Contra-variant form on Kac's modules and some series compositions (Q1908479)

The representation theory of classical Lie superalgebras \(G= G^0 \oplus G^1\) begins with what is called the ``Kac module''. For a dominant weight \(\varphi\), the Kac module is the module \(K_\varphi= U(G) \bigotimes_{U(P)} V^0_\varphi\), where \(V^0_\varphi\) is the simple \(G^0\)-module of \(\varphi\), \(U\) is for the universal enveloping algebra, \(P= G^0\oplus (B\cap G^1)\) and \(B\) is the Borel subalgebra of \(G\). The dominant weight \(\varphi\) is called typical if \(K_\varphi\) is irreducible. In the paper under review the author studies the case \(G= sl(n,m)\), the Lie superalgebra of all linear transformations with supertrace equal to zero in the \(n+m\)-dimensional vector space \(V\). In this case \(\varphi\) is called tensorial if \(K_\varphi\) is embeddable into a tensor power of \(V\). The main results of the paper are the following: (i) Let \(\tau^*\) be a dual basis of a basis of the Cartan algebra \(H\) of \(G\). For every dominant weight \(\varphi\) there exists \(r>0\) such that \(\varphi+ (r+i) \tau^*\) is a typical tensorial weight for each natural \(i\). (ii) If \(\varphi\) is a typical tensorial weight, then the Kac module \(K_\varphi\) carries a contravariant symmetric form which is positive and gives a lot of information on \(K_\varphi\). (iii) The series compositions of \(K_{\varphi+ x\tau^*}\) is obtained for any \(x\), where \(\varphi\) is the trivial weight of \(sl (n) \oplus sl (m) \subset G^0\).