Computing Cohomology Spaces of Left-Invariant Involutive Structures on  \(\textrm{SU}(2)\)

Involutive systems of complex left-invariant vector fields are investigated on \(\textrm{SU}(2)\) . We prove closedness of the range (in the smooth topology) of the annihilation operator \(\partial _-\) (which spans the CR structure of \(\textrm{SU}(2)\) ) and other vector-valued operators associated to it. We also compute the cohomologies of certain (elliptic) corank 1 systems associated to \(\partial _-\) .