Modular forms and representations of groups. (Russian) Zbl 0771.11022

Let \(G_ 0\) be a simple Lie group with Lie algebra Lie\((G_ 0)\) of even rank. Let \(G\) be such a finite subgroup of \(G_ 0\), every element \(g\) of which has in the adjoint representation a rational characteristic polynomial \(\prod_ k(x^{a_ k}-1)^{t_ k}\), with which we associate the modular form \(\eta_ g(z)=\prod_ k \eta(a_ k z)^{t_ k}\). On the group \(G\) we define for every odd prime \(p\): \(\psi_ p(g)=p^{k(g)-1} \chi_ g(p)\), when \(k(g)\) and \(\chi_ g(p)\) are the weight and character of the form \(\eta_ g(z)\).
The main result of the paper then is the following: If \(p\) is relatively prime to the order of \(g\), we have \[ \psi_ p(g)=\left({{-1} \over p}\right)^{{{\dim G_ 0} \over 2}} p^{{r\over 2}-1} ch_{(p- 1)\rho}(g), \] when \(r\) is the rank of the algebra Lie\((G_ 0)\) and \(ch_{(p-1)\rho}\) is the Weil character of an irreducible representation of \(G_ 0\) with highest weight \((p-1)\rho\), where \(\rho\) is the sum of the positive roots of Lie\((G_ 0)\).


11F22 Relationship to Lie algebras and finite simple groups
11F20 Dedekind eta function, Dedekind sums