Arghya Mondal
TIFR, Mumbai
September 27, 2018
Components of harmonic Poincare duals of special cycles: The de Rham complex of a compact locally symmetric space $\Gamma\setminus G /K$ is isometric to the cochain complex $C^*(\mathfrak{g}, K; C^\infty(\Gamma\setminus G)_K)$ of the relative Lie algebra cohomology of $(\mathfrak{g}, K)$ with coefficients in $C^\infty(\Gamma\setminus G)_K$. This gives an orthogonal decomposition of the space of harmonic forms on $\Gamma\setminus G /K$ into cochain groups of the form $C^*(\mathfrak{g}, K;V)$, where $V$ is an isotypical sub-representation of $C^\infty(\Gamma\setminus G)_K$ on which the Casimir operator acts trivially. Using representation theoretic methods, we will deduce some conditions for vanishing of a component of the harmonic Poincare dual of a special cycle $\Gamma'\setminus G'/K'$ with respect to this decomposition. For certain special cycles, when $G=SU(p,q)$, these conditions can be applied to deduce which components do not vanish.
TIFR, Mumbai
September 27, 2018
Components of harmonic Poincare duals of special cycles: The de Rham complex of a compact locally symmetric space $\Gamma\setminus G /K$ is isometric to the cochain complex $C^*(\mathfrak{g}, K; C^\infty(\Gamma\setminus G)_K)$ of the relative Lie algebra cohomology of $(\mathfrak{g}, K)$ with coefficients in $C^\infty(\Gamma\setminus G)_K$. This gives an orthogonal decomposition of the space of harmonic forms on $\Gamma\setminus G /K$ into cochain groups of the form $C^*(\mathfrak{g}, K;V)$, where $V$ is an isotypical sub-representation of $C^\infty(\Gamma\setminus G)_K$ on which the Casimir operator acts trivially. Using representation theoretic methods, we will deduce some conditions for vanishing of a component of the harmonic Poincare dual of a special cycle $\Gamma'\setminus G'/K'$ with respect to this decomposition. For certain special cycles, when $G=SU(p,q)$, these conditions can be applied to deduce which components do not vanish.