A new kind of automorphic form and a proof of the essential transformation laws
Michael Andrew Henry
TL;DR
Problem: construct a vector-valued analogue of automorphic forms for arbitrary Hecke triangle groups from quasiautomorphic data. Approach: build the Hecke vector-form $\mathbf{F}_U(z)$ from a quasiautomorphic form $U_{\mathfrak{t}_\mu,w,r}$ using a hauptbuch $\mathbf{G}_U(z)$ and a transfer-convolution framework with the creation matrix $A_r$ and generalized Pascal matrix $P_r(z)$ so that $\mathbf{F}_U(z)=e^{z A_r}\mathbf{G}_U(z)$ and $\mathbf{F}_U(z)=P(\mathbf{G}_U)(z)\nu_r(z)$. Main results: explicit transformation laws under the generators $T$ and $S$, namely $\mathbf{F}_U(Tz)=e^{\varpi_\mu A_r}\mathbf{F}_U(z)$ and $\frac{\mathbf{F}_U(Sz)}{z^{w-r}}=d_r^{\mathbf{y}}(a_i)\mathbf{F}_U(z)$, holding modulo $\mathfrak{t}_\mu$, thereby giving a robust vector-automorphic framework. Significance: extends automorphic/quasimodular structures to nonclassical triangle groups with concrete, computable transformation laws and a new vector-analytic perspective.
Abstract
We utilize the structure of quasiautomorphic forms over an arbitrary Hecke triangle group to define a new vector analogue of an automorphic form. We supply a proof of the functional equations that hold for these functions modulo the group generators.