40 pages, AMS-LaTeX, uses Xy-pic 3.7 package; v2: minor typos fixed, minor improvements in exposition, treatment in 10.4 of homotopy in neighborhoods corrected and simplified (requiring small changes throughout section 10), section 13 expanded; v3: minor typos fixed, corrected statement of Theorem 1
Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a combinatorial invariant that to a great extent characterizes the cohomology of the associated sheaf. The theory has been successfully applied to solve a number of problems concerning the intersection cohomology and weighted cohomology of the reductive Borel-Serre compactification [math.RT/0112251], as well as the ordinary cohomology of X [math.RT/0112250]. In this paper we extend the theory so that it covers L^2-cohomology. In particular we construct an L-module whose cohomology is the L^2-cohomology of X and we calculate its micro-support. As an application we obtain a new proof of the conjectures of Borel and Zucker.
The goal of the course was a review of results mainly due to M. Olbrich and the first author. We consider a discrete cocompact subgroup $Γ$ of a semisimple Lie group $G$. We relate the group cohomology of $Γ$ with coefficients in the maximal globalization of a representation of $G$ with the multiplicities of unitary representations of $G$ in $L^2(G/Γ)$. Explicit calculations are given in the case $G=SL(2,R)$. In the rank-one case we state a version of the Selberg trace formula, discuss the singularities of the Selberg zeta-functions, and state their relation with the cohomology of $Γ$ in princ…
This expository article is an expanded version of talks given at the "Current Developments in Mathematics, 2002" conference. It gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the intersection cohomology of a real equal-rank Satake compactification of a locally symmetric space with that of the reductive Borel-Serre compactification. We motivate the conjecture with examples and then give an introduction to the various topics that are involved: intersection cohomology, the derived category, and compactifications of a locally symmetric spa…
We show that the continuous cohomology groups of a $ p $-adic reductive group with coefficients in the locally analytic vectors of an admissible $ \mathbb{Q}_p $-Banach space representation are homeomorphic to those with coefficients in the Banach space representation itself. Moreover, we deduce that the canonical topologies on those continuous cohomology groups are Hausdorff and are the uniquely determined finest locally convex topologies.
We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction of discrete series representations on spaces of square integrable harmonic forms with values in holomorphic vector bundles over flag domains. Some special cases had been described by Satake in 1971 and the author in 1975. Here we develop a theory of pseudo--riemannian nilmanifolds of complex type and the nilmanifold versions of flag domains. We construct the…
We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight $k$, we show its corresponding Galois representation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.
In this paper one proves a special case of a conjecture by Nicolas Bergeron. This conjecture is a kind of automorphic Lefschetz property. It relates the primitive cohomology of a locally symmetric manifolds modeled on $U(p,q+r)$ to the primitive cohomology of some of its totally geodesic submanifolds that are locally symmetric and modeled on $U(p,q)$.
Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(Σ_d, S^λ). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a "generic cohomology" result for Specht …
The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\GL_n$ over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\GL_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of …
We relate the generating functions of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups. We show that the first Hochschild cohomology of a positive defect block of a symmetric group is non-zero, answering in the affirmative a question of the third author. To do this, we prove a formula expressing the dimension of degree one Hochschild cohomology as a sum of dimensions of centres of blocks of smaller symmetric groups. This in turn is a consequence of a general formula that makes more precise a theorem of our prev…
We use the dimension and the Lie algebra structure of the first Hochschild cohomology group to distinguish some algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type. In particular, we complete the classification of algebras of dihedral type that was mostly determined by Zhou and Zimmermann.
In this paper we will derive an explicit description of the genuine projective representations of the symmetric group $S_n$ using Dirac cohomology and the branching graph for the irreducible genuine projective representations of $S_n$. In 2015 Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of $S_n$ are related to the characters of elliptic graded modules. We derived the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties $\mathcal{B}_e$ of $\mathfrak{g}$ and were able to use Dirac coh…
We develop the theory of locally analytic representations of compact $p$-adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard's isomorphisms between continuous, locally analytic and Lie algebra cohomology to solid representations. We also prove a comparison result between the group cohomology of a solid representation and of its analytic vectors.
Let p be a prime. Let FpSp be the group algebra of the symmetric group over the finite field Fp with |Fp|=p$. Let Fp be the trivial FpSp-module. We present a projective resolution PRes Fp of the module Fp and equip the Yoneda algebra Ext(Fp,Fp) with an A(oo)-structure such that Ext(Fp,Fp) becomes a minimal model of the dg-algebra Hom(PRes Fp, PRes Fp).
Let $A$ be a finite dimensional symmetric Hopf algebra over a field $k$. We show that there are $A$-modules whose Tate cohomology is not finitely generated over the Tate cohomology ring of $A$. However, we also construct $A$-modules which have finitely generated Tate cohomology. It turns out that if a module in a connected component of the stable Auslander-Reiten quiver associated to $A$ has finitely generated Tate cohomology, then so does every module in that component. We apply some of these finite generation results on Tate cohomology to an algebra defined by Radford and to the restricted u…
We give a formula for the character of the representation of the symmetric group $S_n$ on each isotypic component of the cohomology of the set of regular elements of a maximal torus of $SL_n$, with respect to the action of the centre.
We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.
Let $Γ\backslash G/K$ be a compact Hermitian locally symmetric space, where $G$ is simple. We study the components of a de Rham cohomology class of $Γ\backslash G/K$, with respect to the Matsushima decomposition, where the class is obtained by taking Poincaré dual of a totally geodesic complex analytic submanifold. Using an extension of the vanishing result of Kobayashi and Oda, we specify the existence of certain components of such cohomology classes when $G=\text{SU}(p,q), 5\le p\le q$.
We show that dualising transfer maps in Hochschild cohomology of symmetric algebras commutes with Tate duality. This extends a well-known result in group cohomology.
Let $Σ_d$ denote the symmetric group of degree $d$ and let $K$ be a field of positive characteristic $p$. For $p>2$ we give an explicit description of the first cohomology group $H^1(Σ_d,{\rm{Sp}}(λ))$, of the Specht module ${\rm{Sp}}(λ)$ over $K$, labelled by a partition $λ$ of $d$. We also give a sufficient condition for the cohomology to be non-zero for $p=2$ and we find a lower bound for the dimension. Our method is to proceed by comparison with the cohomology for the general linear group $G(n)$ over $K$ and then to reduce to the calculation of ${\rm{Ext}}^1_{B(n)}(S^d E,K_λ)$, where $B(n)…
In this paper we deal with the action of the symmetric group on the cohomology of the configuration space $C_n(d)$ of $n$ points in $\mathbb{R}^d$. This topic has been studied by several authors (see the introduction). On the cohomology algebra $H^*(C_n(d); \mathbb{C})$ there is, in addition to the natural $S_n$-action, an extended action of $S_{n+1}$; this was first shown for the case when $d$ is even by Mathieu, Robinson and Whitehouse and the second author. For the case when $d$ is odd it was shown by Mathieu (anyway we will give an elementary algebraic construction of the extended action f…
We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection gr…
A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to non-trivial coefficient systems, in the form of $\mathrm{FI}$-modules over a field, though one now obtains periodicity of the cohomology instead of stability. In this paper, we further refine these results. Our main theorem states that if $M$ is a finitely generated $\mathrm{FI}$-module over a noetherian ring $\mathbf{k}$ then $\bigoplus_{n \ge 0} \mathrm{H}^t(S_n, M_n)$ admits the structure of a $\mathbf{D}$-module, where $\mathbf{D}$ is the divided power alg…
An FI-module $V$ over a commutative ring $\bf{k}$ encodes a sequence $(V_n)_{n \geq 0}$ of representations of the symmetric groups $(\mathfrak{S}_n)_{n \geq 0}$ over $\bf{k}$. In this paper, we show that for a "finitely generated" FI-module $V$ over a field of characteristic $p$, the cohomology groups $H^t(\mathfrak{S}_n, V_n)$ are eventually periodic in $n$. We describe a recursive way to calculate the period and the periodicity range and show that the period is always a power of $p$. As an application, we show that if $\mathcal{M}$ is a compact, connected, oriented manifold of dimension $\ge…
For $φ$ a normalized positive definite function on a locally compact abelian group $G$, we consider on the one hand the unitary representation $π_φ$ associated to $φ$ by the GNS construction, on the other hand the probability measure $μ_φ$ on the Pontryagin dual $\hat{G}$ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,π_φ)$ and reduced 1-cohomology $\bar{H}^1(G,π_φ)$. For example, $\bar{H}^1(G,π_φ)=0$ if and only if either $Hom(G,\mathbb{C})=0$ or $μ_φ(1_G)=0$, where $1_G$ is the trivial character of $G$.
In this note we compute low degree rational Pontryagin classes for every closed locally symmetric manifold of noncompact type. In particular, we answer the question: Which locally symmetric M have at least one nonzero Pontryagin class?
