Even unimodular lattices associated with the Weil representation of the finite symplectic group
The irreducible components of the Weyl representation of the group \(G=Sp(2n,q)\) where \(q=p^ d\), \(p>2\) is a prime are considered. The Weyl representation W of G is a complex representation of degree \(q^ n\) obtained from the action of G on an extraspecial group of order \(pq^{2n}\). See, for example, the articles of \textit{I. M. Isaacs} [Am. J. Math. 95, 594-635 (1973; Zbl 0277.20008)], \textit{G. M. Seitz} [J. Lond. Math. Soc., II. Ser. 10, 115-120 (1975; Zbl 0333.20039)] and \textit{H. N. Ward} [J. Algebra 20, 182-195 (1972; Zbl 0239.20013)]. W is the sum of two irreducible components of degrees \((q^ n-1)/2\) and \((q^ n+1)/2\), one of them is faithful and the other is not. Define the characters of the faithful and unfaithful components by \(\psi_ 1\) and \(\psi_ 2\), respectively. Let \(q=p\equiv 3 (mod 4)\), \(H\supset G\) be the group of skew symplectic transformations, Q be the rational field. Then \(| H:G| =2\), it is proved that the induced character \(\psi^ H_ 1\) is a character of an absolutely irreducible Q-representation \(\theta\) of H. Let M be the QH-module corresponding to \(\theta\). The main result of the paper (Theorem 3.2) shows that for even n there exists an H-invariant integral lattice in M that supports an even symmetric positive definite unimodular H-invariant form. The author points out that the argument in Theorem 3.2 is derived from \textit{J. G. Thompson}'s work [J. Algebra 38, 523-524 (1976; Zbl 0344.20001)]. It is crucial that \(\psi_ 1\) defines an irreducible Brauer character modulo any prime r. This was shown by \textit{Seitz} (cited above) for \(r\neq p\) and by \textit{A. E. Zalesskij} and the reviewer [Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1987, No.6, 9-15 (1987)] for \(r=p.\) The paper under review also gives some information of independent interest on Schur indices of irreducible characters of Sp(2n,q). Theorem 1.4. Let \(q\equiv 3 (mod 4)\), \(\lambda\in Irr G\) be a not real-valued character, and let \(\lambda\) (1) be relatively prime to p. Then \(\lambda\) has Schur index 1 over Q. Lemma 1.6. Let \(q\equiv 3 (mod 4)\). Then \(Q(\psi_ i)=Q(\sqrt{-p})\), \(i=1,2\), and the \(\psi_ i's\) both have Schur index 1 over Q. In [J. Algebra 96, 249-274 (1985; Zbl 0576.20026)] the author has shown that if \(q\equiv 1 (mod 4)\), then any \(\lambda\in Irr G\) is real-valued and if \(\lambda\) is faithful, it has Schur index 2 over Q. The author emphasizes that an analogue of theorem 3.2 for \(p\equiv 1 (mod 4)\) and arbitrary n cannot be proved by his methods because in this case \(\psi_ 1\) has Schur index 2 over Q. For \(G=Sp(2,p)\), \(p\equiv -1 (mod 4)\) Theorem 4.1 gives such an analogue.
even symmetric positive definite unimodular H-invariant form
scientific article; zbMATH DE number 4112885
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