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Finite groups and Hecke operators. (English) Zbl 0636.10021

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 \(\prod (1-k i)^{e_i}\) 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 \(\tilde T_p\) are constructed on \(M(\rho)\) which, under suitable circumstances, “lift” the usual Hecke operators \(T_p\) in the sense that \(\tilde T_p\Gamma)_g = T_p\Gamma_g\).
What is surprising is that the operators \(\tilde T_p\) are related to the oriented Bott cannibalistic class: on \(q\)-expansions \(\tilde T_p\) 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 \(\mathrm{Spin}(n,\mathbb{R})\).
Various applications are given: there are examples of “Thompson eigenforms” where for certain sporadic simple groups \(G\), \(\Omega_G\) is invariant under all \(\tilde T_p\) and hence the corresponding Dirichlet series has an Euler product. “Thompson-Eisenstein” series are also constructed.
Reviewer: Geoffrey Mason

MSC:

11F22 Relationship to Lie algebras and finite simple groups
11F25 Hecke-Petersson operators, differential operators (one variable)
20D08 Simple groups: sporadic groups

Citations:

Zbl 0548.20005
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References:

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