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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 \(\prod (1-k\quad i)^{e_ i}\) 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)

Citations:

Zbl 0548.20005
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References:

[1] Atiyah, M.F., Tall, D.O.: Group representations, ?-rings and thej-homomorphism. Topology 8, 253-297 (1969) · Zbl 0176.52701
[2] Biagioli, A.J.: Products of transforms of the Dedekind eta function. Ph.D. Thesis, U. Wisconsin (Madison), 1982
[3] Bott, R.: Lectures onK(X). New York: Benjamin 1969 · Zbl 0194.23904
[4] Broué, M.: Groupes finis, séries formelles et fonctions modulaires. Sémin. Jacques Tits, Collège de France, 1982
[5] Conway, J.H.: Three lectures on exceptional groups. In: Powell-Higman, Finite Simple Groups (pp. 215-247). London: Academic Press 1971
[6] Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc.11, 303-339 (1979) · Zbl 0424.20010
[7] Frenkel, I., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular functionj as character. M.S.R.I. reprint, Berkeley, 1984 · Zbl 0543.20016
[8] Griess, R.: Schur multipliers of some sporadic simple groups. J. Alg.32, 445-446 (1974) · Zbl 0303.20010
[9] Hecke, E.: Mathematische Werke, 2nd ed. Göttingen: Vandenhoeck u. Ruprecht 1970 · Zbl 0205.28902
[10] Husemöller, D.: Fibre bundles, 2nd Ed. (Graduale Texts in Mathematics, Vol. 20) Berlin Heidelberg New York: Springer 1975
[11] Koike, M.: On McKay’s conjecture, to appear · Zbl 0548.10018
[12] Kisilevsky, H., McKay, J.: Multiplicative products of ?-functions. Contemp. Math.45 (1985) · Zbl 0578.10028
[13] Lang, S.: Introduction to modular forms. New York: Springer 1976 · Zbl 0344.10011
[14] Mason, G.: Applications of quasi-invertible characters. J. Lond. Math. Soc. · Zbl 0588.20008
[15] Mason, G.: Frame-shapes and rational characters of finite groups. J. Alg.89, 237-246 (1984) · Zbl 0548.20005
[16] Mason, G.:M 24 and certain automorphic forms. Contemp. Math.,45, 223-244 (1985) · Zbl 0578.10029
[17] Mason, G.: Modular forms and the theory of Thompson series. In: Proceedings of the Rutgers Theory Year, 1983-1984 (Aschbacher, M., et al., eds.), pp. 391-407. Cambridge: Cambridge University Press 1984
[18] Mason, G.: Discriminants and the spinor norm. To appear in Proc. Lond. Math. Soc. · Zbl 0685.20005
[19] Ogg, A.: Modular forms and Dirichlet series. New York: Benjamin 1969 · Zbl 0191.38101
[20] Rademacher, H.: Topics in analytic number theory. New York: Springer 1973 · Zbl 0253.10002
[21] Schoeneberg, B.: Elliptic modular functions. New York: Springer 1974 · Zbl 0285.10016
[22] Serre, J.-P.: A course in arithmetic. Berlin Heidelberg New York: Springer 1973 · Zbl 0256.12001
[23] Thompson, J.G.: Finite groups and modular functions. Bull. Lond. Math. Soc.11, 374-351 (1979) · Zbl 0424.20011
[24] Tom Dieck, T.: Transformation groups and representation theory. S.L.N. 766, New York, 1979 · Zbl 0445.57023
[25] Zassenhaus, H.: On the spinor norm. Arch. Math.13, 434-451 (1962) · Zbl 0118.01804
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