Papers

We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $\pi$ of $\mathrm{GL}(n,\mathbb{Z})\backslash\mathrm{GL}(n,\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $\pi$, including its spectral parameters and the dimension of its minimal $K$-type. As a key ingredient, we establish new uniform decay bounds for generalized spherical functions.

For a semisimple Lie group $G$ satisfying the equal rank condition, the most basic family of unitary irreducible representations is the Discrete Series found by Harish-Chandra. In this paper, we study the structure of symmetry breaking operators for Discrete Series when restricted to a subgroup $H$ of the same type by combining classical results with recent work of T. Kobayashi, Nakahama and Pevzner. This we do by using reproducing kernels for the representations and our previous duality principle in order to find some explicit details on the nature of the differential operators representing symmetry breaking operators, in particular to what extent they are given by differentiations in normal directions to the $H$-orbit in $G/K$.

This paper solves the Wehrl-type entropy problem for arbitrary compact connected semisimple Lie groups. Let $G$ be a compact connected semisimple Lie group, and let $\pi:G\to U(V_\lambda)$ be a finite-dimensional irreducible unitary representation associated with the highest weight $\lambda$. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on $V_\lambda$. They also uniquely maximize every Husimi power moment of order $p>1$. The proof uses a second-variation at the extremizer by perturbation in directions associated with Killing fields, similar to the strategy of Frank and Lieb used in \cite{FrankLieb}. Then the problem reduces to the extreme-moment property of highest-weight vectors.

Let $k$ be a field and let $\Lambda=\left(\begin{smallmatrix}A&N\\M&B\end{smallmatrix}\right)_{\phi,\psi}$ be a finite-dimensional Morita context algebra. We introduce a bilateral approximation construction which glues support $\tau$-tilting modules over $A$ and $B$ by alternately correcting the two corner components through minimal approximations and pushouts. When this process terminates, it yields a support $\tau$-tilting $\Lambda$-module with the prescribed componentwise torsion class. The one-sided case recovers Zhang's triangular-matrix construction, while the two-sided compatibility conditions give direct corner induction and, for radical-valued connecting maps, are also necessary, extending the Gao–Huang criterion. Examples show that the bilateral correction process can terminate even when neither one-sided compatibility condition is satisfied, while in other examples the process never terminates.

Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},\mu_h^{-1})$. For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius–Hecke correspondences on its compactly supported $\ell$-adic cohomology, with coefficients in an $\ell$-adic local system. The formula, obtained by modifying the Langlands–Kottwitz method, expresses these Lefschetz numbers as elliptic stable geometric distributions on endoscopic groups of $G$, placing the cohomological traces in a form suitable for comparison with automorphic spectral data. We also give a group-theoretic criterion determining which elliptic endoscopic groups can contribute to a given Newton stratum. For certain unitary Shimura varieties, we exhibit an intermediate Newton stratum with no non-trivial endoscopic contribution, although there are non-trivial endoscopic contributions in the cohomology of the ambient Shimura variety.

This paper investigates generalized conformal modules over the Virasoro conformal algebra $Vir$, a notion originally defined by V.~Kac but largely unexplored in the literature. First, we present a twisted construction that produces new generalized conformal modules over any Lie conformal algebra from existing conformal modules. Then we give a complete classification of non-trivial generalized conformal modules over $Vir$ that are free of rank one over $\mathbb C[\partial]$, which in turn yields a classification of modules over the Lie algebra \(W_1=\bigoplus_{i\geq -1}\C L_i\) of vector fields on a line that are free of rank one over $\mathbb C[L_{-1}]$ with explicit actions expressed in terms of Bell polynomials. For arbitrary finite rank $n$, we construct a class of generalized conformal modules $V_{a,b,C(\partial)}$ and establish a complete irreducibility criterion in terms of the differential operator $\mathcal L=\frac{d}{d\partial}+C(\partial)$ acting on $\mathbb{C}(\partial)^n$. In the case of $n=2$, this criterion reduces to the absence of rational solutions of a Riccati equation. Then we prove the existence of irreducible generalized conformal modules of arbitrary finite rank, revealing a striking contrast with the conformal setting. We also construct and classify a family of infinite torsion generalized conformal modules over $Vir$, which are non-trivial and irreducible under suitable conditions.

We study the geometry of Shimura varieties associated with a Hermitian form of signature $(p,q)$ over an imaginary quadratic field $E$, where $2\leq p\leq q$. We prove that when $p\geq w_E$ and $(p,q)\neq(2,2),(2,3)$, where $w_E:=\#\mathscr{O}_E^\times$, there exists a toroidal compactification with at worst canonical singularities. As an application, combining this singularity analysis with Arthur's multiplicity formula, we prove that only finitely many pairs $(p,q)$ satisfying the above conditions and $p+q\equiv1\pmod{w_E}$ give rise to unitary Shimura varieties that are not of general type. Our method also improves the singularity bound of Gritsenko–Hulek–Sankaran (Invent.\ Math., 2007) for $\mathrm{O}^+(2,n)$ to the range $n\geq6$ and shows that this bound is sharp.

We study the vacuum structure of maximally supersymmetric Yang–Mills theory in two dimensions. At low energies, the theory decomposes into superselection sectors consisting of orbifold conformal field theories and massive vacua. The sectors are indexed by the cuspidal data appearing in Lusztig's generalized Springer correspondence. This suggests an enhancement of the correspondence to holomorphic factorization algebras.

A qKZ-type discrete flat connection with multiplicative step $p$ has monodromy operators when $p$ is a root of unity. The monodromy operators are commuting endomorphisms of the connection. We construct their eigensections and eigenvalues. The construction is based on the presentation of the discrete flat connection as a discrete Gauss-Manin connection.

We show that the universal rigid monoidal category on one object has an abelian envelope. Since it was proven by Coulembier-Etingof-Ostrik-Pauwels (2023) that this category cannot have an abelian envelope with the quotient property, this disproves the conjecture in [op.cit.] saying that every abelian envelope has the quotient property. Monoidal Ringel duality (Flake-Gruber arxiv:2512.19558) yields a candidate envelope as a lower finite highest weight category. Our proof that this is an abelian envelope relies on Coulembier-Etingof's (2024) continuants to show the required universal property. AI was used to find these results.

We construct an irreducible character of $\operatorname{GL}_{n}(q)$ whose square contains every irreducible character that is trivial on the centre of $\operatorname{GL}_{n}(q)$. This character is related to, but in general not equal to, the Steinberg character. A key ingredient, which we also prove, is that every nonlinear irreducible character of $\operatorname{GL}_{n}(q)$ that is trivial on the centre contains the trivial character when restricted to the diagonal subgroup $T$.

We use an analogue of the Springer resolution to describe the $G$-module structure on the ring of regular functions on the universal cover $\widetilde{\mathcal{O}}$ of any nilpotent orbit for $G = SL_n$. Building on previous work on the extended Springer resolution, we construct a variety $\widetilde{\mathcal{M}}$ that is finite over the cotangent bundle of a partial flag variety $G/P$, and proper and birational over the affinization $\mathcal{M}$ of $\widetilde{\mathcal{O}}$. We use techniques in birational geometry to show that $\widetilde{\mathcal{M}}$ has rational singularities, which provides the cohomology vanishing needed to describe the ring of functions on $\widetilde{\mathcal{O}}$ as an induced representation from a Levi subgroup of $G$. Our results also yield a description of the structure of $R(\widetilde{\mathcal{O}})$ as a graded $G$-module. We describe the minimal embedding of $\mathcal{M}$, study the lifting of characters of the component group of $\widetilde{\mathcal{O}}$ to parabolics and Levi subgroups, and make a more general vanishing conjecture.

We give a computer-assisted classification, up to isomorphism, of all nontrivial finite simply reducible groups of order at most 2000. There are 7889 such groups, all of even order. We provide representatives and a table of their numbers at each order. Order 1024, which is not available as a catalogue of group objects in the SmallGroups library of the computer algebra system GAP, requires a separate construction. Combining descendant generation with a classification of quadratic maps over $\mathbb F_2$, we obtain exactly 803 groups of this order, including 221 of nilpotency class two. We also derive exact counting formulas on several infinite families of orders and explicit bounds from structural subclasses.

We develop an extension of Deligne–Lusztig theory to certain (possibly infinite type) disconnected reductive groups arising from the special fibers of point stabilizers in the Bruhat–Tits building, which we call paraductive. We then compute explicit lower bounds for the Tate cohomology of representations of paraductive groups, relating these to Shintani descent, Lusztig restriction, and the Glauberman correspondence. As an application, using Feng's modular functoriality and Scholze's independence of $\ell$, we compute the Fargues–Scholze L-parameters of non-singular depth $0$ cuspidal representations of a (possibly wildly ramified) reductive group over a nonarchimedean local field.

We prove a stronger form of Cowling's $L^p$-integrability conjecture for an admissible class of Kunze–Stein groups containing all $S$-algebraic groups and all closed CCR boundary-transitive subgroups of tree automorphism groups. We also disprove a cyclic version formulated by Samei–Wiersma.

We prove that the Auslander–Reiten conjecture, the generalised Nakayama conjecture, the Nakayama conjecture, and each of the two Tachikawa conjectures are globally equivalent for Artin algebras over any fixed commutative artinian ring. For finite dimensional algebras over a fixed field, we also prove that the derived invariance of the Nakayama conjecture is equivalent to the truth of the Nakayama conjecture for all such algebras, answering a question raised by Chen and Xi.

This is the second of two papers dedicated to the explicit computation of the Fargues–Scholze correspondence. We compute the Tate cohomology of cuspidal representations arising from Yu's construction. Combining this with modular functoriality in the Local Langlands Correspondence, independence of $\ell$, and the partial characterization of the Local Langlands Correspondence established in the first paper, we compare the Fargues–Scholze and Kaletha parametrizations. Among the consequences, we deduce that the Fargues–Scholze correspondence has finite fibers and is surjective onto inertial L-parameters.

This is the first of two papers dedicated to the explicit computation of the Fargues–Scholze correspondence. In this first part, we construct (inspired by, and extending, Kaletha's explicit Local Langlands Correspondence for non-singular supercuspidal representations) semisimple inertial L-parameters for all cuspidal representations arising from Yu's constructions, with arbitrary coefficients. We then establish a characterization of this correpondence in terms of certain functoriality properties. This characterization will be used in the sequel paper in order to compute the Fargues–Scholze correspondence after restriction to inertia.

We establish bijections among spherical orbits on loop symmetric spaces, principal bundles on the twistor $\mathbb P^1$, and tempiric Langlands parameters. Using Vogan's theory of minimal $K$-types, we further establish bijections among irreducible equivariant local systems on loop symmetric spaces, irreducible local systems on principal bundles on the twistor $\mathbb P^1$, and irreducible representations of all strong inner forms of the dual symmetric subgroup. The latter bijections may be viewed as a combinatorial shadow of geometric Langlands duality for real reductive groups.

Let $k$ be a field of characteristic $p>0$, $\mathcal{F}$ a saturated fusion system over a finite $p$-group $P$, and $V$ an indecomposable capped endopermutation $kP$-module. Let $D_k^\Omega(P)$ be the subgroup of the Dade group $D_k(P)$ generated by all the relative syzygies. It is known that $V$ has an endosplit $p$-permutation resolution if and only if the Dade class $[V]$ belongs to $D_k^\Omega(P)$. We show that the resolution can be chosen to be $\mathcal{F}$-stable if and only if $V$ is $\mathcal{F}$-stable. As an application, we prove the following folklore result: if two blocks of finite groups are Morita equivalent via a bimodule with an endopermutation $kP$-source $V$ such that $[V]\in D_k^\Omega(P)$, then these two blocks are splendidly Rickard equivalent.

We construct the KLR algebra (in type $A$) as affine sections of a coherent sheaf $\mathcal{E}$ on a product of projective spaces $\mathbb{P}^n$. It was shown by Elias-Qi that the Lie algebra $\mathfrak{sl}_2$ acts on such KLR algebras; our construction geometrically explains this action as stemming from the $\mathfrak{sl}_2$-action on the twisting sheaves $\mathcal{O}(k)$ of $\mathbb{P}^1$ and gives the maximal finite-dimensional $\mathfrak{sl}_2$-submodule of KLR as global sections of $\mathcal{E}$. We furthermore extend the Elias-Qi action to a Witt action. Kleshchev-Loubert-Miemietz showed that the KLR of type $A$ has a graded affine cellular structure; our construction also gives a geometric incarnation of this.

Let $P$ be a finite connected poset and let $\Lambda_P$ be the opposite endomorphism algebra of the direct sum of all interval representations of $P$ over a field. Via projectivization, this algebra governs resolutions relative to interval-decomposable representations, which arise naturally in persistence theory. We first show that $\Lambda_P$ carries a Reedy algebra structure in the sense of Dalezios–Šťovíček. Its Reedy degree is given by the cardinality of the indexing interval, and the induced quasi-hereditary order is given by reverse interval cardinality. With respect to the resulting quasi-hereditary structure, we give a concrete combinatorial description of the standard modules and construct explicit projective resolutions of these modules. Using these resolutions, we reduce the calculation of standard–simple Ext groups to the reduced cohomology of simplicial complexes determined by the interval combinatorics. Order reversal gives the corresponding simple–costandard formula. Building on these calculations, we determine all simple–simple Ext groups. These groups are one-dimensional in a unique degree when the corresponding pair of intervals is saturated, and vanish otherwise. As a consequence, we obtain an exact combinatorial formula for the global dimension of $\Lambda_P$, which in particular shows that it is independent of the coefficient field. As an application, for the $m$ by $\ell$ grid $G_{m,\ell}$ with $m\geq\ell\geq2$, we give the explicit formula $\operatorname{gldim}\Lambda_{G_{m,\ell}}=\min\{2\ell,m+\ell-2\}$. This also gives an explicit formula for the interval-resolution global dimension of these grids, settling the corresponding grid conjectures of Asashiba–Escolar–Nakashima–Yoshiwaki and determining the stable value and the precise stabilization threshold.

Semigroups in which right divisibility is a partial order, that is, $\mathscr R$-trivial semigroups, have been studied by people in algebraic combinatorics in connection with Markov chains (especially left regular bands). Margolis and the author showed that the quiver of the algebra of an $\mathscr R$-trivial monoid is field independent. We prove here that the integral semigroup ring of an $\mathscr R$-trivial semigroup has a bound quiver presentation provided that it is unital. This then gives a uniform quiver presentation over any field via extension of scalars.

Borcherds-Kac-Moody (BKM) Lie algebras are constructed by combining a number of $\mathfrak{sl}_2$-s and $3$-dimensional Heisenberg algebras in a certain way, generalizing the celebrated Kac-Moody algebras. Such algebras have a wide range of applications, including the examples of the monster and fake monster Lie algebras being BKM algebras. Among their main features are existence of an invariant bilinear form, denominator identity and Weyl-Kac character formulas for irreducible representations. We introduce a construction of closely related algebras that we call Virasoro-Borcherds-Kac-Moody (VBKM) algebras: if it happens that certain $3$-dimensional Heisenberg algebras inside a BKM algebra form the infinite-dimensional Heisenberg algebra, we replace it by the Virasoro algebra in the construction. We construct a family of Lie algebras over $\mathbb{P}^1$, where the generic fiber is a VBKM algebra and a special fiber is a BKM algebra. This allows us to visualize VBKM algebras as deformations of BKM algebras and prove structural properties about VBKM algebras similar to those of BKM algebras, despite the absence of an invariant bilinear form. Those include denominator identity and character formulas of irreducible representations. We give examples of realizations of such $\mathbb{P}^1$-families of VBKM algebras in lattice vertex operator algebras. In particular, we include the fake monster Lie algebra into such a family.

We describe the ring of Laurent symmetric polynomials in terms of generators and relations, giving an analogue of the Fundamental Theorem of Symmetric Polynomials. We extend the Hall inner product, and give an algebraic proof that the Laurent Schur polynomials form an orthonormal basis of this ring. In the case of infinitely many variables, we realize the ring of Laurent symmetric functions as a direct limit of inverse limits, and relate them to characters of rational and algebraic representations of general linear groups.

LCD codes over finite commutative chain rings are known to be free. However, if the ring is not indecomposable, there always exists an LCD ideal that is projective but not free. We prove that all two-sided LCD codes are projective. We generalise the characterisation of finite commutative Frobenius rings by means of the size condition of a code and its dual to the noncommutative setting. This result is applied to prove that group codes are LCD if and only if they are generated by a central idempotent.

We construct, for every odd prime $p$, a bijection between the $p'$-spin characters $\widetilde S_n$ and those of a Sylow normalizer which, in particular, is equivariant under the Galois action detecting $p$-rationality. An analogous result holds for $\widetilde A_n$. We also obtain a blockwise refinement for height-zero spin characters and their Brauer correspondents.

Quasiparabolic sets extend parabolic coset spaces while retaining a length filtration and natural Hecke algebra deformations. Gelfand models built from induced linear characters ask when such spaces can account for every irreducible representation exactly once. We classify, up to equality of the individual induced characters, all quasiparabolic Gelfand models for finite irreducible Coxeter groups with inducing characters restricted from their ambient parabolic subgroups, including the additional models of type \(D_{4r+2}\) and \(B_3\). We derive the associated finite Gelfand-pair and commutant consequences, construct canonical Hecke structures for the additional classical models.

For the Jordan quiver, we construct a radial quantum trace morphism from the quantum preprojective algebra to the algebra of invariant differential operators on the Cartan subalgebra. Its classical limit recovers the classical trace morphism associated with the commuting quotient, and we prove that both the quantum and classical trace morphisms are surjective. We further relate noncommutative quantizations of the Jordan preprojective algebra to the spherical rational Cherednik algebra of type A. Finally, we derive explicit formulas for the Harish–Chandra homomorphism and its radial part in power-sum coordinates.

For the Iwahori-Hecke algebras $\mathcal H_q(\mathfrak S_n)$ of the symmetric group $\mathfrak S_n$ at a primitive $e$-th root of unity, James and Mathas proved a theorem which relates $v$-decomposition numbers $d_{\lambda\mu}^{e}(v)$ for different values of $e$, by adding “empty runners” to the abacus display for the labelling partitions $\lambda,\mu$. Fayers proved a similar theorem, which involves adding “full” runners to these abacus displays. In this paper we use Uglov's map from the set of $r$-abaci for $r$-partitions to the set of $e$-abaci for partitions to extend these theorems to the cyclotomic Hecke algebras of type $G(r,1,n)$.