The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants (Q1909238)

Let \(G\) be a connected reductive group defined over an algebraically closed field \(k\) of characteristic zero, and let \(\mathfrak g\) be its Lie algebra. Fix a maximal torus \(T\) with Lie algebra \(\mathfrak t\). Let \(W\) denote the Weyl group defined by \(W=N_GT/T\). Clearly, \(k({\mathfrak g})^G\) is a polynomial ring generated by algebraically independent homogeneous invariants \(f_1,\dots,f_r\) (\(r=\text{rank}({\mathfrak g})\)) and \(k({\mathfrak g})^G\cong k({\mathfrak t})^W\). Write it as \(B\). For any \(G\)-module \(M\), let \(B(M)\) denote \((k({\mathfrak g})\otimes M)^G\). Let now \(M\) be any \(G\)-module with zero-weight multiplicity \(\ell(\neq 0)\) and let \(e_1(M)\leq e_2(M)\leq\dots\leq e_\ell(M)\) be the ordered sequence of degrees of any homogeneous system of generators of the graded \(B\)-module \(B(M)\). These are called (Kostant's) generalized \(G\)-exponents for \(M\). Write \(E(M)\) for their sum. Similarly, for any \(G\)-module \(M\), \(E_W(M^T)\) is defined in this paper. Let \(\phi_1,\dots,\phi_s\) be the roots that are dominant. In this paper, it is shown that the restriction induces an isomorphism of free graded \(B\)-modules \(\rho_M:\text{Mor}_G({\mathfrak g},M)\to\text{Mor}_W({\mathfrak t},M^T)\) if and only if the weights \(2\phi_i\) (\(1\leq i\leq s\)) do not occur at \(T\)-weights in \(M\), and formulas for the sum \(E(M)\) and \(E_W(M^T)\) are given.