Bruhat-Tits buildings, harmonic cocycles and an Eichler-Shimura map in the function field setting

In this talk I will discuss some results about the action of certain arithmetic congruence subgroups Γ on Bruhat-Tits buildings BT_r for general linear groups GL_r over fields in positive characteristic. Grayson-Quillen established an equivariant homotopy equivalence between the unstable region of BT_r and the rational Tits building at the boundary, where instability is defined via Harder-Narasimhan instability of vector bundles. Adapting their proof strategy we obtain an analogous result, where instability is defined in terms of simplex-stabilizers relative to Γ. We use this to deduce results on Γ-invariant harmonic cocycles on BT_r, known by Teitelbaum for GL_2. Conjecturally, these harmonic cocycles form the bridge between Drinfeld cusp forms and function field motives on Drinfeld modular varieties.