i= P O C4x E be the group generated by E and the central produ*
*ct P O C4 of
P and the cyclic group C4 = * Cx with Z=2 amalgamated. The image of G in PGL*
* (n + 1, C)
is V and q(vCx) = v2, v 2 G, is a quadratic form on V such that q(uCx + vCx) = *
*q(uCx) +
q(vCx) + [uCx, vCx] for all uCx, vCx 2 V .
5.11. Lemma. A(GL (n + 1, C))(G, G) ! A(PGL (n + 1, C))(V, V ) is surjective.
Proof.Suppose that B 2 GL (n + 1, C) is such that V BCx = V . Then GB G . Cx:*
* For any
g 2 G there exist h 2 G and z 2 Cx such that gB = hz. But since G has exponent *
*4, z4 = 1 so
z 2 C4 and gB 2 G.
A monomorphic conjugacy class :V ! PGL (n + 1, C)is said to be a (2d + r, r*
*) object of
A(PGL (n+1, C)) if the underlying symplectic vector space of (V, ) is isomorph*
*ic to V = HdxV ?
where H denotes the symplectic plane over F2 and dimFpV ?= r [16, II.9.6] (so t*
*hat dimFpV =
r + 2d). An (r, r) object is the same thing as an r-dimensional toral object. W*
*e write Sp(V ) or
Sp(2d + r, r) (abbreviated to Sp(2d) if r = 0) for the group of linear automorp*
*hisms of V that
preserve the symplectic form.
5.12. Corollary. Suppose that n + 1 = 2dm for some natural numbers d 1 and m *
* 1.
(1)There is up to isomorphism a unique (2d, 0) object Hd of A(PGL (n + 1, C)*
*), and
A(PGL (n + 1, C))(Hd) = Sp(2d), CPGL(n+1,C)(Hd) = Hd x PGL(m, C)
for this object.
20 J.M. MØLLER
(2)Isomorphism classes of (2d + r, r), r > 0, objects V of A(PGL (2dm, C)) c*
*orrespond biject-
ively to isomorphism classes of (r, r) objects V ? of A(PGL (m, C)), and
` '
A(PGL (2dm, C))(V ) = Sp(2d)*A(PGL (m0, C))(V ?)
CPGL(2dm,C)(V ) = V=V ?x CPGL(m,C)(V ?)
for these objects.
Proof.1. The group 21+2d+O4 has [17, 7.5] 21+2dcharacters of degree 1 and 2 irr*
*educible characters
of degree 2d (interchanged by the action of Out(21+2d+O 4) ~=Sp(2d) x Aut(C4) [*
*11, pp. 403-404])
given by (
d~(g) g 2 C4
Ø~(g) = 2
0 g 62 C4
where ~: C4 ! Cxis an injective group homomorphism (~(i) = i). The linear char*
*acters vanish
on the derived group 2 = [21+2d+O 4, 21+2d+O 4] but the irreducible characters *
*of degree 2d do not.
Thus the only faithful representations of 21+2d+O 4 with central centers are mu*
*ltiples mØ~ of Ø~ for
a fixed ~. Phrased slightly differently, GL (m2d, C) contains up to conjugacy a*
* unique subgroup
with central center isomorphic to 21+2d+O 4. For this group and its image Hd in*
* PGL (2dm, C) we
have
A(GL (m2d, C))(21+2d+O 4, 21+2d+O 4) ~=Sp(2d) ~=A(PGL (m2d, C))(Hd, Hd)
CGL(m2d,C)(21+2d+O 4) ~=GL(m, C), CPGL(m2d,C)(Hd) ~=Hd x PGL(m, C)
where the last isomorphism is a consequence of [29, 5.9].
2. The (2d+r, r) object (V, ) of A(PGL (2dm, C)) and the (r, 0) object (V ?, *
*? ) of A(PGL (m, C))
correspond to each other iff there is an m-dimensional representation ~: V ?! G*
*L(m, C)such that
C2d ~ is a lift of |V ? and ~ a lift of ? . According to 5.10 any lift of |*
*V ? has this form for
some ~ uniquely determined up to the action of (V ?)_.
We use 5.11 to calculate the Quillen automorphism group of a (2d + r, r) obje*
*ct Hd x V ? of
A(PGL (2dm, C)). Let Hd x V ? be covered by the group P O C4 x V ? as in 5.10. *
*Let ff be an
automorphism of P OC4, let fi be any homomorphism of the form P OC4 ! Hd ! V ?,*
* and let fl be
any Quillen automorphism of (V ?, ? ). Choose a homomorphism i1:P OC4 ! HdxC4=*
*C2 ! C4
such that ~(i1(x)ff(x)) = ~(x) for all x 2 C4 and a homomorphism i2:V ?! C4 su*
*ch that
~(i2(v))~(fl(v)) = ~(v) for all v 2 V ?. Then the automorphism of P O C4 that *
*takes (x, v) to
(i1(x)i2(v)ff(x), fi(x) + fl(v)) preserves the trace of Ø~#~ and therefore the *
*automorphism in-
duced on the quotient is a Quillen automorphism of Hd x V ?. Conversely, any a*
*utomorphism
of P O C4 x V ? takes the center C4 x V ? isomorphically to itself and hence it*
* is of the form
(x, v) ! (i(x, v)ff(x), fi(x) + fl(v)) for some automorphism ff of P O C4, some*
* homomorphism
fi :P O C4 ! Vv?anishing on C4, and some homomorphism i :P O C4x V ?! C4. Such *
*an auto-
morphism preserves the trace of Ø~#~ iff ~(i(x, v)ff(x)) = ~(fl(v)) for all (x,*
* v) 2 Z(P OC4xV ?) =
C4x V ?. But this means that the induced automorphism of Hd x V ?is of the stat*
*ed form.
5.13. Example. (Oliver's cochain complex [32]) The non-toral objects of A(PGL (*
*2m, C)) of rank
4 are
o One (2, 0) object H, A(PGL (2m, C))(H) = Sp(2) = GL(2, F2), ß0 = H.
o P (m, 2) (3, 1) objects V , A(PGL (2m, C))(V ) = Sp(3, 1),`ß0 = V=V ?or V*
* . '
o P (m, 3)+P (m, 4) (4, 2) objects E, A(PGL (2m, C))(E) = Sp(2)*A(PGL (m0,*
* C))(E? ),
A(PGL (m, C))(E? ) = 1, C2, GL(E), ß0 = E=E? , E=E? or E=L, E=E? or E.
o One (4, 0) object if m is even.
The (2, 0) object H contributes
Hom Sp(2)(St(H), H) ~=F2
The (3, 1) objects V contribute
HomSp(3,1)(St(V ), V ) ~=HomSp(3,1)(St(V ), V=V ?) ~=F2
2-COMPACT GROUPS 21
The (4, 2) objects E with A(PGL (m, C))(E? ) = 1 contribute
Hom0 1(St(E), E=E? ) ~=F22
@ Sp(2) 0A
* 1
and the (4, 2) objects E with A(PGL (m, C))(E? ) = C2 contribute
Hom 0 1(St(E), E=L) ~=Hom0 1(St(E), E=E? ) ~=F2
@Sp(2) 0 A @ Sp(2) 0 A
* C2 * C2
The (4, 0) object (if it exists) and the (4, 2) objects with A(PGL (m, C))(E? )*
* = GL (E) do not
contribute to the cochain complex for the corresponding Hom -groups are trivial*
*. Thus the cochain
complex for computing higher limits of the functor ß1(BZCPGL(2m,C)) will have t*
*he form
1Y ffi2
(5.14)0 ! Hom Sp(2)(St(H), H) ffi-! HomSp(3,1)(St(V ), V=V ?) -!
[m=2]
Y Y
Hom 0 1(St(E), E=E? ) x Hom 0 1(St(E), E=E? ) ! .*
* . .
@Sp(2) 0A @Sp(2) 0 A
* 1 * C2
To show vanishing of the relevant higher limits it suffices to show that ffi1 i*
*s injective and that the
rank of ffi2 is P (m, 2) - 1.
6.N-determinism of the A-family
By inductively applying 3.3 and 4.6 we show that the 2-compact groups PGL (n *
*+ 1, C), n 1,
are uniquely N-determined.
6.1. Lemma. Suppose that n + 1 = 2m 2 is even.
(1)There is a unique monomorphism conjugacy class ~: Z=2 ! PGL (n + 1, C)wit*
*h discon-
nected centralizer. The centralizer of this monomorphism is GL(m, C)2=Cx *
*o Z=2
(2)There is a unique monomorphism conjugacy class :H ! PGL (n + 1, C), H =*
* (Z=2)2,
such that is non-toral. The centralizer of this monomorphism is H x PGL*
* (m, C) and
the Quillen automorphism group is GL(H).
Proof.Use that any monomorphism of Z=2 into PGL (n + 1, C) lifts to ~: Z=2 ! GL*
*(n + 1, C).
The only possibility is that ~ = m . regis a direct sum of regular representati*
*ons. The result for
non-toral rank 2 objects in A(PGL (n + 1, C)) is a special case of 5.10.
6.2. Lemma. Suppose that PGL (r + 1, C) is uniquely N-determined for all 0 r *
*< n. Then
PGL (n + 1, C), n 1, satisfies conditions 4.6.( 1), 4.6.( 2), and 4.6.( 3).
Proof.We shall verify 4.6.(1) and 4.6.(2) by establishing the alternative two c*
*onditions from 4.8.
Let (V, ) be a toral elementary abelian 2-subgroup of PGL (n + 1, C) of rank*
* 2 and C( ) =
CPGL(n+1,C)( ) its centralizer. We have seen that C( ) is LHS (2.20) and that *
*Z~(C( )0) =
~Z(N0(C( ))) as C( )0 does not contain a direct factor isomorphic to GL(2, C)=G*
*L (1, C) = SO(3)
(2.24, 5.7). The identity component C( )0 has ß*(N)-determined automorphisms ac*
*cording to 3.2
and 3.4, and C( ) has N-determined automorphisms by 3.1. The identity componen*
*t C( )0 is
N-determined according to 4.3 and 4.4, and C( ) is N-determined by 4.1. Thus C(*
* ) is LHS and
totally N-determined.
The functor H1(W=W0; ~T0W) is zero on A(PGL (n + 1, C)) t2except on the objec*
*t (V, ) =
(i0, i0, i0, i0), when n + 1 = 4i0, where it has value Z=2. However, this objec*
*t has Quillen auto-
morphism group GL(V ) and since the only GL(V )-equivariant homomorphism St(V )*
* = V ! Z=2
is the trivial homomorphism, lim1(A(PGL (n+1, C)) t2; H1(W=W0; ~T0W)) = 0 follo*
*ws from Oliver's
cochain complex [32].
When n + 1 = 2m is even, we verify condition 4.6.(3) by applying 4.11. Let X0*
* be a connected
2-compact group with maximal torus normalizer j0:N(PGL (n + 1, C) !.XSince the *
*first item
in 4.11 is satisfied by 4.12 and 6.1, it suffices to show that the isomorphism *
*(from 4.6.(3))
f ,L:CPGL(2m,C)(H) = H x PGL(m, C) ! CX0(H, 0)
22 J.M. MØLLER
defined by choosing one of the three lines L in H, is C3-equivariant. Now [24]
Aut(H x PGL(m, C)) = GL(H) x Aut(PGL (m, C))
so that f ,Lis C3-equivariant if ß0f ,Land the restriction of f ,Lto the identi*
*ty components
are C3-equivariant. Here, Aut(PGL (m, C)) = Zx2(or Zx2={ 1} if m = 2) since PGL*
* (m, C) has
ß*(N)-determined automorphisms by induction hypothesis so C3 must act trivially*
* on the identity
components for purely group theoretic reasons. The commutative triangle (4.14)
ß0(H)N
~=mmmmmm NNN~=NN
mmm NNNN
vvmmmm &&N
ß0(CPGL(2m,C)(H, ))___ß_________//ß0(CX0(H, 0))
0(f ,L)
in which the slanted arrows, representing the canonical factorizations, are C3-*
*equivariant (even
GL(H)-equivariant) shows that ß0(f ,L) is C3-equivariant.
We shall next compute the higher limits from 3.3.(2) and 4.6.(4) by means of *
*5.4 and the cochain
complex 5.14 from [32]. As 5.4 is not valid for PGL (2, C) we first consider th*
*is case separately.
6.3. Proposition. The 2-compact group PGL (2, C) is uniquely N-determined.
Proof.The functor CPGL(2,C)takes the Quillen category of PGL (2, C), consisting*
* (5.7, 5.13, 6.1)
of one toral line, L, and one non-toral plane, H,
__________________________________*
*_____________________________
(6.4) L ___________//_HbbGL(H)__________________________*
*________________________________________________________
to the diagram
_____________________________*
*__________________________________
(6.5) GL (1, C)2=GL (1, C) ooC2o_______H bb_GL(H)op___________________*
*______________________________________________________________
of uniquely N-determined 2-compact groups. The 2-compact toral group to the lef*
*t is uniquely
N-determined (4.2) because H1(C2; Z=21 ) = 0 for the non-trivial action. The c*
*enter functor
takes this diagram back to the starting point (6.4) for which the higher limits*
* vanish [29, 12.7.4].
PGL (2, C) is thus uniquely N-determined by 3.3 and 4.6.
6.5. Lemma. The low degree higher limits of the functors ßj(BZCPGL(n+1,C)), j =*
* 1, 2, are:
(1)limi(A(PGL (n + 1, C)), ß1(BZCPGL(n+1,C))) = 0 for i = 1, 2,
(2)limi(A(PGL (n + 1, C)), ß2(BZCPGL(n+1,C))) = 0 for i = 2, 3,
for all n 1.
Let V = F2e1+ F2e2+ F2e3 be a 3-dimensional vector space over F2 with basis {*
*e1, e2, e3} and
(degenerate) symplectic inner product matrix
0 1
0 1 0
@1 0 0A
0 0 0
Let F2[1] be the 21-dimensional F2-vector space on all length one flags [P > L]*
* and F2[0] the 14-
dimensional F2-vector space on all length zero flags, [P ] or [L], of non-trivi*
*al and proper subspaces
of V . The Steinberg module St(V ) over F2 for V is the 23 = 8-dimensional_kern*
*el of the linear
map d: F2[1] ! F2[0]given by d[P > L] = [P_] + [L]. Define f1 = f1| St(V ): St(*
*V )a!sVthe
restriction to St(V ) of the linear map f1:F2[1] ! Vwith values
(
__ L P \ P ?= 0
f1[P > L] =
0 otherwise
on the basis vectors.
Let E = F2e1 + F2e2 + F2e3 + F2e4 be a 4-dimensional vector space over F2 wit*
*h basis
{e1, e2, e3, e4} and (degenerate) symplectic inner product matrix
0 1
0 1 0 0
BB1 0 0 0CC
@ 0 0 0 0A
0 0 0 0
2-COMPACT GROUPS 23
Let F2[2] be the 315-dimensional F2-vector space on all length two flags [V > P*
* > L] and F2[1]
the also 315-dimensional F2-vector space on all length one flags, [P > L] or [V*
* > L] or [V > P ],
of non-trivial, proper subspaces of E. The Steinberg module St(E) over F2 for E*
* is the 26 = 64-
dimensional kernel of the linear_map d: F2[2] ! F2[1]given by d[V > P > L] = [P*
* > L] + [V >
L]_+ [V > P ]. Define F1 = F1| St(E): St(E) !aEs the restriction to St(E) of th*
*e linear map
F1:F2[2] ! Ewith values
(
__ L P \ P ?= 0, V \ V ?= F2e3
(6.6) F1[V > P > L] =
0 otherwise
__
on the basis elements. Define F2 = F2| St(E): St(E) !sEimilarly but`replace'th*
*e condition
V \ V ? = F2e3 by V \ V ? = F2e4. The linear maps F1 and F2 are Sp(2)* 01-equ*
*ivariant
because this group preserves the symplectic inner product on E and preserves V *
*? = F2
pointwise.
6.7. Lemma. Let f1 and F1, F2 be the linear maps defined above.
(1)The vector f1 is a basis vector for
Hom0 1(St(V ), V ) ~=Hom0 1(St(V ), V=V ?) ~=F2
@ Sp(2) 0A @ Sp(2) 0A
* 1 * 1
(2)The set {F1, F2} is a basis for
Hom0 1(St(E), E) ~=Hom0 1(St(E), E=E? ) ~=F22
@ Sp(2) 0A @ Sp(2) 0A
* 1 * 1
The sum F1+ F2 is the linear map defined as in ( 6.6) but with condition *
*V \ V ?= F2e3
replaced by V \ V ?= F2(e3+ e4).
Proof.This can be directly verified by machine computation.
6.8. Proposition. The differentials in the cochain complex 5.14 are given as fo*
*llows:
(1)Let H be the (2, 0) object and V a (3, 1) object of A(PGL (2m, C)). The V*
* -component of
the coboundary map
ffi1V:Hom Sp(2)(St(H), H) ! Hom 0 1(St(V ), V )
@Sp(2) 0A
* 1
is an isomorphism of 1-dimensional F2-vector spaces.
(2)Let V be the (4, 2) object of A(PGL (2m, C)) corresponding ( 5.12, 5.7) t*
*o the two dimen-
sional toral object (1, i - 1, m - i, 0) of A(PGL (m, C)), 1 < i m=2, m*
* 4. Then
ffi2E(xi) = (x1+ xi)F1+ (x1+ xi-1)F2 where
Q
ffi2E: 1 i m=2Hom0 1(St(Vi), Vi) ! Hom 0 1(St(E), E)
@ Sp(2) 0A @Sp(2) 0A
* 1 * 1
Q
is the E-component of the coboundary map and (xi) 2 1 i m=2Hom Sp(3,1)(S*
*t(V ), V ).
Proof.1. The non-zero vector in Hom Sp(2)(St(H), H) is the restriction to St(H)*
* F32of the linear
map F2[0] = F32! H that takes a basis vector [L] in F32to L 2 H. In the composi*
*tion
M M +
St(V ) ! St(P ) ! P -! V
V >P V >P
the middle maps St(P ) ! P equal the map just described if P < V is non-toral, *
*P \ P ?= 0, and
are trivial if P < V is toral, P \ P ?= P . This is precisely the map f1.
2. For any non-toral three dimensional subspace V of E we have either
o V \ V ?= F2e3, and then V = Vi, or,
o V \ V ?= F2e4, and then V = Vi-1, or,
o V \ V ?= F2(e3+ e4), and then V = V1,
24 J.M. MØLLER
and thus the composite linear map
M L xi M +
St(E) ! St(V ) ---! V -! E
E>V
equals xiF1+ xi-1F2+ x1(F1+ F2) = (x1+ xi)F1+ (xi-1+ x1)F2.
Proof of Lemma 6.5.Since we already know that these higher limits vanish when n*
*+1 is odd (5.4)
we can assume that n + 1 = 2m is even.
1. In Oliver's cochain complex 5.14, the coboundary map ffi1 is injective and k*
*erffi2 is 1-dimensional
by 6.8 when m 4. See 6.3 for the case m = 1. For m = 2 and m = 3, the cochain*
* complexes
5.14 reduce to
1
0 ! Hom Sp(2)(St(H),-H)ffi!HomSp(3,1)(St(V ),!V=V0?)
1
0 ! Hom Sp(2)(St(H),-H)ffi!HomSp(3,1)(St(V ),!V=VH?)om0 1(St(E), E=E? *
*) = 0
@Sp(2) 0 A
* GL (E? )
with two non-trivial groups, both 1-dimensional F2-vector spaces, and with just*
* one differential ffi1
which is an isomorphism (6.8). Thus the higher limits vanish in these cases as *
*well.
2. Oliver's cochain complex for computing these higher limits over A(PGL (2m, C*
*)) involve the
Z2-modules
Hom0 1 (St(E), ß2(BZCPGL(2m,C)(E))), dimF2 E = 3, 4,
@ Sp(2) 0 A
* A(PGL (m, C))(E?
that are submodules of finite products of Z2-modules of the form
Hom 0 1(St(E), Z2), dimF2 E = 3, 4,
@Sp(2) 0A
* 1
where the action on Z2 is trivial. According to the computer program magma, the*
*se latter modules
are trivial.
Proof of Theorem 1.1.By induction over n using 3.3 and 4.6. The start of the in*
*duction is provided
by 6.3. Use (2.7) to compute the automorphism group.
Proof of Corollary 1.2.The connected 2-compact group GL (n, C) is uniquely N-de*
*termined be-
cause (3.2, 4.3) its adjoint form PGL (n, C) is (1.1). Since the maximal torus*
* normalizer for
GL(n, C) is a split extension, we get (2.7) that Aut(GL (n, C)) is isomorphic t*
*o Z( n)\ AutZ2 n(Zn2).
This finishes the discussion of the 2-compact groups in the A-family. The rel*
*evance of these are
that they occur as centralizers of elementary abelian subgroups of many other 2*
*-compact groups.
Here is a result illustrating this.
6.9. Theorem. [34, 1.3] The simple 2-compact group G2 is uniquely N-determined *
*and its auto-
morphism group Aut(G2) equals Zx\Zx2x C2.
Proof.The Quillen category A(G2) is equivalent to the category A(GL (V ), V ) o*
*f all non-trivial
subspaces of V = F32[12, 6.1] [10, 1.6] [9, 5.3] and the value of centralizer f*
*unctor BCG2 on the
three isomorphism classes of objects L, P , V is SL(4, R), T o Z=2, V . The ran*
*k one centralizer,
SL(4, R) = SL(2, C)OSL (2, C), is uniquely N-determined (6.3, 3.2, 3.4, 4.3, 4.*
*4). Condition 4.6.(2)
is satisfied because H1(W (X); ~T(X)) = 0 for X = G2, SL(4, R) [13], 4.6.(1) an*
*d 4.6.(3) because
the only rank two object in G2 is toral and its centralizer is a 2-compact tora*
*l group. The functor
ß1(BZCG2) is the identity functor and ß2(BZCG2) the zero functor so the obstruc*
*tion groups
vanish. Now 3.3 and 4.6 show that G2 is uniquely N-determined. The short exact *
*sequence (2.7)
can be used to calculate the automorphism group. We have Aut(G2) = W (G2)\NGL(2*
*,Z2)(W (G2))
as the extension class e(G2) = 0 [3]. Using the description of the root system *
*from [2, VI.4.13]
2-COMPACT GROUPS 25
with short root ff1 = "1- "2 and long root ff2 = 2" - "2- "3 generating the int*
*egral lattice in Z32
one finds that ` '
x ff p___ 0 3
NGL(2,Z2)(W (G2)) = Z2, A, W (G2), A = -3 1 0
and therefore Aut(G2) = Zx2=Zx x C2 where the cyclic group of order two is gene*
*rated by the
exotic automorphism A interchanging the two roots.
7.Miscellaneous
This section contains auxiliary results that are used at various places in th*
*e main argument of
this paper.
7.1. The 2-compact toral groups O(2) and Pin(2). Let H = {a + bj|a, b 2 C}, whe*
*re j2 = -1
and ja = _aj for a 2 C, be the quaternion algebra. The normalizer of Cx in Hx *
*is the Lie
group NHx (Cx) = *