Fueter-Regular Discete Series for Sp(1,1)

Abstract

We show that the quaternionic discrete series on G=Sp(1,1) with minimal K-type of dimension n+1 can be realized inside the space of Fueter-regular functions on the quaternionic ball B in H, with values in H^n. We then consider the corresponding $H^n$-valued Fueter-regular automorphic forms on G. For a fixed level $\Gamma$, we construct a non-trivial map from the space of pairs of such automorphic forms, to closed $M_n(H)$-valued differential $3$-forms on $B$, which transform under $\Gamma$ according to a cocycle condition.