## 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$$(G_ 0)$$.

### MSC:

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

### Keywords:

Lie groups; eta-function; Leech lattice