4 an odd lattice in1 1 n 1 is reective if and only if

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Unformatted text preview: M )+ = O(M ) ∩ O(n, 1)+ , O(M ) = O(M )+ × {±1}, where O(n, 1)+ is the subgroup of index 2 of O(n, 1) deﬁned in section 2.2. Note that every Refk (M ) is a normal subgroup of O(M )+ . Let P be a fundamental polyhedron of Refk (M ) in H n . Since O(M ) leaves invariant the set of root vectors α with ﬁxed (α, α), it leaves invariant the set of reﬂection hyperplanes of Refk (M ). Hence, for any g ∈ O(M )+ , there exists s ∈ Refk (M ) such that g (P ) = s(P ). This shows that (4.5) O(M )+ = Refk (M ) S (P ), where S (P ) is the subgroup of O(M ) which leaves P invariant. Example 4.5. Let M = Ep,q,r with ﬁnite Ref(M ). Then Ref(M ) = W (p, q, r ) from Example 2.7, where (p, q, r ) = (1, 1, n)(An ), (2, 2, n − 2)(Dn ), (2, 3, 3)(E6 ), (2, 3, 4) (E7 ), (2, 3, 5)(E8 ). We have S (P ) = Z/2Z(An , E6 , Dn , n ≥ 5), S (P ) = S3 (D4 ) and S (P ) is trivial for E7 , E8 . The standard notations for the ﬁnite reﬂection groups W (p, q, r ) are W (T ), where T = An , Dn , E6 , E7 , E8 . The corresponding lattices Ep,q,r are called ﬁnite root 22 IGOR V. DOLGACHEV lattices. Their reﬂection groups are the Weyl groups of the corresponding root systems. In general Ref(Ep,q,r ) is larger than the group W (p, q, r ) (see Example 4.11). Example 4.6. Let M = II25,1 be an even unimodular hyperbolic lattice of rank 26. According to Theorem 4.4 ∼ II25,1 = U ⊥ E 3 . 8 The lattice II25,1 contains as a direct summand an even positive deﬁnite unimodular lattice Λ of rank 24 with (v, v ) = 2 for all v ∈ Λ. A lattice with such properties (which determine uniquely the isomorphism class) is called a Leech lattice. Thus II25,1 can also be described as (4.6) II25,1 = U ⊥ Λ. The description of Ref(II25,1 ) = Ref2 (II25,1 ) was given by J. Conway [22]. The group admits a fundamental polytope P whose reﬂection hyperplanes are orthogonal to the Leech roots, i.e. root vectors of the form (f − (1 + (v,v) )g, v ), where 2 v ∈ Λ and f, g is a basis of U with Gram matrix ( 0 1 ). In other words, a choice 10 of a decomposition (4.6) deﬁnes a fundamental polyhedron for the reﬂection group with fundamental roots equal to the Leech vectors. We have where S (P ) ∼ Λ = O(II25,1 )+ = Ref(II25,1 ) O(Λ). S (P ), Deﬁne a hyperbolic lattice M to be reﬂective if its root vectors span M and Ref(M ) is of ﬁnite covolume (equivalently, its index in O(M ) is ﬁnite).6 In the hyperbolic case the ﬁrst condition follows from the second one. It is clear that the reﬂectivity property of M is preserved when we scale M , i.e. replace M with M (k) for any positive integer k. The following nice result is due to F. Esselmann [42]. Theorem 4.7. Reﬂective lattices of signature (n, 1) exist only if n ≤ 19 or n = 21. The ﬁrst example of a reﬂective lattice of rank 22 was given by Borcherds [12]. We will discuss this lattice later. An important tool in the classiﬁcation (yet unknown) of reﬂective lattices is the following lemma of Vinberg [112...
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