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In this paper we are interested in investigating the principal series representations of a certain rank 2 Kac–Moody group. This group (denoted bySL   ˜   2    (q    (   (  z  )   )   )${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$   ) is introduced as a central extension by  q  *    ${\mathbb {F}_q^*}$    of the group of two-by-two unimodular matrices over the non-archimedean local field,  q    (   (  z  )   )${\mathbb {F}_q((z))}$   . We begin by determining a formula for a cocycle that defines this central extension. This is used to classify all irreducible representations of the centrally extended torus insideSL   ˜   2    (q    (   (  z  )   )   )${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$    and, as a consequence of this classification, we are able to determine the irreducible unramified principal series representations ofSL   ˜   2    (q    (   (  z  )   )   )${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$   .

Unramified representations of the universal central extension of SL 2 (𝔽 q ((z)))