Related papers: The dualizing module and top-dimensional cohomolog…

Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual duality group of dimension $n \choose 2$ and the Steinberg module $\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of the…

Borel-Serre proved that the integral symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ is a virtual duality group of dimension $n^2$ and that the symplectic Steinberg module $\operatorname{St}^\omega_n(\mathbb{Q})$ is its dualising…

Denote the virtual cohomological dimension of SL_n(Z) by t=n(n-1)/2. Let St denote the Steinberg module of SL_n(Q) tensored with Q. Let Sh_* denote the sharbly resolution of the Steinberg module St. By Borel-Serre duality, the…

Borel--Serre proved that for a number ring $R$ with fraction field $K$, the symplectic group $\text{Sp}_{2n}(R)$ is a virtual duality group of degree quadratic in $n$, and that the symplectic Steinberg module $\text{St}^\omega_{2n}(K)$ is…

We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral…

Let St denote the Steinberg module of $SL_n(Q)$ tensored with Q. Let Sh denote the sharbly resolution of St. By Borel-Serre duality, $H^{n(n-1)/2-i}(SL_n(Z),Q)$ is isomorphic to $H_i(SL_n(Z),St)$. The latter is isomorphic to the homology of…

Let $K$ be a number field with ring of integers $R = \mathcal{O}_K$. We show that if $R$ is not a principal ideal domain, then the symplectic group $\operatorname{Sp}_{2n}(R)$ has non-trivial rational cohomology in its virtual cohomological…

The two isomorphic Borel subalgebras of gl(n), realized on upper and lower triangular matrices, allow us to consider the gl(n) \opus t_n algebra as a self-dual Drinfeld double. Compatibility conditions impose the choice of an orthonormal…

Let $\mathcal{O}$ be the ring of integers of a number field, and let $n\geq 3$. This paper studies bi-interpretability of the ring of integers $\mathbb{Z}$ with the special linear group $\text{SL}_n(\mathcal{O})$, the general linear group…

We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the…

For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical…

We define two subalgebras which can be seen as the quantization of the coordinate rings of the unipotent radical of the standard positive (respectively negative) Borel subgroup of $SL_{n+1}$. We give a presentation for these algebras and…

For a field $k$, we prove that the $i$th homology of the groups $GL_n(k)$, $SL_n(k)$, $Sp_{2n}(k)$, $SO_{n,n}(k)$, and $SO_{n,n+1}(k)$ with coefficients in their Steinberg representations vanish for $n \geq 2i+2$.

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are…

We study a version of the BGG category O for Dynkin Borel subalgebras of root-reductive Lie algebras g, such as gl(\infty). We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In…

In the first half of the present paper, we study higher-level generalizations of differential modules in positive characteristic. These objects may be regarded as ring-theoretic counterparts of vector bundles on a curve equipped with an…

Let $O$ be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by $T$ the maximal torus of diagonal matrices in GL(n). With every $a\in O\cap…

We confirm a conjecture of Quillen in the case of the mod $2$ cohomology of arithmetic groups ${\rm SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m})}[\frac{1}{2}]\thinspace)$, where ${\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\thinspace)}$ is an…

We study three related homological properties of modules in the BGG category O for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and…

We consider local Gorenstein duality for cochain spectra $C^*(BG;R)$ on the classifying spaces of compact Lie groups $G$ over complex orientable ring spectra $R$. We show that it holds systematically for a large array of examples of ring…