Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.00249

Vogan's Conjecture on local Arthur packets of $p$-adic $\mathrm{GL}_n$ and a combinatorial Lemma

Chi-Heng Lo
View original source ↗Read source document ↗Export metadata ↓

Abstract

For $\mathrm{GL}_n$ over a $p$-adic field, Cunningham and Ray proved Vogan's conjecture, that is, local Arthur packets are the same as ABV packets. They used the endoscopic theory to reduce the general case to a combinatorial lemma for irreducible local Arthur parameters, and their proof implies that one can also prove Vogan's conjecture for $p$-adic $\mathrm{GL}_n$ by proving a generalized version of this combinatorial lemma. Riddlesden recently proved this generalized lemma. In this paper, we give a new proof of it, which has its own interest.

Explore related subjects

math.RTmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chi-Heng Lo. 2023-11-01. Vogan's Conjecture on local Arthur packets of $p$-adic $\mathrm{GL}_n$ and a combinatorial Lemma. https://doi.org/10.2140/pjm.2024.333.331

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Intertwining Algebras and Affine Hecke Algebras for Finite Central Extensions of Classical $p$-adic Groups with Application to Metaplectic Groups

For a finite central extension $\tilde{G}$ of a classical $p$-adic reductive group, we consider the endomorphism algebra of some induced projective generator à la Bernstein of the category of smooth representations of $\tilde{G}$. In the case where the Levi subgroups decompose, we compute this algebra to get a result similar to the one previously obtained by the first author for classical $p$-adic groups, showing that this intertwining algebra is a twisted semi-direct product of an affine Hecke algebra with parameters by a twisted finite group algebra. We discuss also the case with non-decomposed Levi subgroups. We then give an application to the category of genuine representations of a $p$-adic metaplectic group. Using results of C. Mœglin relative to the Howe correspondence, we show that the Bernstein components of these groups are equivalent to tensor products of categories of unipotent representations of classical groups. This generalizes a previous result of the first author. It implies an equivalence of categories between the category of genuine representations of the $p$-adic metaplectic group and the direct sums of those of smooth representations of the corresponding split odd special orthogonal group and its pure inner form.

math.RT↗
arXiv2026

Spectral Invariants of Complex Solvable Lie Algebras: Nilradical Weights and Hyperplane Arrangements

A central open problem in Lie theory is the classification of finite-dimensional complex solvable Lie algebras, but classification up to isomorphism becomes increasingly difficult as the dimension grows and the number of non-isomorphic families explodes. This motivates a classification method by studying coarser spectral invariants arising from the characteristic polynomial of the adjoint representation. We prove that the characteristic polynomial is determined by the generalized weights of the adjoint action on the nilradical. This addresses the problem of expressing the spectral index in Lie-theoretic terms, gives sharp bounds in terms of the nilradical and quotient dimensions, and characterizes spectral equivalence. We then resolve a second problem concerning the higher Betti numbers of the eigenvariety complement. We endow the distinct non-$z_0$ factors of the characteristic polynomial with a natural matroid structure, which we call the spectral matroid, and use the Orlik--Solomon algebra of the associated hyperplane arrangement to determine the Betti numbers and Poincaré polynomial combinatorially and to prove log-concavity of the Betti sequence. Finally, we apply our theory to solvable Lie algebras with abelian nilradical to obtain explicit characteristic polynomials and spectral-equivalence criteria, including examples of non-isomorphic algebras with identical characteristic polynomials.

math.RT↗
arXiv2026

Global dimension of dg algebras via compact silting objects

We introduce a notion of global dimension for a triangulated category relative to a compact silting object. We prove that the finiteness of this dimension is an intrinsic property of the triangulated category itself and, therefore, independent of the choice of the silting object. Focusing on the setup of connective differential graded (dg) algebras, we analyse the behaviour of global dimension under dg algebra homomorphisms and establish explicit bounds. This allows us to deduce a bound for the global dimension of certain dg quiver algebras. We also relate the regularity of the big singularity category of a proper connective dg algebra to the finiteness of its global dimension.

math.RT↗