Papers

We prove the full generating series arithmetic Siegel–Weil formula on the product of two modular curves, with arbitrarily singular coefficients. This is the case $\operatorname{GSpin}(2,2)$ in Kudla's program.

Let theta be an ordinary character or a Brauer character of a finite group N. We show that theta extends to every overgroup $G=G_\theta\trianglerighteq N$ if and only if Z(N) <= Ker(theta) and the deflated character of Inn(N) extends to Aut(N)_theta. This equivalence is compatible with fields of values. A similar investigation for extensions from subnormal subgroups leads to a new characteristic subgroup of N.

Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators \(C_1,\ldots, C_L\), constructed from a degeneracy graph which records how the eigenspaces of \( C_1\) are successively split by \( C_2,\ldots, C_L\). They showed that the basis property is equivalent to the non-vanishing of the determinant of the matrix of values of these monomials on the projector basis, and conjectured a factorised formula for this determinant, as a product of powers of eigenvalue differences. We prove this formula for any number of generators. The proof uses graph analogues of Newton polynomials. Their evaluation matrix is not triangular in general, but expanding the rows of the children of each node of the penultimate layer in a Newton basis of their eigenvalues makes it block triangular, with diagonal blocks given by smaller degeneracy graphs in which the final layer is removed. The formula then follows by induction on the number of layers. We show that the exponents in the determinant are overlap volumes of higher-dimensional Young diagrams attached to the nodes of the graph: intervals, ordinary Young diagrams, plane partitions and their higher analogues. Finally, the block triangular structure gives an explicit recursive inversion of the evaluation matrix, expressing the projectors in the monomial basis.

Let $W$ be a complex reflection group and $X$ the De Concini-Procesi wonderful model of its reflection arrangement. Inside Etingof's sheaf of Cherednik algebras on $X$ we define a subsheaf generated by the Dunkl operators that extend the standard ones on the hyperplane complement. We conjecture that at $t=0$ the image of the logarithmic vector fields in a natural module over this subsheaf is the sheaf of Gaudin algebras on $X$ defined by Aguirre, extending a construction of Bonnafé and Rouquier from the hyperplane complement to $X$. We prove the conjecture for $W=S_{3}$.

The Fomin-Kirillov algebras $\mathcal{FK}_n$ are shown to be infinite-dimensional for $n\geq 6$. By computer calculations, we find an explicit nonzero primitive element of degree 14 in the identity component of $\mathcal{FK}_6$. For every $n\geq 6$, the existence of such an element implies that $\mathcal{FK}_n$ contains a polynomial subalgebra and is therefore infinite-dimensional, and that the canonical homomorphism from $\mathcal{FK}_n$ onto its associated Nichols algebra is not an isomorphism.

Let $\DD$ be $\R$, $\C$, or $\HH$, and let $H=\GL_p(\DD)\times\GL_q(\DD)$ be the block diagonal subgroup of $G=\GL_{p+q}(\DD)$. We prove that $\dim\Hom_H(\pi,\eta)\leq1$ for every irreducible Casselman–Wallach representation $\pi$ of $G$ and every smooth character $\eta$ of $H$.

Let $(W,S)$ be a Coxeter system of finite rank, let $T$ be its set of reflections and let $c \in W$ be a Coxeter element. For $x$ in the interval $[1,c]_T$ of the absolute order, let $P(x)$ be the subgroup generated by the reflections $t\le_T x$. We prove that $x$ is a Coxeter element of $P(x)$ with respect to the canonical Coxeter generators of $P(x)$, that $P(x)$ is generated by the factors of any reduced reflection factorization of $x$, and that the Hurwitz action on the reduced reflection factorizations of $x$ is transitive. The proof reduces the general case to the case of a Coxeter system whose Coxeter matrix has entries in $\{1,2,3,\infty\}$, which was treated by Hubery and Krause.

Let $p$ be an odd prime, and let $\rho\colon G\to \GSp_{2n}(\Z_p)$ be an irreducible symplectic residually multiplicity free representation. We define the symplectic extension graph $\Gamma_\rho^\GSp$ associated with $\rho$ to be the directed graph whose vertices are the residual factors of $\rho$ and a directed edge between two vertices corresponds to a nonsplit extension between the vertices realized by $\rho$. Using the Euclidean Bruhat–Tits building associated with $\GSp_{2n}(\Q_p)$, under some assumptions, we prove the existence of certain paths in $\Gamma_\rho^\GSp$ and show that $\Gamma_\rho^\GSp$ is a disjoint union of nonempty strongly connected subgraphs. Moreover, we provide examples to show that the result we obtain is optimal.

We introduce a $q$-analogue of the bicolored Eulerian quasisymmetric functions by incorporating an additional grading arising from affine length, obtained by interpreting bicolored permutations as affine permutations. We also provide an intrinsic description of this refinement in terms of bicolored permutations by introducing notions of alignments and crossings, together with an explicit bijection identifying these statistics with the alignments and crossings in the existing literature. Finally, we prove that our refinement is symmetric by expressing it in terms of the Theta operators appearing in the Theta coinvariant Conjecture and the modified Macdonald symmetric functions. Assuming the Theta coinvariant conjecture implies the Schur-positivity of our refinement.

Let $p$ be a prime $\ge 5$. We determine the semisimplifications of the mod $p$ reductions of irreducible two-dimensional crystalline representations of the Galois group of ${\mathbb Q}_p$ of slope $2$ and weights $k\ge3p+4$ with $k \not \equiv 6 \bmod (p-1)$. The proof relies on the compatibility between the $p$-adic and the mod $p$ Local Langlands Correspondences for $\mathrm{GL}_2(\mathbb{Q}_p)$.

Motivated by the 3d-3d correspondence, we introduce an index $\mathcal{I}_g^\phi$ for a mutation loop $\phi$ of a quiver and an integer $g \geq 0$. We prove its integrality $\mathcal{I}_g^\phi \in \mathbb{Z}$ under certain conditions, which extends the integrality conjecture of Gang–Kim–Yoon for $3$-manifolds to the broader framework of mutation loops in cluster algebras. Our index also has an interpretation as the genus-$g$ partition function of a $(1+1)$-dimensional TQFT. We provide explicit computations for mutation loops of period one. In particular, in the case of the generalized Kronecker quivers, we give an explicit formula expressing the generating function of the indices for all genera $g$ as a ratio of polynomials. Altogether, we provide a new insight to the interplay between the geometry of $3$-manifolds and cluster algebra.

We give a general formula for minimal projective presentations and for minimal discrete-injective presentations of finite-dimensional modules over basic pseudocompact algebras. We also show that the representation-theoretic g-vector of a module coincides with its DWZ-g-vector. These results generalize the description that Derksen, Weyman and Zelevinsky gave of minimal projective presentations for modules over certain finite-dimensional Jacobian algebras. In particular, we can drop their finiteness assumption, and we may work over fields of arbitrary characteristic.

We introduce a diagram algebra $\mathcal{K}_L$ based on Kuperberg's $A_2$-webs and prove a Schur–Weyl duality $\mathcal{K}_L \cong \operatorname{End}_{U_q(\mathfrak{sl}_3)}((V^+ \oplus V^-)^{\otimes L})$, where $V^\pm$ are the fundamental representations of $U_q(\mathfrak{sl}_3)$. Thus, $\mathcal{K}_L$ is a natural $A_2$ analogue of the Temperley–Lieb algebra. We construct an explicit cellular structure for $\mathcal{K}_L$, with the web basis as its cellular basis and cell modules indexed by dominant $U_q(\mathfrak{sl}_3)$-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of $\mathcal{K}_L$ and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate $\mathcal{K}_L$, and identify an additional local generator. Finally, we derive a bimodule decomposition of $(V^+ \oplus V^-)^{\otimes L}$, yielding a complete set of irreducible $\mathcal{K}_L$-modules for formal or generic $q$.

The Okubo algebra brings together ternary gradings, integral root frames and rank-three geometry. We examine how these structures extend to the complex Exceptional Periodicity algebras. With $N=4(n+1)$, a tetracode character gives eight nonzero homogeneous components of dimension $t_n=8n+3+2^{4n}$. On a tip, a compatible grading has zero component of dimension $8n-5$ and eight nonzero components of dimension $1+2^{4n-3}$. The arithmetic behaves differently: for $n\geq2$, the lattice $D_N^+$ has no full-rank orthogonal frame of unscaled $A_2$ lattices. We exhibit failures of the Jacobi and Jordan identities and determine the derivations and full automorphism group of the ambient algebra above level one. We also give a Clifford-module proof, with the normalization fixed by root products, of the published realization of EP tips as special Vinberg $H$-algebras. This identifies a positive real form and yields explicit formulas for its cone, cubic, Cholesky coordinates and rational dual invariant. The cone is symmetric exactly at $n=1$. Its determinant-one hypersurface belongs to the known $L(8n,1,0)$ series, whose $r$- and $c$-maps give homogeneous scalar manifolds for supergravity theories with eight real supercharges.

We introduce river ribbon tableaux, where a river of non-trivial ribbons is placed between two banks of single boxes. Compared to other ribbon peelings, this reduces the geometric differences across diagrams of different shapes, and the maximal number of ribbon configurations. This allows us to prove a uniform character bound for symmetric groups which, specialized to balanced diagrams, proves the conjecture of Moore and Russell. As an application, we prove a conjecture of Dousse and Féray on the number of standard tableaux of skew shapes.

We study automorphisms of Hausel's medium algebras by identifying them with equivariant cohomology algebras of affine Schubert varieties of a complex reductive group $G$. The automorphisms in question are then given by the action of either complex automorphisms or real structures of $G$ on affine Schubert varieties. We focus on describing the fixed points of those automorphisms on the spectra of medium algebras, and prove that the torsion-free quotients of the corresponding coinvariant algebras are isomorphic to equivariant cohomology rings of fixed loci in affine Schubert varieties. In the case of real structures, this naturally leads to and involves the study of affine Grassmannians for quasi-anisotropic (also called quasi-compact) real reductive groups, closely related to recent work of O'Brien. We also discuss relations to automorphisms of Higgs bundle moduli and to symmetric pair variants of medium algebras.

We give two counterexamples to Stapledon's Effectiveness Conjecture. Each counterexample is a $6$-dimensional lattice polytope with an order-two symmetry whose equivariant $H^*$-series $\varphi[t]$ is a polynomial with one coefficient that is not effective. For polytopes with a symmetry of order two, the goal of this paper is to build up a picture of the landscape of results for when the Effectiveness Conjecture holds. Our main result is a criterion, in terms of the faces of $P$, that determines when the Ehrhart ring of $P$ admits an invariant linear system of parameters of the correct representation type. We use these results to reduce the space of possible counterexamples to one of two cases. A randomised computer search produced counterexamples in each case. We also give an example where the equivariant $H^*$-series is effective but admits no invariant linear system of parameters, neither for $P$ nor any invariant subdivision of it, answering a question of Stapledon in the case of groups of order two.

Let $F$ be a non-Archimedean local field of characteristic zero. We prove multiplicity one for every irreducible smooth admissible representation in the split forms of Wan-Zhang Models 6-10. Combining the Gelfand-Kazhdan method with nilpotent-wavefront restrictions on Bernstein-finite equivariant distributions, we establish the required symmetries through rational orbit analysis, parabolic descent, stabilizer characters, and conormal arguments. The type-$D_6$ analysis treats both the vector-similitude group of Model 9 and the semispin form arising in the exceptional $E_7$ model. For Model 10, the distribution theorem gives a product bound for periods; the individual multiplicity bound follows from Langlands classification and a vanishing theorem for standard modules induced from nongeneric tempered data on the maximal $E_6$ Levi.

We develop an algebraic theory of local intertwining periods for symmetric pairs $(G,H,\theta)$ over non-archimedean local fields $F$ of characteristic different from $2$, and finite-length representations of Levi subgroups of $G$ with coefficients in an algebraically closed field $R$ of characteristic different from the residual characteristic of $F$. For open orbits this algebraic version is essentially contained in the work of Blanc and Delorme. The algebraic theory relies on a crucial new uniqueness result, extending a result of Blanc and Delorme for open periods, which asserts that an admissible intertwining period is the only $H$-equivariant linear form extending the corresponding integral on the space of well-supported sections. Moreover the algebraic construction does not require the unimodularity hypothesis imposed in the work of Matringe–Offen–Yang. When $R=\mathbb{C}$, we also remove this assumption in the classical non-archimedean analytic setting, and prove convergence of intertwining periods for all symmetric spaces. At the same time we extend the context of the whole theory in the following manner: we allow $F$ to be of positive odd characteristic, we allow twisting characters in the periods, and we allow $H$ to lie between $(\mathbb{G}^\theta)^\circ(F)$ and $G^\theta$. Finally, in the algebraic setting, as in the work of Blanc and Delorme on open periods, we obtain explicit bounding polynomials for the singularities of the intertwining period. Such bounds are a key ingredient of a new proof of the cuspidal case of a conjecture of D. Prasad on quaternionic symplectic models.

Replacing the circle by a manifold leads to current Lie algebras and their central and derivation extensions. We separate the coefficient algebra, harmonic analysis and geometric data. Differentials modulo exact forms describe central extensions; divergence-free vector fields give explicit cocycles, whose compatible derivations satisfy a stabilizer equation. Compact examples include the three-sphere and its current–Virasoro extension. We also explain the analytic qualifications for SL(2,R) and its symmetric space and the role of transported multiplication in soft geometries. The recent harmonic constructions of Campoamor-Stursberg, Marrani and Rausch de Traubenberg are compared with the cohomological work of Coquereaux, Frappat, Ragoucy and Sorba and Castellani's loop geometry and infinite-dimensional free differential algebras. A coefficient-algebra construction connects currents to higher brackets and clarifies the additional requirements of a dynamical theory. For Lie algebras of order F, central extensions obey coupled binary and symmetric F-ary conditions, with explicit examples and obstructions.

We give two types of examples where two non isomorphic finite quandles have isomorphic enveloping groups. This implies that two non-isomorphic finite quandles can share isomorphic representation categories (as additive and tensor categories simultaneously).

Let $\mathfrak p$ be the $(-1)$-eigenspace of a complex semisimple symmetric pair. For every $r\ge2$, we prove $\dim C_r(\mathcal N(\mathfrak p))\le \lfloor r\dim\mathfrak p/2\rfloor$ and show that every irreducible component is isotropic for a natural family of alternating forms. For a nonzero Hermitian pair with $\mathfrak g$ simple and $\mathfrak p=\mathfrak p^-\oplus\mathfrak p^+$, put $d=\dim\mathfrak p^+$ and $\ell=\operatorname{rk}(\mathfrak g,\mathfrak k)$. Then \[ \dim C_r(\mathcal N(\mathfrak p))=rd,\qquad \dim C_r(\mathfrak p)=\max\{rd,\,2d+(r-1)\ell\}. \] In this case we also determine the maximal-dimensional components of $C_r(\mathfrak p)$; for $r\ge3$, the only maximal-dimensional nilpotent components are $(\mathfrak p^+)^r$ and $(\mathfrak p^-)^r$. For arbitrary symmetric pairs, $\dim C_r(\mathcal N(\mathfrak p))=ra+m$ for all large $r$, where $a$ is the largest dimension of an abelian subspace of $\mathfrak p$ consisting of nilpotent elements and $m$ is the dimension of the variety of such subspaces. Applications give dimension and component formulas for equivariant punctual Quot schemes under explicit framing conditions, and bounds for punctual Gorenstein loci refined by socle parity.

We study certain simple affine vertex algebras $L_{-1-h/6}(\mathfrak{g})$ from rank-one Argyres-Douglas theories, where $\mathfrak{g}$ is a non-simply-laced simple Lie algebra and $h$ is its Coxeter number. For $L_{-3}(F_4)$, we determine the maximal ideal of $V^{-3}(F_4)$, Zhu's algebra, and the associated variety. We also classify the irreducible highest-weight $L_{-3}(F_4)$-modules and determine the characters of its irreducible ordinary modules. The associated variety is $\overline{\mathbb{O}}_{\widetilde A_1}$, and the level $-3$ is totally collapsing for the orbit $\mathbb{O}_{\widetilde A_1}$. The Poisson coinvariants of the $C_2$-algebra of $L_{-3}(F_4)$ are isomorphic to $\mathbb{C}[\ell]/(\ell^5)$, where $\ell$ is induced by the conformal vector. Consequently, the characters of ordinary $L_{-3}(F_4)$-modules satisfy a fifth-order modular linear differential equation. We also give modular linear differential equations of orders five and eleven satisfied by the characters of ordinary modules for $L_{-2}(B_3)$ and $L_{-2}(G_2)$ respectively.

By equipping the category $\mathcal{M}$ of modules over a finite dimensional algebra with the action of a finite tensor category $\mathcal{C}$, we introduce a subcategory $\mathcal{PM}$ satisfying numerous remarkable properties: it is generating and cogenerating, functorially finite, closed under Auslander–Reiten translations (i.e. precluster tilting), and it computes the finitistic dimension. Through a functor category perspective, $\mathcal{PM}$ is identified with the $\mathcal{C}$-module functors that are projective, and we also prove a $\mathcal{C}$-equivariant version of the Auslander(–Solberg) correspondence. As examples, we show that $\mathcal{PM}$ contains only finitely many indecomposable modules for some tensor product algebras. For some family of 1-Iwanaga–Gorenstein algebras introduced by Geiss–Leclerc–Schröer, we show that $\mathcal{PM}$ is exactly the subcategory of $\tau$-locally free modules.

Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.

Let \(\Lambda\) be a \(\Z\)-graded Artin algebra which is Iwanaga–Gorenstein. We prove that \(\Lambda\) is CM-finite if and only if the category of gradable Gorenstein-projective modules has finite type, and if and only if the lengths of the indecomposable gradable Gorenstein-projective modules are bounded. Moreover, if \(\Lambda\) is CM-finite, then every Gorenstein-projective \(\Lambda\)-module is gradable. These results provide Gorenstein-projective analogues of the corresponding theorems of Gordon–Green for graded Artin algebras of finite representation type.

Let $G$ be a split connected semi-simple group over a local non-archimedean field $F$. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack $\bpo$ of $G$-bundles on ${\mathbb P}^1$ endowed with a trivialization at $0$ and $\infty$. This space is a representation of $G(F)\times G(F)$ which contains the regular representation of $G(F)$. We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation $\pi$ of $G(F)$ defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of $G(F)$ with an Iwahori fixed vector defines a ${\mathbb C}$-point on the Calogero-Moser space associated to $G$. We also show that for $G=PGL(N)$ the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of $\pi$ on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when $F$ is a finite field.

We study various symmetry conditions related to the Auslander-Reiten formula for finite-dimensional algebras. In particular, we define a new class of algebras which we call $E$-symmetric. A theorem by Derksen-Weyman-Zelevinksy implies that all Jacobian algebras are $E$-symmetric. We prove the converse for the class of monomial algebras.

We show that microlocal Arthur packets can be reconstructed from an otherwise underdetermined system of linear equations arising from Kashiwara's local index theorem. The key is a geometric rigidity property of Arthur orbits which, together with Fourier symmetry, uniquely determines the microlocal multiplicities defining these packets. This realizes the associated Arthur distribution as the unique normalized element in the intersection of two filtered spaces of standard stable distributions related by Fourier transform. Under standard enhanced local Langlands hypotheses and Aubert compatibility, this gives an alternative representation-theoretic definition of microlocal Arthur packets for general connected reductive $p$-adic groups that does not use twisted endoscopy and ensures stability through the stability of tempered packet distributions. In the unipotent setting, we use this characterization to prove agreement between endoscopic and microlocal Arthur packets for split $\mathrm{SO}_{2n+1}$ when the residue characteristic is at least $6n+4$. We further use this characterization to compute all unipotent Arthur packets with unramified real infinitesimal character for inner-to-split adjoint exceptional $p$-adic groups. In this exceptional setting, we prove the microlocal weak Arthur packet conjecture and give counterexamples to the microlocal analogues of the least- and greatest-parameter conjectures.

Let $F$ be a nonarchimedean local field of characteristic zero and residue characteristic $p$. Let $E$ be a quadratic field extension of $F$. Let $D$ be the unique quaternion division algebra defined over $F$. Let $G=\GL_{2}(E)$ and $H=D^\times$. This paper studies the modulo distinction problem for the pair $(G,H)$ with $\ell\neq p$, which gives a complete classification of $H$-distinguished representations of $G$, including the cases $\ell=2$ and $p=2$. In terms of type theory, we give a complete classification of $H$-distinguished supercuspidal representations when $p$ is odd.