Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0211138

Geometric rationality of equal-rank Satake compactifications

Leslie Saper
View original source ↗Read source document ↗Export metadata ↓

Abstract

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Satake compactification of a Hermitian locally symmetric space; Baily and Borel proved this is always geometrically rational. Satake compactifications for which all the real boundary components are equal-rank symmetric spaces are a natural generalization of the Baily-Borel-Satake compactification. Recent work (see math.RT/0112250, math.RT/0112251) indicates that this is the natural setting for many results about cohomology of compactifications of locally symmetric spaces. In this paper we prove any Satake compactification for which all the real boundary components are equal-rank symmetric spaces is geometrically rational aside from certain rational rank 1 or 2 exceptions; we completely analyze geometric rationality for these exceptional cases. The proof uses Casselman's criterion for geometric rationality. We also prove that a Satake compactification is geometrically rational if the representation is defined over the rational numbers.

Explore related subjects

math.RTmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Leslie Saper. 2004-09-18. Geometric rationality of equal-rank Satake compactifications. https://arxiv.org/abs/math/0211138

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

KEEP EXPLORING

Related papers

arXiv2026

A local relative trace formula for F*\SL(2,F)

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula.

math.RT↗
arXiv2026

Semi-infinite parabolic IC-sheaf

Let G be a connected reductive group, P its parabolic subgroup. We consider the parabolic semi-infinite category of sheaves on the affine Grassmanian of G and construct the parabolic version of the semi-infinite IC-sheaf of each orbit. We establish some of its properties and relate it to sheaves on the Drinfeld compactification of the moduli stack Bun_P of P-torsors on a curve. We also relate the parabolic semi-infinite IC-sheaf with the dual baby Verma object on the spectral side.

math.RT↗
arXiv2026

The Grothendieck group of an extriangulated category

In this paper, we investigate the split Grothendieck group $K^{\rm sp}_{0}(\mathcal{M})$ of a $d$-rigid subcategory $\mathcal{M}$ in an extriangulated category $\mathscr{C}$. As applications, we prove the following results: (1) If $\mathcal{M}$ is a silting subcategory, then the Grothendieck group $K_{0}(\mathscr{C})$ is isomorphic to $K_{0}^{\rm sp}(\mathcal{M})$; (2) If $\mathcal{M}$ is a $d$-cluster tilting subcategory, then $K_{0}(\mathscr{C})$ is isomorphic to the index Grothendieck group $K_{0}^{\rm in}(\mathcal{M})$; (3) Let $\mathcal{C}_{A_{n}}^{d}$ be the $d$-cluster category of type $A_n$. If $d$ is even, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}/(n+1)\mathbb{Z}$. If $d$ is odd, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}$ if $n$ is odd; $K_0(\mathcal{C}_{A_{n}}^{d})\cong 0$ if $n$ is even.

math.RT↗