A vertex operator algebra related to E8 with automorphism group O+(10, 2)

We study a particular VOA which is a subVOA of the E8-lattice VOA and determine its automorphism group. Some of this group may be seen within the group E8(C), but not all of it. The automorphism group turns out to be the 3-transposition group O(10, 2) of order 2357.17.31 and it contains the simple group Ω(10, 2) with index 2. We use a recent theory of Miyamoto to get involutory automorphisms associated to conformal vectors. This VOA also embeds in the moonshine module and has stabilizer in M I , the monster, of the form 2.Ω(10, 2). Hypotheses We review some definitions, based on the usual definitions for the elements, products and inner products for lattice VOAs; see [FLM]. Notation 1.2. Φ is a root system whose components have types ADE, g is a Lie algebra with root system Φ, Q := QΦ , the root lattice and V := VQ := S(Ĥ−) ⊗ C[Q] is the lattice VOA in the usual notation. Remark 1.3. We display a few graded pieces of V (⊗ is omitted, and here Q can be any even lattice). We write Hm for H ⊗ t in the usual notation for lattice VOAs (2.1) and Qm := {x ∈ Q | (x, x) = 2m}, the set of lattice vectors of type m. V0 = C, V1 = H1, V2 = [S H1 +H2] +H1CQ1 + CQ2, V3 = [S H1 +H1H2 +H3] + [S H1 +H2]CQ1 +H1CQ2 + CQ3, V4 = [S H1 + S H1H2 +H1H3 + S H2 +H4]+ [SH1 +H1H2 +H3]CQ1 + [S H1 +H2]CQ2 +H1CQ3 + CQ4. Remark 1.4. Let F be a subgroup of Aut(g), where g is the Lie algebra V1 = H1 + CQ1 with 0 binary composition. The fixed points V F of F on V form a subVOA. We have an action of N(F )/F as automorphisms of this sub VOA. 44 R. L. Griess, Jr. Notation 1.5. For the rest of this article, we take Q to be the E8-lattice. Take F to be a 2B-pure elementary abelian 2-group of rank 5 in Aut(g) ∼= E8(C); it is fixed point free. Let E := F ∩ T where T is the standard torus and where F is chosen to make rank(E) = 4. Let θ ∈ F rE; we arrange for θ to interchange the standard Chevalley generators xα and x−α . See [Gr91]. The Chevalley generator xα corresponds to the standard generator e of the lattice VOA VQ . Notation 1.7. L := Q ∼= √ 2Q denotes the common kernel of the lattice characters associated to the elements of E; in the [Carter] notation, these characters are h(E); in the root lattice modulo 2, they correspond to the sixteen vectors in a maximal totally singular subspace. Then (1.7.1) V F 1 = 0 and (1.7.2) V F 2 = S H1 + 0 + CL θ 2, where the latter summand stands for the span of all e + e, where λ runs over all the 15 ·16 = 240 norm 4 lattice vectors in L. Thus, V F 2 has dimension ( 9 2 ) + 240 2 = 36 + 120 = 156 and has a commutative algebra structure invariant under N(F ) ∼= 2 ·GL(5, 2). We note that N(F )/F ∼= 2:GL(5, 2) [Gr76][CoGr][Gr91]. We will show (6.10) that Aut(V F ) ∼= O(10, 2). 2. Inner Product. Definition 2.1. The inner product on SHm is 〈xn, xn〉 = n!mn〈x, x〉n . This is based on the adjointness requirement for h⊗tk and h⊗t−k (see (1.8.15), FLM,p.29). When k > 0, h ⊗ t acts like multiplication by h ⊗ t and, when h is a root, h⊗ t acts like k times differentiation with respect to h. When n = 2, this means 〈x2, x2〉 = 2m2〈x, x〉. In V F 2 , m = 1. Definition 2.2. The Symmetric Bilinear Form. Source: [FLM], p.217. This form is associative with respect to the product (Section 3). We write H for H1 . The set of all g and xα spans V2 . (2.2.1) 〈g, h〉 = 2〈g, h〉, whence (2.2.2) 〈pq, rs〉 = 〈p, r〉〈q, s〉 + 〈p, s〉〈q, r〉, for p, q, r, s ∈ H. (2.2.3) 〈xα , x+β 〉 = { 2 α = ±β 0 else (2.2.4) 〈g, x+β 〉 = 0. A VOA related to E8 with automorphism group O (10, 2) 45 Notation 2.3. In addition, we have the distinguished Virasoro element ω and identity I := 1 2ω on V2 (see Section 3). If hi is a basis for H and h ∗ i the dual basis, then ω = 12 ∑ i hih ∗ i . Remark 2.4. (2.4.1) 〈g, ω〉 = 〈g, g〉 (2.4.2) 〈g, I〉 = 1 2 〈g, g〉 (2.4.3) 〈I, I〉 = dim(H)/8 (2.4.4) 〈ω, ω〉 = dim(H)/2 If {xi | i = 1, . . . l} is an ON basis,