# Colloquium abstracts

**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 G /K is isometric to the cochain complex C*(g, K; C^infty(Gamma G)_K) of the relative Lie algebra cohomology of (g, K) with coefficients in C^infty(Gamma G)_K. This gives an orthogonal decomposition of the space of harmonic forms on Gamma G /K into cochain groups of the form C*(g, K;V), where V is an isotypical sub-representation of C^infty(Gamma 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' 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.