Generalized exponents and forms.
Let \(V=\mathbb{C}^l\) or \(\mathbb{R}^l\) be a complex or real vector space. A reflection is an element of \(\text{GL}(V)\) whose fixed point set is a hyperplane in \(V\). Let \(G\) be a reflection group, i.e., a finite subgroup of \(\text{GL}(V)\) generated by reflections. We assume all \(G\)-modules are \(\mathbb{C} G\)-modules. For any \(G\)-module \(U\) and irreducible \(G\)-module \(M\), let \(U^M\) be the isotypic component of \(U\) of type \(M\), i.e., the direct sum of those \(G\)-submodules of \(U\) isomorphic to \(M\). Let \(U^G:=\{u\in U:gu=u\) for all \(g\in G\}\) denote the set of \(G\)-invariants. The reflection group \(G\) acts contragradiently on \(V^*\) and thus on the symmetric algebra \(S:=S(V^*)\), which we identify with the algebra of polynomial functions on \(V\). Let \(I\subset S\) be the Hilbert ideal generated by the invariant polynomials of positive degree. Then \(S^G=\mathbb{C}[f_1,\dots,f_l]\) for some homogeneous polynomials \(f_1,\dots,f_l\) called basic invariants. The algebra \(S/I\) is called the coinvariant algebra. \(S/I\) is isomorphic to the regular representation and \(S\simeq S^G\otimes S/I\) as \(G\)-modules. For any irreducible \(G\)-module \(M\), the isotypic component \((S/I)^M\) decomposes as \(M_1\oplus M_2\oplus\cdots\oplus M_{\dim M}\) for some homogeneous subspaces \(M_i\simeq M\) of degree \(e_i(M)\). We call \(e_1(M),e_2(M),\dots,e_{\dim M}(M)\) the \(M\)-exponents. Let \(m_1,\dots,m_l\) be the \(V\)-exponents, called the exponents of the group. Similarly, let \(m_1^*,\dots,m_l^*\) be the \(V^*\)-exponents, called the coexponents of the group. The exponents and coexponents of the group indicate the invariant theory of differential forms and derivations. In this paper, the author investigates twisted reflection representations (\(V\) tensor a linear character) using the theory of semi-invariant differential forms. The author uses information about semi-invariant polynomials, forms, and derivations to describe generalized exponents for \(\chi V:=V\otimes\mathbb{C}_\chi\) and generalized coexponents for \(\chi V^*:=\mathbb{C}_\chi\otimes V^*\), i.e, \(\chi V\)-exponents and \(\chi V^*\)-exponents, where \(\chi\) is a linear character of \(G\). The author applies Springer's theory of regular numbers to reflection groups generated by \(\dim V\) reflections. Although her arguments are case-free, the author also includes explicit data and gives a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.
scientific article; zbMATH DE number 2174455