# Orthogonal group of the lattice $I_{p,q}$?

Here $I_{p,q}$ is the unique-up-to-isometry unimodular lattice of signature $(p,q)$, whose Gram matrix is diagonal with $p$ 1s and $q$ -1s.

In his paper "ON GROUPS OF UNIT ELEMENTS OF CERTAIN QUADRATIC FORMS", Vinberg gives a description of the automorphism group of the lattice $I_{p,1}$. It is a semi-direct product of the subgroup generated by reflections, which is a hyperbolic Coxeter group that can be effectively described, and a subgroup of the symmetries of the fundamental polyhedron for this Coxeter group.

Are there similar descriptions of the orthogonal group of $I_{p,q}$? What about the special case $q=2$?

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Thanks. I have been looking at Allcock's papers, which are very interesting, but haven't found anything besided the hyperbolic/Lorentzian case. Do you have a particular paper in mind? –  A. Pascal Aug 17 '11 at 21:19
Yes, it's annoying that the Russian version is free but you have to pay for the translation. At least I hope some of that money goes to support Russian mathematics. –  A. Pascal Aug 17 '11 at 21:23
My campus math library has an article copying service. If they have it (at a separate storage facility a few miles away) one can request a scanned pdf, although I suppose they would refuse for something book length or too recent, and they do have translated Mat. Sbornik. The one he recommended for me was "new complex- and quaternion-hyperbolic reflection groups", but actually his email was more informative. This refers to my several MO questions on covering radius and class number for positive forms/lattices. Even the standard, Neumaier and Siedel 1983, is just Lorentzian. –  Will Jagy Aug 17 '11 at 22:38
Did you read his Fall 2011 class description, ma.utexas.edu/users/allcock/teaching/Coxeter.html where he recommends Humphries, "Reflection Groups and Coxeter Groups". –  Will Jagy Aug 17 '11 at 22:40
I see, he mistyped, the book is by Jim Humphreys, an MO user. –  Will Jagy Aug 18 '11 at 1:48

The subgroup generated by reflections is normal, and therefore is finite-index by the Margulis normal subgroup theorem (as long as the rank is $\geq 2$, so $|p|\geq 2, |q|\geq 2$).

The conjugate of a reflection is a reflection. In fact, a reflection may be defined as a matrix element $A$ such that $I−A$ has rank 1. This is clearly conjugacy invariant. Also, if you conjugate a reflection in the vector $v$ by a matrix $B$, then you get a reflection in $Bv$.