Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1901.11501

The product of lattice covolume and discrete series formal dimension: p-adic GL(2)

Lauren C. Ruth
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $F$ be a nonarchimedean local field of characteristic $0$ and residue field of order not divisible by $2$. We show how to calculate the product of the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension of a discrete series representation of $GL(2,F)$. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.

Explore related subjects

math.RTmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lauren C. Ruth. 2019-01-31. The product of lattice covolume and discrete series formal dimension: p-adic GL(2). https://arxiv.org/abs/1901.11501

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

KEEP EXPLORING

Related papers

arXiv2026

On the first relative Hochschild cohomology

In this paper we investigate the Lie algebra structure of the first relative Hochschild cohomology. Let $A,B$ be finite-dimensional basic $k$-algebras over an algebraically closed field of characteristic zero, such that $Q_B$ is a subquiver of $Q_A$. We show that if the complement of $Q_A$ by the arrows of $Q_B$ is a simple directed graph, then the first relative Hochschild cohomology $\mathrm{HH}^1(A|B)$ is a solvable Lie algebra. We also compute the Lie algebra structure of the first relative Hochschild cohomology for radical square zero algebras and for dual extension algebras of directed monomial algebras.

math.RT↗
arXiv2026

Dirac operators for infinite-dimensional color Lie algebras

We develop the Dirac formalism for infinite-dimensional quadratic $\mathbb{Z}$-graded color Lie algebras with finite-dimensional components. Cubic Dirac operators are defined in completions of the quantum Weil algebra determined by the $\mathbb{Z}$-grading. The same grading fixes the normal-ordering convention. Normal ordering introduces a cohomological obstruction to the construction, measured by a color analogue of the Kac-Peterson class. When this class is trivial, we construct cubic and relative cubic Dirac operators satisfying the expected invariance properties and Parthasarathy-type square formulas. We further extend the Chern-Weil homomorphism to completed $\mathfrak{g}$-differential algebras and use it to identify the classical precursor of the cubic Dirac operator with the Chern-Simons element associated with the invariant quadratic polynomial determined by the quadratic structure. As applications, we consider symmetrizable Kac-Moody superalgebras. In this setting, the Kac-Peterson class is trivial, with primitive given by the Weyl vector, which yields the linear correction defining the cubic Dirac operator. We then use the relative Dirac operator to extract representation-theoretic information from highest weight supermodules. As an explicit example, for the affine Kac-Moody superalgebra associated with $\mathfrak{osp}(1\vert 2n)$, we compute the kernel of $\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}$ on integrable highest weight supermodules. Finally, for unitarizable highest weight supermodules, we explain why the usual Dirac inequality is not available in the affine setting.

math.RT↗
arXiv2026

Functions on Nilpotent Orbit Covers and Birational Geometry

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.

math.RT↗