The cohomology of the cotangent bundle of Heisenberg groups

L. CAGLIERO; P. TIRAO

ADVANCES IN MATHEMATICS

Elsevier

Año: 2004 vol. 181 p. 276 - 307

0001-8708

Given a parabolic subalgebra $\frak g_1\times\frak n$ of a semisimple Lie algebra, the $\frak g_1$-invariants have been computed in the cohomology group of $\frak n$ with exterior adjoint coefficients (Kostant, Griffiths). Bott proved that this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. The aim of the article is to explicitly compute the module structure of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients, over the symplectic group. The authors prove that this is the cohomology of the cotangent bundle of the Heisenberg group. \par Nomizu proved that the cohomology group $\text {H}^\ast$ of the compact homogeneous space $X=N/\Gamma$, where $N$ is a connected and simply connected nilpotent Lie group and $\Gamma$ is a discrete subgroup of $N$, is canonically isomorphic to the cohomology group $\text {H}^\ast(\frak n)$ of the Lie algebra $\frak n$ of $N$. Results of Bott, Kostant and Griffiths prove that this cohomology (with sufficient conditions) can be computed by the cohomology of the Heisenberg Lie algebras $\cal H =V\oplus\Bbb{C}z$ with $z$ central, $V$ a symplectic complex vector space with symplectic form $(\cdot,\cdot)$, as an $\text {Sp}(V)$-module. Here $\text {Sp}(V)$ is the symplectic group of $V$, that is the the subgroup of $\text {GL}(V)$ preserving the symplectic form. \par The main goal of the article is to give an explicit description of the above $\text {Sp}(V)$-cohomology modules, with coefficients in the full exterior algebra $\Lambda\cal{H}$. $\text {H}^p(\cal{H},\bigwedge^q\cal{H})$ is determined for all $p$ and $q$. The result is expressed in the Young diagram notation. \par The symplectic preliminaries are treated in an elementary manner, so is the theory of weights of $\text {Sp}(V)$-modules and their representation by Young diagrams. The tensor product and exterior algebra structure of $\text {Sp}(V)$-modules are given for computational reasons. All the nice basic theory leads up to the computation of at first the Lie algebra cohomology, then the trivial and the exterior adjoint (co) homology of $\cal{H}$. In addition to the main result, the article contains a lot of other explicit, nice results on the subject.