arXiv subjects

Aaron Slipper

Publications and source records attributed to Aaron Slipper.

7 recordsLinked to original sources

arXiv2026

Bosonic codes from compact phase spaces

We present the algebraic structure of bosonic quantum error-correcting codes on genus-two Riemann surfaces. We explicitly construct the code words as automorphic forms and analytically generate the full tower of code spaces at all weights. We prove a fundamental no-go theorem: for any genus greater than one, the stabilizer group is non-amenable, forcing a strictly positive spectral gap in the stabilizer Hamiltonian. Consequently, no normalizable quantum state can satisfy all stabilizer conditions. This sharply contrasts with standard Gottesman-Kitaev-Preskill (GKP) codes, where the amenability of the stabilizer group $\mathbb{Z}^2$ permits approximate code words with arbitrary precision.

quant-ph↗

arXiv2026

Quasi-Classical Braverman--Kazhdan Intertwiners via Quiver Varieties

We show that Braverman--Kazhdan normalized intertwiners for $SL_n(\mathbf{C})$ have a quasi-classical incarnation governed by type $A$ quiver varieties. More precisely, for standard parabolic subgroups $P$ and $P'$ with conjugate Levi subgroups, we construct $SL_n\times L^{\mathrm{ab}}$-equivariant isomorphisms $\Phi(P,P'):\overline{T^*(SL_n/[P,P])}^{\mathrm{aff}}\rightarrow\overline{T^*(SL_n/[P',P'])}^{\mathrm{aff}}$ between the affinizations of the cotangent bundles of the corresponding Braverman--Kazhdan spaces, and we prove that these isomorphisms satisfy Coxeter relations. The construction uses $SL$-gauge analogues of Lusztig--Maffei--Nakajima reflection functors, thereby extending Wang's quiver-variety realization of the quasi-classical Gelfand--Graev action from the Borel case to arbitrary parabolic subgroups. In this way, we complete the quasi-classical Braverman--Kazhdan intertwiner story for $SL_n(\mathbf{C})$ and obtain a systematic source of non-isomorphic varieties whose affinized cotangent bundles are isomorphic.

math.RT↗

arXiv2026

Geometrization of the Schr\"odinger Model for the Minimal Representation of an Even Orthogonal Group: The de Rham Setting

We construct and compare three $D$-module models for the minimal representation of the conformal group of an even-dimensional quadratic space. Let $V$ be a quadratic space over a field $\kappa$ of characteristic $0$, $C\subset V^*$ be the isotropic cone, $\Delta\in D_V$ be the associated Laplace--Beltrami operator, $G$ be the conformal group of $V$, and $D_C$ be the algebra of Grothendieck differential operators on $C$. We prove that the category of finitely generated $D_C$-modules is equivalent both to a Kazhdan--Laumon glued category attached to the smooth locus $C^o$ and to a category of ``harmonic'' twisted $D$-modules on the projective conformal compactification $G/P \supset V$. The gluing is governed by the quadric Fourier transform, while the harmonic model is built from a distinguished $G$-equivariant sheaf $H$ on $G/P$ extending the local quotient $D_V/D_V\Delta$. We prove a new geometric interpretation of higher symmetries of the Laplacian as global sections of $H$, and use this connection to give a geometric proof of the theorem of Levasseur, Smith, and Stafford that $D_C$ is Noetherian despite the singularity of $C$. We also study, via a descent procedure we call ``$F$-moment descent,'' the algebraic geometry of the closure of the minimal nilpotent orbit of $G$, which is the quasiclassical analogue of the minimal representation. Finally, we analyze the filtered structure of $D_V/D_V\Delta$ as a right $D_C$-module, identifying its associated graded layers through a flat degeneration of an affine flag multicone whose special fiber is the Rees space of a natural ideal in $\kappa[\overline{O}_{min}]$.

math.RT↗

arXiv2025

The $\rho$-Fourier transform

Let $G$ be a reductive group over a local field $F$ and let $\rho:{}^LG \to \mathrm{GL}_{V_{\rho}}(\mathbb{C})$ be a representation of its $L$-group satisfying suitable assumptions. Braverman, Kazhdan and Ng\^o conjectured that one has a $\rho$-Fourier transform on $L^2(G(F))$ and a $\rho$-Schwartz space $\mathcal{S}_{\rho}(G(F))<L^2(G(F))$ fixed under the Fourier transform that satisfies certain desiderata. We construct the Fourier transform for arbitrary fields. Over non-Archimedean fields we construct the Schwartz space, and in the Archimedean case we construct an approximation to it. This proves a large portion of their conjectures. Our methods are spectral in nature.

math.NT↗

arXiv2025

Modulation groups

Conjectures of Braverman and Kazhdan, Ng\^o and Sakellaridis have motivated the development of Schwartz spaces for certain spherical varieties. We prove that under suitable assumptions these Schwartz spaces are naturally a representation of a group that we christen the modulation group. This provides a broad generalization of the defining representation of the metaplectic group. The example of a vector space and the zero locus of a quadric cone in an even number of variables are discussed in detail. In both of these cases the modulation group is closely related to algebraic groups, and we propose a conjectural method of linking modulation groups to ind-algebraic groups in general. At the end of the paper we discuss adelization and the relationship between representations of modulation groups and the Poisson summation conjecture.

math.NT↗

arXiv2025

Logarithmic Geometry and Geometric Class Field Theory

We demonstrate an application of logarithmic geometry in the context of geometric Langlands, by providing a logarithmic upgrade of Deligne's geometric class field theory for tamely ramified Galois groups. In particular, we define a framed logarithmic Picard space, and show that a logarithmic compactification of the classical tamely ramified Div-to-Pic map has, for sufficiently large degree, log-simply connected fibers given by logarithmically compactified vector spaces. This provides a canonical bijection between local systems on the curve with divisorial log structure and multiplicative local systems on the framed logarithmic Picard, a logarithmic version of the Hecke eigensheaf correspondence of geometric Langlands for GL_1. We use this to re-derive tamely ramified global Artin reciprocity for function fields, and show that logarithmic geometry allows for a geometric interpretation of local-to-global compatibility at all places, in addition to the unramified places.

math.AG↗

arXiv2024

Parabolic Kazhdan-Laumon and the Kloosterman Fourier Transform for Quadric Cones

Let $G$ be a split reductive group over $F_q$, and let $M$ be a standard Levi subgroup of $G$. Let $Par_M(G)$ denote the set of parabolic subgroups of $G$ with Levi factor $M$. For $P$ and $P'$ in $Par_M(G)$, we let $U=R_u(P)$ and $U'=R_u(P')$ denote the unipotent radicals, and we denote by $\overline{G/U}$ and $\overline{G/U'}$ the affinizations of the corresponding homogeneous spaces. Extending the work of Kazhdan-Laumon and Braverman-Kazhdan (arXiv:math/9809112, arXiv:math/0206119) to general parabolic basic affine, or paraspherical, spaces, we propose a construction for certain intertwining operators $F_{P',P}: S(\overline{G/U}(F_q),C) \to S(\overline{G/U'}(F_q),C)$ for suitable function spaces $S$, defined via kernels analogous to those appearing in those works. We then study the extent to which these intertwiners are normalized. We show that, for opposite $(n-1)+1$ parabolics of $SL_n$, our transform reduces to the classical linear Fourier transform, and that, for opposite unipotents in $SL_3$ or opposite Siegel parabolics in $Sp_4$, our transforms are given by a Fourier transform on a quadric cone, with kernel coming from a Kloosterman sum. We prove Fourier inversion for this transform on a natural subclass of functions on the quadric cone, establishing a finite-field analogue of the quadric Fourier transform of Gurevich-Kazhdan, Getz-Hsu-Leslie, and Kobayashi-Mano (arXiv:2304.13993, arXiv:2103.10261, arXiv:0712.1769).

math.RT↗