Finite groups and Hecke operators (Q1097911)

For a finite group G, the author defines a Thompson series to be a formal power series \(\Gamma =\sum \gamma _ nq\) n with coefficients in the ring RG of virtual characters of G with the property that for each \(g\in G\), the series \(\Gamma _ g=\sum \gamma _ n(g)q\) n is the q-expansion of a (meromorphic) modular form with signature (weight, level and character) depending on g. The paper initiates a systematic study of such series, the main results concerning the possibility of introducing a theory of Hecke operators for Thompson series. Let \(\rho\) : \(G\to O(n,{\mathbb{Q}})\) be a rational orthogonal representation of G. In earlier papers [J. Algebra 89, 237-246 (1984; Zbl 0548.20005); Discriminants and the spinor norm, to appear in Proc. Lond. Math. Soc.] the author introduced and studied a Thompson series \(\Omega\) for which each \(\Omega _ g\) is the eta-product \(\prod _{i}\eta (q\) \(i)^{e_ i}\) where \(\prod (1-k\quad i)^{e_ i}\) is the characteristic polynomial of \(\rho\) (g). Let M(\(\rho)\) be the space of Thompson series \(\Gamma\) for which each \(\Gamma _ g\) has signature equal to that of \(\Omega _ g\). For a prime p, operators \~T\({}_ p\) are constructed on M(\(\rho)\) which, under suitable circumstances, ``lift'' the usual Hecke operators \(T_ p\) in the sense that (\~T\({}_ p\Gamma)_ g=T_ p\Gamma _ g.\) What is surprising is that the operators \~T\({}_ p\) are related to the oriented Bott cannabalistic class: on q-expansions \~T\({}_ p\) acts via \(U_ p+\Psi _ pV_ p\) where for odd p, \(\Psi _ p\) is essentially the restriction to \(\rho\) (G) of the oriented Bott class of O(n,\({\mathbb{R}})\). For \(p=2\), \(\rho\) must be a spin representation of G, in which case \(\Psi _ 2\) is the restriction to \(\rho\) (G) of the half-spin character of Spin(n,\({\mathbb{R}})\). Various applications are given: there are examples of ``Thompson eigenforms'' where for certain sporadic simple groups G, \(\Omega _ G\) is invariant under all \~T\({}_ p\) and hence the corresponding Dirichlet series has an Euler product. ``Thompson- Eisenstein'' series are also constructed.