Basic sums of coadjoint orbits of the unitriangular group
Let \(K = \overline {\mathbf F}_q\), \(U_n(K)\) the \(n \times n\) unitriangular group over \(K\), \(F : U_n (K) \to U_n(K)\) the standard Frobenius map and \(U = U_n(q)\) the finite group of \(F\)-stable points of \(U_n(K)\). Thus \(U\) is the group of \(n \times n\) unipotent upper triangular matrices over \({\mathbf F}_q\). The author studies the complex characters of \(U\). The classification of the irreducible characters of \(U_n(q)\) has long been known to be a difficult problem and this paper makes a nice contribution to the problem, when \(q = p^e\) and \(p \geq n\). The condition \(p \geq n\) is necessary since the author makes use of the Kirillov theory of coadjoint orbits, which was adapted to the case of \(U_n(q)\) by Kazhdan. Now \(U_n(K)\) acts on \(u_n(K)^*\), the dual of the Lie algebra \(u_n(K)\) of \(n \times n\) nilpotent matrices over \(K\), by the coadjoint representation. By the Kirillov theory the irreducible characters of \(U\) are in bijection with the \(F\)-stable \(U_n(K)\)-orbits on \(u_n(K)^*\), and the degree of an irreducible character is \(q^{{1\over 2} \dim O}\) where \(O\) is the corresponding orbit. The author defines certain ``orbit sums'' as follows. Let \(\Phi(n) = \{(i,j)\mid 1 \leq i < j \leq n\}\). A subset \(D\) of \(\Phi(n)\) is called a basic subset of \(|D \cap \{(i,j) : i < j \leq n\}|\leq 1\) for all \(i\), \(1 \leq i < n\) and \(|D \cap \{(i,j) : 1 \leq i < j\}|\leq 1\) for all \(j\), \(1 < j \leq n\). Let \(e^*_{ij} \in u_n (K)^*\) be defined by \(e^*_{ij} (a) = a_{ij}\) where \(a = (a_{ij}) \in u_n(K)\). For each \(\alpha \in K^\#\), let \(O_{ij}(\alpha)\) denote the \(U_n(K)\)-orbit of \(\alpha e^*_{ij}\). The author then defines for any non-empty basic subset \(D\) of \(\Phi(n)\) and a map \(\phi : D \to K^\#\), the ``basic sum'' \(O_D (\phi)\) to be the subset \(O_D(\phi) = \sum_{(i,j) \in D} O_{ij} (\phi(i,j))\) of \(u_n (K)^*\). Now for each \((i,j) \in \Phi(n)\) one has a subgroup \(U_{ij} (q) = \{x = (x_{ab}) \mid x_{ib} = 0\), \(i < b < j\}\) of \(U\), and if \(\alpha \in {\mathbf F}^\#_q\) we have a linear character \(\lambda_{ij} (\alpha)\) of \(U_{ij} (\alpha)\) defined by \(\lambda_{ij} (\alpha) (x) = \psi(\alpha x_{ij})\) where \(x = (x_{ij})\) and \(\psi\) is a fixed non-trivial additive character of \({\mathbf F}_q\). The induced characters \(\xi_{ij} (\alpha)\) of the \(\lambda_{ij} (\alpha)\) to \(U\) are irreducible, and have been studied by \textit{G. I. Lehrer} [Compos. Math. 28, 9-19 (1974; Zbl 0306.20007)]. The author now defines, for each non-empty basic set \(D\) and map \(\psi : D \to F^\#_q\), a product of such characters by \(\xi_D (\phi) = \prod_{(i,j) \in D} \xi_{ij} (\phi (i,j))\). The \(\xi_D (\phi)\) are called basic characters of \(U\). He studies the geometry of the subsets \(O_D (\phi)\) of \(u_n (K)^*\). He shows that \(O_D(\phi)\) is an irreducible subvariety of \(u_n(K)^*\) and thus obtains a decomposition of \(u_n(K)^*\) as a disjoint union of irreducible subvarieties. He introduces two subsets \(R(D)\) and \(S(D)\) of \(\Phi(n)\) associated with a basic set \(D\), and finds the dimension of \(O_D (\phi)\) in terms of \(S(D)\). He also shows that the sum \(\sum \chi(1)^2\) where \(\chi\) runs over the irreducible constituents of a fixed \(\xi_D (\phi)\) is equal to \(q^{|S(D)|}\). Finally he studies the ring of \(U_n (K)\)-invariant polynomial functions defined on \(u_n(K)^*\) and shows that it is a polynomial ring over \(K\).
disjoint union of irreducible subvarieties
scientific article; zbMATH DE number 815878