Invariant polynomials under the maximal unipotent subgroup of a classical group
Let \(G\) be a classical simple group, i.e., \(G = SL (n, \mathbb{C})\), \(SO (n, \mathbb{C})\), \(Sp (2n, \mathbb{C})\). Let \(g\) be the Lie algebra of \(G\), \(h\) a Cartan subalgebra of \(g,H\) the corresponding connected subgroup of \(G\). Let \(\Delta\), be the root system attached to \(h\), \(\Pi\) a basis of simple roots, \(\Delta^+\) the set of positive roots with respect to \(\Pi\). Then we have a decomposition: \(g = n_- \oplus h \oplus n_+\), where \(n_+ = \sum_{\alpha \in \Delta^+} g_\alpha\), \(g_\alpha\) the root spaces. Let \(N\) be the subgroup of \(G\) corresponding to \(n_+\) under the exponential map. If \(V\) is an irreducible representation of \(G\) then we have a character \(\Xi\) associated to it, i.e., if \(\nu\) is a highest weight vector of \(V\) then \(h. \nu = \Xi (h) \nu\). Let \(\Psi_0\) and \(\Psi_0^*\) be the characters associated to the natural representation and its dual respectively and let \(\Psi = \Psi_0. \Psi_0^* \). Let \(L = \text{Ker} \Psi \times N\), it acts on \(S'(g) = S(g^*)\) by the restriction of the action of \(G\) induced by the coadjoint action on \(g^*\). The purpose of this paper is to prove the following: Theorem. \(S(g^*)^L\) is a polynomial ring on \(d(G)\) variables, where \(d(SL (n, \mathbb{C})) = 2(n - 1)\), \(d(SO (n, \mathbb{C})) = n - 1\), \(d(Sp (2n, \mathbb{C})) = 2n\). This theorem is a generalization to all the classical groups of the one proved by the author [Rev. Unión Mat. Argent. 35(1990), 163-167 (1989; Zbl 0815.20030)] for \(SL (n, \mathbb{C})\). In fact the method involved in the proof is the same. The proof consists of the following steps: first we give a set \(A(g)\) of \(d(G)\) polynomials, then we prove that they are \(L\)-invariant and algebraically independent, finally we compute the dimension of \[ S' (g)^L_{\psi^m} = \bigl\{ p \in S'(g)^L : h.p = \Psi (h)^m p\;\forall h \in H \bigr\}, \] as an \(S'(g)^G\)-module and we can see that it has the right value, i.e., it coincides with the dimension of the homogeneous polynomials of degree \(m\) in \(d(G) - rk(g)\) variables.
scientific article; zbMATH DE number 854422