Modular forms and representations of the dihedral group. (English. Russian original) Zbl 0923.11070

Voskresenskaya, G. V.

Math. Notes 63, No. 1, 115-118 (1998); translation from Mat. Zametki 63, No. 1, 130-133 (1998).

The author continues her study of the relationship between representations of finite groups and multiplicative eta products. (Cf. the preceding review Zbl 0923.11069). Theorem: For the dihedral groups \(D_n\) with \(3\leq n\leq 23\), \(n\notin\{13, 17, 19\}\) (and only for these \(n)\), there exists a faithful representation \(\Phi\) of \(D_n\) such that, for any \(g\in D_n\), the eta product \(\eta_g(z)\) corresponding to \(\Phi(g)\) is an eigenfunction of all Hecke operators. A complete list of the functions \(\eta_g(z)\) is given. It contains all the 28 multiplicative eta products with integral weight which are known to exist [D. Dummit, H. Kisilevsky and J. McKay, Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)].

Reviewer: G.Köhler (Würzburg)

MSC:

11F22	Relationship to Lie algebras and finite simple groups
11F20	Dedekind eta function, Dedekind sums
11F11	Holomorphic modular forms of integral weight
11F25	Hecke-Petersson operators, differential operators (one variable)
20C15	Ordinary representations and characters

Keywords:

representations of finite groups; multiplicative eta products; dihedral groups; Hecke operators

Citations:

Zbl 0923.11069; Zbl 0578.10028

References:

[1]	G. V. Voskresenskaya,Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],29, No. 2, 71–73 (1995). · Zbl 0847.11022
[2]	G. V. Voskresenskaya,Mat. Zametki [Math. Notes],60, No. 2, 292–294 (1996).