Character formula for finite-dimensional simple representations of the Lie superalgebra \(\mathfrak{gl}(2,2)\)
The purpose of this paper is to find a decomposition of finite-dimensional simple \(\mathfrak{gl}(2,2)\)-modules as a direct sum of simple \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-modules and to use this decomposition in order to find a combinatorial character formula for simple \(\mathfrak{gl}(2,2)\)-modules of finite dimension. A simple \(\mathfrak{gl}(2,2)\)-module is, by restriction, a \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-module and so, since \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\) is a reductive (complex) Lie algebra, it is completely reducible with respect to this Lie algebra. The Cartan subalgebra of diagonal matrices in \(\mathfrak{gl}(2,2)\) is also a Cartan subalgebra of \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\); hence, a combinatorial character formula may be obtained from a \(\mathfrak{gl}(2,2)\)-module decomposition. The simple \(\mathfrak{gl}(2,2)\)-module \(L_\lambda\) of highest weight \(\lambda\) is the unique simple quotient of the Kac module \(K_\lambda\) [\textit{V. G. Kac}, Differ. Geom. Methods Math. Phys. II, Proc., Bonn 1977, Lect. Notes Math. 676, 597--626 (1978; Zbl 0388.17002)] and is of finite dimension if and only if \(\lambda\) is dominant. When \(\lambda\) is dominant, the author finds a decomposition of \(L_\lambda\) as a direct sum of simple \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-modules via the \(\mathfrak{gl}_2 \times \mathfrak{gl}_2\)-decomposition of \(K_\lambda\). He exploits this decomposition to show that, if \(L_\lambda\) is not of dimension 1, then the conjecture of \textit{J. Bernstein} and \textit{D. Leites} [C. R. Acad. Bulg. Sci. 33, 1049--1051 (1980; Zbl 0457.17002)] for the character of \(L_\lambda\) is verified.
scientific article; zbMATH DE number 5712614
scientific article