The cohomology of the cotangent bundle of Heisenberg groups

LEANDRO CAGLIERO, PAULO TIRAO

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2004 vol. 181 p. 276 - 307

0001-8708

The Heisenberg Lie algebra may be written in the form $scr{H}=V oplusBbb Cz$, where $z$ is a central element, $V$ is a symplectic complex vector space with symplectic form $(·,·)$, and the bracket is defined by $[v,w]=(v,w)z$ for all $v,win V$. The symplectic group ${ m Sp}(V)$ acts naturally on $scr{H}$ and induces an action on the exterior algebra $igwedgescr H$. This paper gives the ${ m Sp}(V)$-module structure of the homology space $H(scr{H},igwedge scr{H})$. The main tool is Howe duality ef[R. E. Howe, in The Schur lectures (1992) (Tel Aviv), 1--182, Bar-Ilan Univ., Ramat Gan, 1995; MR1321638 (96e:13006)]. This work generalizes the calculation of the adjoint cohomology $H(scr{H},scr{H})$ given by L. Magnin ef[Comm. Algebra 21 (1993), no. 6, 2101--2129; MR1215560 (94d:17027)] and the cohomology with trivial coefficients given by L. J. Santharoubane ef[Proc. Amer. Math. Soc. 87 (1983), no. 1, 23--28; MR0677223 (84b:17010)]. The paper concludes with an explicit description of the ${ m { m Sp}}(V)$-module structure of the cohomology group of the Lie algebra of the cotangent bundle of the Heisenberg group. The paper continues the theme of successfully applying group representation theory to the computation of the homology of nilpotent Lie algebras; previous studies of the ${ m GL}(V)$-module structure of the homology of free 2-step nilpotent Lie algebras include those of S. Sigg ef[J. Algebra 185 (1996), no. 1, 144--161; MR1409979 (97i:17014)], L. Cagliero and P. Tirao ef[Q. J. Math. 53 (2002), no. 2, 125--145; MR1909506 (2003e:17030)] and J. Grassberger, A. D. King and Tirao ef[J. Algebra 254 (2002), no. 2, 213--225; MR1933866 (2003h:17026)].