This article is devoted to the character theory of simple Lie superalgebras (over algebraically closed field of characteristic zero). A structure theory of Lie superalgebras and the theory of their finite-dimensional representations is developed in the author’s article “Lie superalgebras” [Adv. Math. 26, 8–96 (1977)]. Let \(G =G_0+G_1\) be a Lie superalgebra with a nondegenerate invariant bilinear form, which admits a decomposition on mutually orthogonal subspaces. \(G=N^{-}+H+N^{+}\), where \(H\) is a diagonalizable subalgebra, \(N^{-}\) and \(N^{+}\) are nilpotent \(H\)-invariant subalgebras. Let \(W\) be the Weyl group of \(G_0\). Let \(\Delta^+_0\) and \(\Delta^+_1\) be roots of \(H\) in \(N^+\cap G_0\) and \(N^+\cap G_1\), respectively and let \(\bar\Delta^+_1=\{\alpha\in \Delta^+_1\mid (\alpha,\alpha) =0\}\). Let \(\rho\) be the difference between the half-sum of roots from \(\Delta^+_0\) and the half-sum of roots from \(\Delta^+_1\). We set:
\[ L= \prod_{\alpha\in \Delta^+_0} \left(e^{\alpha/2} - e^{-\alpha/2}\right) \bigg / \prod_{\alpha\in \Delta^+_1}\left(e^{\alpha/2} - e^{-\alpha/2}\right). \]
For the subalgebra \(B=H+N^+\) and a linear form \(\Lambda\in H^*\) we define an \(f\)-dimensional even \(B\)-module: \(h(v_\Lambda)=\Lambda(h| v_\Lambda)\), \(h\in H\), \(N^+ (v_\Lambda) =0\). The induced \(G\)-module \(\tilde V(\Lambda)\) contains a unique maximal submodule \(I(\Lambda)\); we set \(V(\Lambda)= \tilde V(\Lambda)/I(\Lambda)\). We have the weight decomposition with respect to \(H: V(\Lambda) =\oplus V_\lambda\). We set \(\mathrm{ch}\, V(\Lambda) =\sum (\dim V_\lambda) e^\lambda\).Any finite-dimensional irreducible \(G\)-module \(V\) is isomorphic to one of \(V(\Lambda)\).
We call \(V\) typical if it splits in any finite-dimensional representation. Equivalent definition is that any finite-dimensional irreducible \(G\)-module \(V\)’, such that infinitesimal characters of \(V\) and \(V'\) are equal, is isomorphic to \(V\).
Two main statements are the following.
Statement 1. A \(G\)-module \(V(\Lambda)\) is typical if and only if \((\Lambda+\rho, \alpha)\neq 0\) for any \(\alpha\in\bar\Delta^+_1\).
Statement 2. \(\mathrm{ch}\, V(\Lambda)=L^{-1} \sum_w (\det w) e^{w(\Lambda+\rho)}\).
The proof of these statements is based on Statement 3. Let \(B\) be the algebra of \(W\)-invariant polynomials on \(H^*\) and \(A\) be the image of \(S(G)^G\) by restriction homomorphism. We set \(Q= \prod_{\alpha\in\bar\Delta^+_1} h_\alpha\). Then \(Q\in A\) and for any \(P\in B\) there exists \(k\), such that \(Q^kP\in A\).
In the present article statement 3 is proved in a weaker form. Now I can deduce this statement from the following generalization of Chevalley theorem for arbitrary Lie algebras.
Statement 4. Let \(G\) be an algebraic Lie algebra, \(N\) be the nilradical of \(G\), \(H\) be a Cartan subalgebra, \(W\) be the Weyl group, and \(k[H]^W\) be the algebra of \(W\)-invariant polynomials on \(H\). For \(\alpha\in H^*\) let \(P_\alpha\in k[H]^W\) be a polynomial of minimal degree divisible by \(L\). Let \(\beta_, \ldots, \beta_k\) be the roots of \(H\) on \(N\) for which the polynomials \(P_{\beta_i}\) are restrictions of semiinvariant polynomials \(Q_i\), \(i=1,\ldots, k\) on \(G\) with respect to the adjoint action. Then the restriction homomorphism \(h[G]\to k[H]\) is injective on \(G\)-semiinvariant polynomials, its image belongs to \(h{H}^*\) and any polynomial from \(h{H}^*\) is the restriction of a rational function \(P/Q_1^{S_1},\ldots, Q_k^{S_k}\), where \(P\) is a semiinvariant polynomial on \(G\).
In the article formulas for supercharacter, for \(\dim V\) and \(dim V_0 - \dim V_1\) in the case of typical \(V\) are also obtained. In some cases an explicit construction of typical representations is given.