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

TL;DR

This paper constructs a realization of quaternionic discrete series representations for Sp(1,1) using Fueter-regular functions and explores associated automorphic forms and their transformation properties.

Contribution

It introduces a novel realization of quaternionic discrete series via Fueter-regular functions and constructs a map to differential forms satisfying a cocycle condition.

Findings

Realization of quaternionic discrete series inside Fueter-regular functions.

Construction of automorphic forms with specific transformation properties.

Development of a map to closed differential forms satisfying cocycle conditions.

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.