Papers

In the context of Fell bundles over discrete groups, we introduce and analyze suitable notions of direct sum and tensor product of actions of Fell bundles on Hilbert bundles. For $C^*$-dynamical systems, where actions arise from equivariant representations, we discuss how the tensor product of two such actions is related to the action associated with the tensor product of the underlying equivariant representations. Likewise, since any action gives rise to a $C^*$-correspondence between the full cross-sectional $C^*$-algebras of the involved Fell bundles, we examine the connection between the $C^*$-correspondence associated with the tensor product of two given actions and the tensor product of the associated $C^*$-correspondences. We also discuss how the tensor product of actions naturally appears in Rieffel-like induction processes. Finally we present related results on the Fourier-Stieltjes and Fourier algebras of a Fell bundle.

We derive approximation rates for Sobolev functions defined on the complex unit sphere and complex projective space, representing quantum states, and on the special and projective unitary groups, representing quantum unitary operations. By using harmonic analysis on compact groups and compact homogeneous spaces, we characterize the corresponding Peter–Weyl decompositions and Sobolev spaces via irreducible representations and the Laplace–Beltrami operator. Combining spectral truncation with finite-depth quantum-circuit approximation, invoking the Solovay–Kitaev theorem, we establish quantitative rates that separate the spectral truncation error from the circuit approximation error.

Let $\hbar$ be a non-zero real number and put $q= e^{-\pi i \hbar}$. Let $G$ be a semisimple split real Lie group of classical type, with Lie algebra $\mathfrak{g}$ and associated Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$. We show that there exists a locally compact quantum group $(L_{\hbar}(B),\Delta)$ which integrates the Borel part $U_q(\mathfrak{b})$ of the Drinfeld–Jimbo quantized enveloping algebra $U_q(\mathfrak{g})$, viewed as a Hopf $*$-algebra. We further compute the associated Haar weights and modular data for $(L_{\hbar}(B),\Delta)$. Up to an excess Cartan part, the Drinfeld double of $(L_{\hbar}(B),\Delta)$ can be seen as an analytic quantization of the function algebra of the totally positive semigroup $G^+\subseteq G$.

Heyer's left adjoint to derived parabolic induction gives rise to higher mod $p$ Satake transforms \[ \mathcal S_V^i: \mathcal H_G(V) \longrightarrow \mathcal H_M\bigl(H^i(K_U,V)\bigr), \] whose extreme degrees recover Herzig's invariant and coinvariant transforms. For $G=\mathbf G(F)$, where $\mathbf G$ is a connected reductive group defined over a finite extension $F/\mathbb{Q}_p$, we describe these maps explicitly using Mackey decompositions and corestriction in group cohomology. For $G=\mathrm{GL}_n(F)$, we evaluate them on the standard Hecke generators, showing that their support lies in a single Weyl orbit and that their values on the individual support cosets are concrete corestriction operators. For $G=\mathrm{GL}_2(F)$, we determine the image in every cohomological degree for each non-Steinberg Serre weight. The endpoint images are the classical polynomial Satake images, while every interior image is of the form \[ k[x,y^{\pm1}]/(x^r) \] for some positive integer $r$. Thus, every interior image is a finite nilpotent thickening of the Laurent polynomial algebra $k[y^{\pm1}]$. We give explicit bounds for its thickness, prove that they are sharp, and show that in the unramified case every interior image is either reduced or a square-zero thickening.

We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot split into several nearby eigenvalues, while an isotypic limiting eigenspace imposes corresponding divisibility constraints on the multiplicities in the approximating cluster. We then construct a one-parameter equivariant Steklov homogenisation on closed orientable surfaces, allowing Laplace spectral representations to be transferred to Steklov spectral clusters. Applied to the hyperbolic surfaces of Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault, this yields first Steklov multiplicities 16 and 21 in genera 10 and 17, and at least 24 in genus 37. These examples give counterexamples to Jammes's conjecture on first Steklov multiplicity.

In characteristic zero, we prove that the Tor algebras of the generic, symmetric, and skew-symmetric determinantal rings are Koszul with respect to a natural regrading. In particular, their Eilenberg–Moore spectral sequence degenerates, which allows us to explicitly determine the Poincaré series of their residue fields and answers a problem raised by Avramov. To do this, we construct the universal hook algebra and prove that it has a quadratic Gröbner basis and thus is Koszul. This leads to a new standard monomial construction of the irreducible polynomial GL-representations, alongside the Schur and Weyl functors, which is of independent interest.

We introduce the global obstruction property (GOP) for Artin algebras: for every finitely generated module of infinite projective dimension, two distinct syzygies have a common nonprojective indecomposable direct summand. We prove that GOP implies the generalized Auslander–Reiten condition, and hence the Auslander–Reiten conjecture. We give criteria for GOP to hold for pullbacks over semisimple algebras and construct an example which is not syzygy-finite. For each Artin algebra $A$, we construct an algebra $\mathcal{G}(A)$ satisfying GOP by using triangular matrix extensions. We show that $\text{fin.dim}\mathcal{G}(A)=\text{fin.dim} A+2$ and that $A$ and $\mathcal{G}(A)$ are singularly equivalent. Finally, we give a pair of derived equivalent finite-dimensional algebras for which GOP holds on only one side.

We give an explicit formula for the Lusztig parameters of determinantial modules over symmetric quiver Hecke algebras. The formula applies to every Bruhat pair and every fixed reduced expression of the upper Weyl group element. In finite types $A$, $D$, and $E$, we identify the parameters of the modules in Ménard's seed using Demazure weaves. We then factor prefix determinantial modules into these seed modules, with multiplicities given by Lusztig cycles. We also give a criterion for recognizing monomials in this fixed seed and an explicit formula recovering their multiplicities from Lusztig parameters.

Nadler and Yun conjectured that the automorphic side of the Betti geomet- ric Langlands correspondence forms a local system of categories over the moduli of curves. We prove this local constancy conjecture using purely microlocal methods, independent of the recently estab- lished Betti geometric Langlands conjecture. In particular, it follows that the Betti automorphic category depends only on the genus of the curve, verifying a conjecture of Ben-Zvi and Nadler.

Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$, and $\kappa$. In this paper, via varying $b$, $\ell$, and $\kappa$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to \[ B_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+7}{18\xi_{\mathrm{new}}+9} =0.874957019420098946\ldots, \] where $\xi_{\mathrm{new}}$ is the unique root in $(0.38858,0.38859)$ of $1188\xi^3-2160\xi^2+171\xi+190=0$. Explicitly, with $\arccos\in[0,\pi]$, \[ \xi_{\mathrm{new}}=\frac{20}{33}+\frac{\sqrt{1391}}{33} \cos\!\left(\frac13\arccos\frac{28470}{1391^{3/2}} -\frac{2\pi}{3}\right). \] The limiting parameters are \[ \begin{gathered} b_{\mathrm{new}}=\frac{8(3\xi_{\mathrm{new}}-1)(12\xi_{\mathrm{new}}^2-40\xi_{\mathrm{new}}+15)} {9(2\xi_{\mathrm{new}}+1)(24\xi_{\mathrm{new}}^2-20\xi_{\mathrm{new}}+5)}, \ell_{\mathrm{new}}=\frac{5-6\xi_{\mathrm{new}}}{9+18\xi_{\mathrm{new}}},\qquad \kappa_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+5}{18\xi_{\mathrm{new}}+9} =2B_{\mathrm{new}}-1. \end{gathered} \] We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].

Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field $F$ of characteristic zero. Let $q_F$ be the cardinality of its residue field. Let $J_s$ be the twisted Jacquet module of the representation induced from $\Delta(\tau,m+i)|\det|^s$. It carries an action of $G\times H$, where $G$ and $H$ are split symplectic or special orthogonal groups. Let $\sigma$ be an admissible representation of $G$ of finite length. We prove that $(J_s\otimes\sigma)_G$ is induced from $\Delta(\tau,i)|\det|^s$ and a fixed conjugate of $\sigma$, outside a finite set of values of $q_F^{-s}$ depending on $\tau$ and $\sigma$. No genericity assumption on $\sigma$ is needed. For irreducible $\sigma$, this determines all irreducible quotients of $J_s$ of the form $\sigma^\vee\boxtimes\pi$. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal $\sigma$ unless $G$ is the split group $\mathrm{SO}_2$. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when $\tau$ is cuspidal on $\mathrm{GL}_n$ and $n$ is larger than the rank of $G$.

For a complex simple Lie algebra $\mathfrak{g}$ with compact conjugation $\tau$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, \tau y)$, it is shown that $H([x,y],[x,y]) \leq (h^{\vee})^{-1}H(x,x)H(y,y)$, where $h^{\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in $\tau$-stable long-root subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$. Equivalently, the comass of the Cartan $3$-form with respect to $H$ equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index $\ell$ the optimal constant is $2/\ell$; this contains the Böttcher–Wenzel and Bloch–Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. The proof uses the Nahm algebra of $\mathfrak{g}$ with the conjugation induced by $\tau$: critical points of the relevant function correspond to $\tau$-twisted idempotents, and its Hessian is governed by $\tau$-twisted multiplication operators, whose anticommutation with multiplication by $i$ turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$ of Dynkin index $j$ give critical points with critical value $1/(jh^{\vee})$, whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

Let $S$ be a Cohen-Macaulay normal domain containing a field $k$ with an action of a finite group $G$ with $|G|$ invertible in $k$. We prove that $G$-equivariant noncommutative crepant resolutions of $S$ induce noncommutative crepant resolutions of $S^G$ if the $G$-action is small. We also discuss similar results for equivariant maximal modification algebras. We introduce the notion of $G$-maximal modifying ($G$-MM) module, and we show that $G$-MM module over $S$ induces MM $S^G$-module in some special cases. Furthermore, we prove that Gorenstein MMAs of three-dimensional $\mathbb{Q}$-Gorenstein Cohen-Macaulay rings are derived equivalent. As an application of these results, we prove the existence of an MM module and derived equivalences of MMAs for three-dimensional terminal singularities of index two.

Specialization maps are natural maps from equivariant vanishing cycle cohomology to the equivariant cohomology of the ambient space. We study their injectivity problem for moduli stacks of representations of quivers with potentials, including those associated with generalized conifolds and finite subgroups of $SO(3,\mathbb{C})$. Applications to the irreducibility of modules of Reshetikhin type shifted Yangians are also given.

We study the quantized $K$-theoretic Coulomb branches of framed star-shaped quivers with central gauge dimension $n\geq2$, $r-1$ one-dimensional outer gauge vertices, and a one-dimensional framing at the central vertex. For $n=2$ and arbitrary $r\geq1$, and for every $n\geq2$ with $1\leq r\leq5$, we prove that the loop-graded Koszul-perverse heart is a monoidal categorification of the associated quantum cluster algebra over $\mathbb{Z}[v^{\pm1}]$. This quantum cluster algebra coincides with its upper quantum cluster algebra. For all $n\geq2$ and $r\geq1$, we construct an explicit initial seed and identify the associated upper quantum cluster algebra with the Grothendieck ring of this heart. Every mutation is realized by a short exact sequence, and every quantum cluster monomial is represented by a simple object.

We construct a finite-dimensional Auslander-regular algebra $R$ with global dimension three, which is right diagonal but not left diagonal; moreover, any simple right $R$-module with grade $i$ has projective dimension $i$, but there is a simple left $R$-module with grade one and projective dimension two. It answers a question raised by Iyama negatively.

A reciprocity between multiplicities of projective direct summands for modular group algebras has been known for decades. Recently, the first author extended it to finite-dimensional symmetric algebras. We further generalize the reciprocity theorem to arbitrary finite-dimensional algebras by establishing a tensor-hom adjunction for stable and costable categories.

We establish a quantum super Schur–Weyl duality for the mirabolic Hecke algebra \(H_n^{\mathrm{mir}}(q)\) with \(q\) indeterminate and determine the corresponding bimodule decomposition. The centralizer of the mirabolic action is shown as a direct sum of homogeneous quantum Schur superalgebras. We describe the tensor-space annihilator and the resulting faithful quotient. When nonzero, the annihilator is generated by an explicit spectral idempotent attached to the smallest rectangle excluded by the hook condition. From the quantum super Schur–Weyl duality, we derive a super Frobenius formula for mirabolic Hecke algebras. This formula then yields a Murnaghan–Nakayama rule and Regev-type formulas for the irreducible characters. We also construct a super mirabolic RSK bijection from words in a super alphabet with an additional even letter to hook semistandard insertion tableaux together with recording pairs that distinguish the positions of the additional letter. This correspondence yields a Roichman formula for irreducible characters of \(H_n^{\mathrm{mir}}(q)\). Finally, we establish a parameter-inversion isomorphism between \(H_n^{\mathrm{mir}}(q^{-1})\) and the \(q\)-rook monoid algebra \(R_n(q)\) and use it to transport the duality, annihilators, and character formulas in both directions. The transported tensor decomposition provides a representation-theoretic interpretation of hook and two-row character sums for \(R_n(q)\).

We develop two new general frameworks for constructing finitely generated residually finite groups that possess infinite girth, satisfy no non-trivial group law, and contain no non-abelian free subgroups. While such groups are rare–with known examples including the first Grigorchuk group, Thompson's group F, and certain iterated wreath products–our approach provides a more explicit construction of such groups realized as subgroups of direct products of symmetric groups. We investigate the relationship between the girth of a group and that of its finite quotients, leading to the introduction of a "marked" girth criterion, which we establish as a necessary and sufficient condition for infinite girth within the class of residually finite groups. Furthermore, we explore the connection between girth and profinite completions and provide a congruence realization result for infinite girth in linear groups. We conclude by posing several open questions that arise naturally from our results.

We give an explicit description of the orbits of the split real form $G_{2(2)}$ on the complex flag manifolds of $G_2$ and study their induced invariant CR structures. In particular we study their Levi forms and show that all possible cases in the simple homogeneous setting occur, namely Levi nondegeneracy, $k$-nondegeneracy for $2\leq k\leq 3$, and holomorphic degeneracy. Moreover, following the classical approach to orbits on flag manifolds, we give equations for these homogeneous CR manifolds.

We construct categorical models and cluster characters for skew-symmetrizable cluster algebras admitting good ramified realizations. Goodness is equivalent to nondegenerate realizability. When a reddening sequence is given, it is enough to check sign coherence of the arrow types along that sequence. Triangular extensions with arbitrary connecting bimodule types preserve goodness when both component matrices admit reddening sequences. Consequently, every matrix obtained from one vertex by triangular extensions and mutations admits a good Jacobi-finite nondegenerate realization for every compatible positive integral symmetrizer. For a good Jacobi-finite nondegenerate realization, the monodromy $F$-polynomials satisfy mutation for all finite decorated representations, without a rigidity hypothesis. A full initial order determines valued universal presentations and their cluster characters. Under the reachability condition, these characters form a basis of the middle cluster algebra for arbitrary geometric coefficients, with frozen variables inverted, and contain all cluster monomials. The same family is an upper-cluster-algebra basis when the middle and upper algebras coincide. For a fixed Jacobian algebra and $\delta$-vector, the Newton polytope of the generic monodromy $F$-polynomial is independent of the order.

Let $F$ be an algebraically closed field of characteristic different from $2$, and let $V$ be a nonnegatively integer-graded vertex operator algebra over $F$. We consider higher Zhu algebras obtained by taking a quotient of $V$ by a relation space containing the vectors $D_V^{(h)}a-\binom{-\mathrm{wt}(a)}{h}a$ for all homogeneous $a\in V$ and integers $h\geq1$, where $D_V^{(h)}a=a_{-h-1}\mathbf1$. For $n\geq1$ and a nonzero unital module $U$ over the corresponding higher Zhu algebra, we study the quotients $L_n(U)$ of induced modules and determine the kernel of the canonical map $U\to\Omega_n(L_n(U))/\Omega_{n-1}(L_n(U))$. This kernel is the largest submodule of $U$ that factors through the adjacent lower-level algebra. When this submodule is zero, the canonical map is an isomorphism, and each $\Omega_r$ is the direct sum of the first $r+1$ homogeneous subspaces. These results are proved using an expansion of zero modes with integer coefficients and the grading of the standard induced modules. Moreover, the construction is natural with respect to module homomorphisms, and $L_n(U)$ is indecomposable in the graded module category whenever $U$ is indecomposable. From nonsplit extensions of modules for higher Zhu algebras satisfying the stated hypotheses, we construct nonsplit short exact sequences of weak $V$-modules.

Motivated by an observation concerning the $S_3$-conjecture, we prove that every abelian composition factor of a finite rational group has order $2$, $3$, or $5$. Thompson previously restricted these orders to $\{2,3,5,7,11\}$; we exclude $7$ and $11$, establishing the bound that he proposed. Together with the work of Feit and Seitz on nonabelian composition factors, this completes the determination of the possible composition factors of finite rational groups.

We study reconstruction of graded surfaces with stops $(S,M,\eta)$ from their mapping class groups and the derived Picard groups of their partially wrapped Fukaya categories. In genus at least two, we prove that the ungraded mapping class group determines the surface with stops $(S,M)$, while the derived Picard group with the shift distinguished determines the underlying graded surface $(S,\eta)$. Nevertheless, the derived Picard group with both the shift and the Serre functor distinguished has countably infinite fibres. We also construct arbitrarily large finite families of pairwise non-equivalent categories with the same invariant and Grothendieck rank. For line fields induced by nowhere-vanishing vector fields, we show that the rational Serre representation determines the surface with stops $(S,M)$ in every genus. For arbitrary line fields in genus at least two, the full integral derived Picard representation, with the shift distinguished, has fibres of cardinality at most four. It completely determines the graded surface with stops $(S,M,\eta)$ when $S$ has one or at least five boundary components, while non-singleton fibres occur for two, three, and four boundary components.

We prove that every cluster monomial of type $A_n$ is log-concave and unimodal with respect to its Laurent expansion in a fan initial cluster. We derive a coefficient formula for laminar interval products arising from compatible diagonals in the polygon model. In even rank, we reduce log-concavity in fan coordinates to log-concavity along integer lines. In odd rank, we use convolution to handle the coefficient sums arising from the fan substitution. This gives partial affirmative answers to the log-concavity conjecture of Chen-Huang-Sun and the unimodality conjecture of Chen. Our result also complements Shen's study on the supports of products of canonical basis elements for cluster X-varieties of type $A_n$.

Recently, Girsch-Kurinczuk have obtained new formulas for local newforms in depth-zero and minimax integral-depth cuspidal representations of $p$-adic $\mathrm{GL}(n)$. One of their main results is an averaging formula which provides an integral representation of the Whittaker newform in terms of the Paškūnas-Stevens Whittaker function. In the present paper, we first prove a reverse averaging formula by establishing an integral representation of the Paškūnas-Stevens Whittaker function in terms of the Whittaker newform. Together, these formulas give an explicit two-way passage between these two distinguished vectors. Finally, we use our averaging formula and earlier work of Kurinczuk-Matringe to give new explicit test vectors in terms of Whittaker newforms for such cuspidal $\mathrm{GL}(n)\times\mathrm{GL}(n)$ pairs, whenever their Rankin-Selberg $L$-factor is not identically equal to $1$.

We provide character expansions of certain specialisations of multiparameter Hall–Littlewood polynomials of types $\mathrm{B}_n$, $\mathrm{C}_n$, and $\mathrm{BC}_n$ for rectangular shapes. We use these expansions to establish the six Jacobi-Trudi-type identities conjectured by Ole Warnaar in 2025.

The solvable degree of a Schur algebra $D$ is the greatest common divisor of all integers $n$ such that $D^{n\times n}$ is a direct summand of the rational group algebra $\Q G$ for some finite solvable group $G$. The main result of this paper gives divisibility conditions on the solvable degree depending on the center and the rational primes that divide the discriminant of $D$.

We study nil-Hecke quotients defined by powers of root coordinates. We identify their central ideals through the determinant components of root-orbit power ideals. Apolarity and discriminant removal reduce the graded dual centers to directional bounds on invariant polynomials. We determine their Hilbert series for every exponent and root orbit in all irreducible crystallographic types except $E_6,E_7,E_8$. Relative discriminants compare centers for reflection subgroups and transfer equivariant resolutions. Coordinate-root centers in types $B$ and $C$ are Grassmannian cohomology rings. We classify self-injectivity in type $A$, in rank two, and in type $F_4$, and determine the first nonsemisimple centers in the remaining classical families. For an explicit open set of reflection parameters, the polynomial image of the rational Cherednik algebra at $t=0$ is the full nil-Hecke algebra. This realizes our algebras as Cherednik quotients and their finite centers as closed subschemes on a section of the generalized Calogero–Moser space.