Roelof Bruggeman

For vector-valued Maass cusp forms for~$SL_2(\mathbb{Z})$ with real weight~$k\in\mathbb{R}$ and spectral parameter $s\in\mathbb{C}$, $\mathrm{Re} s\in (0,1)$, $s\not\equiv \pm k/2$ mod $1$, we propose a notion of vector-valued period functions, and we establish a linear isomorphism between the spaces of Maass cusp forms and period functions by means of a cohomological approach. The period functions are a generalization of those for the classical Maass cusp forms, being solutions of a finite-term functional equation or, equivalently, eigenfunctions with eigenvalue $1$ of a transfer operator deduced from the geodesic flow on the modular surface. We apply this result to deduce a notion of period functions and related linear isomorphism for Jacobi Maass forms of weight $k+1/2$ for the semi-direct product of $SL_2(\mathbb{Z})$ with the integer points $Hei(\mathbb{Z})$ of the Heisenberg group.

We define and study 'non-abelian' Poincar\'e series for the group $G=\mathrm{SU} (2,1)$, i.e. Poincar\'e series attached to a Stone-Von Neumann representation of the unipotent subgroup $N$ of $G$. Such Poincar\'e series have in general exponential growth. In this study we use results on abelian and non-abelian Fourier term modules obtained in arXiv:1912.01334. We compute the inner product of truncations of these series and those associated to unitary characters of $N$ with square integrable automorphic forms, in connection with their Fourier expansions. As a consequence, we obtain general completeness results that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincar\'e series for $\mathrm{SL}(2,\mathbb{R})$.

This work gives a version of the Eichler-Shimura isomorphism with a non-abelian $H^1$ in group cohomology. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. We show Manin's map is injective but far from surjective. By extending Manin's map we are able to construct a bijective map and remarkably this establishes the existence of a non-abelian version of the Eichler-Shimura map.