Characters of irreducible \(G\)-modules and cohomology of \(G/P\) for the Lie supergroup \(G=Q(N)\) (Q1281951)

The authors derive the character formula for any finite-dimensional irreducible representation \(V\) of the Lie superalgebra \({\mathfrak g}=q(n)\). Let \({\mathfrak g}'=q(n-k)\), \(k=1,\dots,n-2\). First, the problem of calculating \(\text{char} V\) has been reduced to calculating the multiplicities of the irreducible \({\mathfrak g}'\)-subquotients of the cohomologies of dominant \({\mathfrak g}'\)-linearized bundles on the \(\Pi\)-symmetric projective superspaces of \(G'=Q(n-k)\). Second, the authors derive recurrence relations and reduce the problem to the calculation of the trivial irreducible representation in the cohomologies of a certain bundle on the \(\Pi\)-symmetric projective superspace, and then calculate the necessary multiplicities.