Along the way, our results provide more evidence for the Mumford-Tate conjecture for Anderson modules.

We compute an explicit contribution of supercuspidal representations arising from Yu's construction to the middle étale cohomology of local Shimura varieties. For the proof, we construct special affinoids in local Shimura varieties whose reductions are isomorphic to Heisenberg Deligne-Lusztig varieties and then apply positive-depth Deligne-Lusztig theory. Along the way, we propose a stacky analogue of special affinoids using the stack of local shtukas and observe that the `general canonical subgroup problem' can be solved purely on the stack of local shtukas, which yields the corresponding claim on Shimura varieties via the Hodge-Tate period map.

In this paper, we give an $A_\infty$-enhancement of the triangulated orbit category $(\mathcal{T}/[1])_{tr}$ when $\mathcal{T}$ is an algebraic triangulated category. We then use this enhancement to give a description of $(\text{per}(A)/[1])_{tr}$ in terms of differential modules and Cohen-Macaulay modules when $A$ is a finite dimensional $\mathbb{Z}$-graded algebra of finite global dimension.

Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Klász and Marczinzik. We use this to show that the Hochschild cohomology of $KG$ is nonzero in every nonnegative degree whenever the characteristic of $K$ divides $|G|$.

This paper studies commutators in finite-dimensional unimodular Hopf algebras and invariants derived from them. For a unimodular Hopf algebra $H$ with the distinguished group-like element $g$, the commutator of $a,b\in H$ is defined by $\{a,b\}=a_{1}S^{2}(b_{1})S(a_{2})S^{-1}(b_{2})g^{-1}$. We show that the span of all commutators is a left coideal and generates a normal left coideal subalgebra $N$ of $H$ for which $H/HN^{+}$ is commutative. The main result asserts that for any central element $a$ and any integral $\Lambda$ of $H$, the commutator $\{a,\Lambda\}$ and the value $\lambda(\{a,\Lambda\})$ for a right integral $\lambda$ of $H^*$ are invariant under twisting. This yields several gauge invariants, including ratios of characters of iterated commutators with $\Lambda$. In the semisimple characteristic-zero case, explicit descriptions of these invariants are given. As applications, we compute some invariants for the group algebra of a finite group, its Drinfeld double, and the restricted quantum group $\overline{U}_{q}(sl_{2})$.

Let \(A\) be a finite-dimensional quantum complete intersection over an arbitrary field. We prove that a finite-dimensional left \(A\)-module \(M\) is projective whenever \(\Ext_A^1(M,M)=\Ext_A^2(M,M)=0\), with no restriction on the multiplicative orders of the commutation parameters. Consequently, \(A\) satisfies the Auslander–Reiten conjecture and Tachikawa's second conjecture. The proof combines the gradability of rigid modules with successive graded twists and the self-extension criterion of Avramov and Buchweitz for commutative complete intersections. We derive a formula comparing ordinary self-extension groups under graded twisting, apply the resulting criterion to the Liu–Schulz algebras, and describe the successive twists explicitly in the four-generator case.

In this article, we first introduce quasi-overlap and quasi-grouping functions on bounded pseudo-ordered sets, respectively, and give their respective preliminary constructions and the induced quasi-overlap functions on quotient posets.. We then explore the invariant properties under weighting and automorphisms. We also introduce the migrativity, the homogeneity, the idempotency, the cancellation law, the Archimedean and limiting properties of quasi-overlap functions and discuss their relations. We finally present two constructing methods of quasi-overlap functions, which are based on boundary-faithful mappings and adjunctions on bounded psosets and the transitive-endpoints of a bounded trellis, respectively. The corresponding statements are valid for quasi-grouping functions as well.

Let $F$ be a singular fibre of a relatively minimal complex elliptic fibration with smooth total space, and let $\Omega$ be a nonvanishing holomorphic two-form near $F$. We show that a small neighbourhood of $F$ is a Weinstein domain for $\mathrm{Re}\,\Omega$ whose completion is a Legendrian surgery, with cocores obtained by completing holomorphic disks transverse to the components of $F$. For every coefficient field and every multiplicative bulk class, we compute the wrapped Floer cohomology of these cocores and prove that it is concentrated in degree zero. The cocores generate, so the bulk-deformed wrapped Fukaya category is equivalent to the category of perfect modules over an explicit algebra: a multiplicative preprojective algebra of affine type for the normal crossing fibres, and a quiver algebra with relations for types $II$, $III$ and $IV$. Applications include formality of the affine plumbing dg algebras; mirrors given by resolved affine surfaces at every classical bulk class for affine D, E and types II, III, IV; and mirrors given by quotient stacks of algebraic tori at root-of-unity bulk classes for the four affine stars.

We construct a bar-type operation for the affine Hecke algebra $\Ha$ attached to a symmetrizable Kac–Moody group. Our main tool is a geometric bimodule defined using the equivariant $K$-theory of a suitable version of the Steinberg variety for this Kac–Moody group. Using elements of this bimodule corresponding to standard and costandard Hodge $D$-modules on the Kashiwara flag variety, we define a bar involution on $\mathcal{H}_{W}^a$ by relating the left and right actions of $\mathcal{H}_{W}^a$ on this bimodule. A central issue is that the resulting formulas for the bar operation do not preserve the algebra itself and naturally produce infinite expansions. To address this, we introduce certain completions of $\mathcal{H}_{W}^a$ on which the bar operation is well-defined. For this construction, we prove a number of combinatorial finiteness results for products in $\mathcal{H}_{W}^a$. In affine type, we further analyze the level-zero part of the algebra and obtain an explicit combinatorial formula for a bar operation on a suitable localization of Cherednik's double affine Hecke algebra. In this case the bar operation itself is not an involution but we also show that this bar operation becomes an involution when the lattice parameter $q$ is set to $1$.

Let $\widetilde G$ be a nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In previous work, we introduced $\prod_{\rho/2}^s(\widetilde G)$, the set of genuine small representations with infinitesimal character $\rho/2$ in the simply laced setting, and showed that, when $G$ is split, these representations are exhausted by the Kazhdan–Patterson lift of the trivial representation. In this paper, we consider nonsplit real Spin groups of type $D_n$. For each $\rho/2$-compatible genuine central character, we determine the irreducible constituents of $\mathrm{Lift}_G^{\widetilde G}(\mathbb C)$. We also describe explicitly the relevant genuine small representations in terms of Langlands parameters. The nonsplit case exhibits a new phenomenon: the lift of the trivial representation need not exhaust $\prod\nolimits^s_{\rho/2}(\widetilde G)$. For the families considered here, equality holds only for $\widetilde{\mathrm{Spin}}(2p+2,2p)$; in the other two families the lift is a proper subset, and in one case it is zero.

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation $\Theta(n^0)$ short closed geodesics, potentially $\Theta(n^0)$ pairs of nearby order-2 cone points, in general $\Theta(n^{\frac{1}{m}})$ cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).

Let $G$ be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish–Chandra centers of the associated Kac–Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group $\check{G}$. Namely, given a noncritical level $\kappa$ for $G$, consider the associated Kac–Moody vertex algebra $V_\kappa(\mathfrak{g})$ defined on a formal disc $\mathscr{D}$, and its Harish–Chandra center of arc group invariants $$V_\kappa(\mathfrak{g})^{\mathscr{J}G} \subset V_\kappa(\mathfrak{g}).$$ Write $\check{\kappa}$ for the dual level for $\check{G}$, $\check{\kappa}^p - \check{\kappa}$ for its image under the Artin–Schreier map, and $\operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}$ for the moduli space of $(\check{\kappa}^p - \check{\kappa})$-opers on the Frobenius twisted disc $\mathscr{D}^{(1)}$. We establish a canonical isomorphism $$\operatorname{Spec} V_\kappa(\mathfrak{g})^{\mathscr{J} G} \simeq \operatorname{Op}_{\check{G}}(\mathscr{D}^{(1)})_{\check{\kappa}^p - \check{\kappa}}.$$ There is a similar identification for the Harish–Chandra center of the filtered complete enveloping algebra at level $\kappa$, where instead of $\mathscr{D}$ we need to consider the punctured disc $\mathscr{D}^\times$. The presence of these Harish–Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod $p$ of the characteristic zero Harish–Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.

We construct a transferred $L_\infty$-structure on the shifted Chouhy–Solotar cochain complex of a finite-dimensional skew-gentle algebra and give explicit basis-level formulas for its binary operation $l_2$ in all shifted degrees. We further prove that the Hochschild dg Lie algebra of an $A_n$-type skew-gentle algebra is homotopy abelian, and hence $L_\infty$-formal, and contrast this with Wong's non-formality result for the dual numbers $\mathbb{K}[\theta]/\langle \theta^2 \rangle$, leading to a general formality classification problem.

We study the behavior of generalized Shalika models under the local theta correspondence for type II pairs of equal rank. This generalizes a result of Gan on the relation between Shalika models and linear models.

We prove the sign-isotypic components of the coinvariant rings $R_n^{(2,1)}$ and $R_n^{(2,0)} \otimes R_n^{(0,1)}$ are isomorphic and show that the triply-graded multiplicity of this sign character is the Schröder polynomial $S_n(q,t,a)$, divided by $1+a$. This settles the sign-character component of a conjecture of Zabrocki (2019) on a module for the Delta theorem and proves a conjecture of F. Bergeron (2020) on the multiplicity of the sign character of $R_n^{(2,1)}$. Finally, using a result of Hogancamp (2017), we enhance a recent result of Gorsky–Mellit (2026) which relates the Khovanov–Rozansky homology of the $(n,n+1)$-torus knot to $R_n^{(2,0)} \otimes R_n^{(0,1)}$, by showing that the associated Poincaré series for this knot can be computed from the sign component of $R_n^{(2,1)}$.

In work of the first author, the geometric Satake equivalence was (non-rigorously) reinterpreted as an equivalence between two algebraically-defined $2$-categories, one built from representations of a Lie algebra, and one built using singular Soergel bimodules. It was then explained how to $q$-deform this equivalence in type $A$, replacing the special linear Lie algebra with its quantum group, and using singular Soergel bimodules for a deformed reflection representation. Both the algebraic reformulation and its $q$-deformation were only proven in types $A_1$ and $A_2$. In this paper, we prove the result in type $A_{n-1}$ for $n \ge 4$, while working generically: more precisely, we work over a field of characteristic zero, where $q$ is not a root of unity, and having adjoined an $n$-th root of $q$. Along the way we generalize certain results in the literature (e.g. the Soergel-Williamson categorification theorem, the Soergel conjecture for spherical elements, various symmetries) to the deformed reflection representation.