Symplectic Groups Lattices

by Rudolf Scharlau and Pham Huu Tiep

Abstract: Let $p$ be an odd prime. It is known that the symplectic group $\SP$ has two (algebraically conjugate) irreducible representations of degree $(p^{n}+1)/2$ realized over $\QQ(\sqrt{\eps p})$, where $\eps = (-1)^{(p-1)/2}$. We study the integral lattices related to these representations for the case $p^{n} \equiv 1 \bmod 4 $. (The case $p^{n} \equiv 3 \bmod 4 $ has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or $p$-modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of the Maslov index.

Key words and phrases: Integral lattice, unimodular lattice, p-modular lattice, finite symplectic group, Weil representation, Maslov index, linear code, self-dual code

1991 Mathematics Subject Classification: Primary 20C10; Secondary 20C15, 20C20, 11E12, 11H31, 94B05.

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To appear in: Transactions of the AMS

Contact: Rudolf.Scharlau@Mathematik.Uni-Dortmund.DE