## 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 $$\prod (1-k\quad i)^{e_ i}$$ 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.
Reviewer: G.Mason

### MSC:

 11F11 Holomorphic modular forms of integral weight 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 20C05 Group rings of finite groups and their modules (group-theoretic aspects)

Zbl 0548.20005
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### References:

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