Locally and Polar Harmonic Maass Forms for Orthogonal Groups of Signature $(2, n)$

Abstract: We generalize the notions of locally and polar harmonic Maass forms to general orthogonal groups of signature $(2, n)$ with singularities along real analytic and algebraic cycles. We prove a current equation for locally harmonic Maass forms and recover the Fourier expansion of the Oda lift involving cycle integrals. Moreover, using the newly defined polar harmonic Maass forms, we prove that meromorphic modular forms with singularities along special divisors are orthogonal to cusp forms with respect to a regularized Petersson inner product. Using this machinery, we derive a duality theorem involving cycle integrals of meromorphic modular forms along real analytic cycles and cycle integrals of locally harmonic Maass forms along algebraic cycles.