Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical Groups

Abstract

Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical GroupsLet $q$ be a power of an odd prime. For the unipotent radicals of maximal rational parabolic subgroups of the finite symplectic, orthogonal and unitary groups, we determine all complex irreducible characters, their values, and the number of characters of each degree. Each of these numbers is a polynomial in $q-1$ with nonnegative integer coefficients, in analogy with a conjecture of Isaacs for unitriangular groups. For the orthogonal radicals, the character table and the power maps do not depend on the Witt type of the residual form, whereas the isomorphism type of the radical does when the isotropic dimension is at least three and the residual dimension is positive and even. Consequently, for $3\le a\le n-1$ the radicals of the stabilizers of totally isotropic $a$-spaces in $\SO^+_{2n}(q)$ and $\SO^-_{2n}(q)$ form Brauer pairs; for $a\le n-2$ both stabilizers are maximal parabolic subgroups. Finally, we express the multiplicities through $R$-polynomials, with the unequal parameters $q^2,\dots,q^2,q$ in the unitary case.