Involutions on the rational K–theory of group rings
of finite groups
Bjørn Jahren
January 8, 2009
Abstract
The ranks of the eigenspaces for the natural (also twisted) involutions on K–theory of group–rings of finite groups is computed. This
is important for the calculation of homotopy groups of automorphism
groups of compact manifolds. The results are based on Borel’s calculation of the real cohomology of arithmetic groups, and (for the
involutions) a careful analysis of the functoriality of these calculations.
1
Introduction, and statement of the main results
The calculation of homotopy groups of automorphism groups of compact
manifolds is well known to reduce (at least in a stable range) to the homotopy
orbit space of a certain involution on the approriate Whitehead space. See
e. g. [4], [6] and most of all the work of Weiss and Williams [13].
The main tool to understand this space is Waldhausen’s A–theory, which
has a compatible involution by [11]. However, in spite of great progress over
the past couple of decades, A–theory is still very difficult to compute, and
the involution seems even more inaccessible, in general.
But even partial results can have interesting consequences. For instance,
the first striking consequence of this theory was the rational calculation by
Farrell–Hsiang, showing that certain homotopy groups of diffeomorphism
groups of large discs and spheres have unexpected infinite order elements [5].
In fact, rationally the problem is reduced to calculating the ±1 eigenspaces
of the involution on K∗ (Z) ⊗ Q, and they achieved this by using Borel’s
calculation of the ranks of these groups.
In this note we observe that similar ideas can be used to compute the
ranks of the eigenspaces of all geometric involutions on Kn (Z[G]) ⊗ Q for
1
finite groups G. By a geometric involutions we here mean one constructed
as follows:
Let ω : G → Z/2 be a homomorphism. (In applications, ω will be the
first Stiefel–Whitney class of a manifold.) The formula
X
X
τ(
ng g) =
ω(g)ng g −1
(1)
defines an anti-involution τ on Z[G], and A 7→ ((τ A)t )−1 then induces an
involution on GL(Z[G]), and hence on K∗ (Z[G]).
homomorphism ω as a twisting of the standard involution
P We think
P of the
−1
ng g 7→
ng g .
Note that even if the involution on Z[G] is trivial, the associated involution on K–theory may be non–trivial. Indeed, Farrell and Hsiang showed
that in the case of the trivial group, the involution on K∗ (Z) ⊗ Q is multiplication by −1. One of our main results is the following generalization in
the case of a trivial twist ω:
Theorem 4.1. The untwisted involution induces multiplication by −1 on
K∗ (Z[G]) ⊗ Q for ∗ > 1.
(In section 4 the result is stated for K∗ (Z[G]) ⊗ R, but this is the same,
since the K–groups are finitely generated.)
The twisted result is more complicated to state. It also requires more
terminology, so we refer to section 5 (Theorem 5.2).
The ranks of the K–groups themselves is of course also interesting, but
a precise, general statement seems to be difficult to find in the literature.
Using Borel’s ideas this is fairly straightforward, and the calculation is included here — also because the methods are needed when we discuss the
involutions. Using a little representation theory, the result can be given a
very nice form, and is given in
Theorem 2.2. Let G be a finite group with r irreducible real representations,
c of them of complex type. For n > 1 we then have


r if n ≡ 1 mod 4
rank Kn (Z[G]) = c if n ≡ 3 mod 4


0 if n is even .
Note that the case n = 1 is different and was computed by Bass in [1].
See, however, the remarks after Theorem 2.2
Theorem 2.2 is proved in section 2, and section 3 disusses certain naturality aspects of the constructions. Naturality is then exploited to study
2
the untwisted involutions in section 4 and the effect of the twisting in section 5. To give a twisting ω is the same as giving an index two subgroup
H ⊂ G, and in section 6 the results of section 5 are reinterpreted in terms
of representations of this subgroup.
2
Ranks of the K-groups
Kn (Z[G]) is finitely generated for every n (e. g. by Suslin [10]), hence the
rank is equal to the dimension of
πn (BGL(Z[G])+ ) ⊗ R ∼
= Prim Hn (GL(Z[G]); R)
∼
= (H n (GL(Z[G]); R)/(decomposables))∗
(2)
∼
= (H n (GLq (Z[G]); R)/(decomposables))∗
for large q.
If G is a finite group, GLq (Z[G]) is an arithmetic subgroup of the linear
algebraic group (defined over Q) whose group of k-points is GLq (k[G]). (See
e. g. [9], 1.2 example (5), and note that GLq (Z[G]) is the multiplicative group
of Mq (Z[G]).) Let us denote this algebraic group by GLG
q .
If Γ is an arithmetic subgroup of a semisimple algebraic group H, Borel
identifies H ∗ (Γ; R)) in a range with the the (Γ, H 0 (R))–invariant forms on
the symmetric space of maximal compact subgroups of H(R). Here H(R)
is the group of real points of H, and H 0 (R) its identiy component. (See
[3, Theorem 7.5]) We would like to apply this result, but GLG
q is not semi–
G
simple, so we need to replace it with some kind of ‘SLq ’.
If the finite group G is commutative, we can use the determinant det :
GLq (k[G]) → k[G] and define SLq (k[G]) = det−1 (1). This is a new algebraic
group defined over Q, with arithmetic subgroup SLq (Z[G]) = GLq (Z[G]) ∩
SLq (Q[G]).
Q
In this case we have k[G]
Q ≈ i ki , where each ki is a finite extension field
of k, and GLq (k[G]) ≈
GLq (ki ). The determinant splits as a product
of determinants GLq (ki ) → ki , so SLq (k[G]) splits as a product of the
SLq (ki )’s. Hence it is semi-simple and also connected, so Borel’s results
apply. In fact, [3, Theorem 11.1] gives isomorphisms H ∗ (SLq (Z[G])) ≈
ISLq (R[G]) below degrees growing to infinity with q, where IH denotes the
graded algebra of H-invariant differential forms on the space of maximal
compact subgroups of H. But R[G] ≈ Rr1 × Cr2 where r1 and r2 are the
numbers of irreducible real representations of real resp. complex type, so
ISLq (R[G]) ≈ ⊗r1 ISLq (R) ⊗r2 ISLq (C) .
3
For a general finite group we have the Wedderburn decomposition
Y
Q[G] ≈
Mni (Di ) ,
(3)
i
where the Di ’s are division algebras over Q, and the factors are matrix
algebras corresponding to the irreducible, rational representations of G. The
center C(Di ) of Di is a finite extension of Q, and the isomorphism in (3)
restricts to an isomorphism
Y
C(Q[G]) ≈
C(Di ) .
(4)
i
It follows that we have a decomposition
Y
GLq (Q[G]) ≈
GLqni (Di ) .
(5)
i
GLqni (Di ) is the group of units in the central simple C(Di )-algebra
Mqni (Di ). But then we can replace the determinant by the reduced norm
defined as follows:
Let E be a splitting field for the division algebra D, i. e. an extension of
C(D) such that E ⊗C(D) D ≈ Md (E). On Md (E) the determinant is defined,
an one shows that if x ∈ D, then det(1 ⊗ x) ∈ C(D), and it is independent
of E. More generally, E will also be a splitting field for the central simple
C(D)-algebra Mn (D), since
E ⊗C(D) Mn (D) ≈ Mn (E ⊗C(D) D) ≈ Mnd (E) .
Again we have det(1 ⊗ A) ∈ C(D) for A ∈ Mn (D) if we identify 1 ⊗ A with
its image in Mn (E) via these isomorphisms.
The reduced norm is the resulting mapping ND : Mn (D) → C(D).
Note that ND is a multiplicative homomorphism, and that ND (cI) = cnd if
c ∈ C(D).
Lemma 2.1. Let A ∈ Mn (D). Then A ∈ GLn (D) if and only if ND (A) 6= 0.
Proof. One implication is an immediate consequence of the multiplicativity.
For the other, let A ∈ Mn (D) be a matrix with ND (A) 6= 0.
Any such A may be written A = ST (use elementary row operations),
where S is a product of elementary matrices and T is a diagonal matrix of
the form diag[1, . . . , a] for some a ∈ D. But then ND (S) = 1 and ND (T ) =
ND (a). Hence ND (a) 6= 0 and therefore also a 6= 0. But then a is invertible
since D is a division algebra. Consequently T is invertible, and since S is,
A will also be invertible.
4
It follows that for division algebras, the reduced norm is a perfectly
good replacement for the determinant. It restricts to a homomorphism ND :
GLn (D) → C(D)∗ and we define
−1
SLn (D) = ker ND = ND
(1) .
Going back to (5) and putting together the NDi ’s for all i, we can define
the map NQ[G] : GLqni (Q[G]) → C(Q[G])∗ such that the diagram
GLqni (Q[G])
NQ[G]
C(Q[G])∗
/
≈
≈
Q
i GLqni (Di )
Q
i N Di
/
∗
i C(Di )
Q
commutes. NQ[G] is again multiplicative, and NQ[G] (I) = 1. (But it is not
homogeneous in the nonabelian case.) We now set
−1
SLq (Q[G]) = ker NQ[G] = NQ[G]
(1) .
G
This defines an algebraic subgroup SLG
q of GLq (defined over Q), with
arithmetic subgroup
SLq (Z[G]) = SLq (Q[G]) ∩ GLq (Z[G]) .
We claim that SLG
q is semisimple:
First observe that in the discussion above, we could have replaced the
ground field Q by any extension F of Q. Thus we have a reduced norm
NF [G] : GLn (F [G]) → C(F [G])∗ for any F , and SLn (F [G]) = ker NF [G] is
the group of F -points of SLG
n.
Q
In particular, take F = C. Then C[G] ≈ i Mni (C) is already split, so
we have
Y
GLq (C[G]) ≈
GLqni (C) ,
i
and NC[G] corresponds to the usual determinant on each factor. Therefore
Y
SLG
SLqni (C) ,
q (C) ≈ SLq (C[G]) ≈
i
which is semisimple.To apply Borel’s results, we analyze SLG
q (R) ≈ SLq (R[G]).
Now the Wedderburn decomposition reads
R[G] ≈
r1
Y
i=1
Mli (R) ×
r2
Y
Mmj (C) ×
j=1
5
r3
Y
k=1
Mnk (H) ,
(6)
where r1 , r2 and r3 are the numbers of real representations of G of real,
complex and quaternionic type, resp. Hence
GLq (R[G]) ≈
r1
Y
GLqli (R) ×
i=1
r2
Y
GLqmj (C) ×
j=1
r3
Y
GLqnk (H) .
(7)
k=1
NR[G] corresponds to the usual determinant on the real and complex factors.
On H it is easy to see that the reduced norm is the square of the usual norm,
and on GLqni (H) it is the determinant after the standard embedding into
GL2qni (C).
SLq (R[G]) ≈
r1
Y
i=1
SLqli (R) ×
r2
Y
SLqmj (C) ×
j=1
r3
Y
SLqnk (H) ,
(8)
k=1
which is a product of connected groups. For the real and complex factors
this is standard. In the quaternionic case, represent a matrix as a product
ST as in the proof of lemma 2.1. Then both factors are clearly in the identity
component, (Note that the set of quaternions of reduced norm equal to one
is precisely the unit sphere in the usual norm on H, and this is connected.)
Now Theorem 11.1 of [3] applies, and we obtain
πn (BSL(Z[G])+ ) ⊗ R ∼
= Prim Hn (SL(Z[G]); R)
n
∼
= (H (SL(Z[G]); R/(decomposables))∗
n
∼
/dec.)∗
= (ISL(R[G])
n
∗
∗
∗
∗
∼
⊗r2 ISL(C)
⊗r3 ISL(H)
/dec.
= ⊗r1 ISL(R)
∗
n
n
n
∼
/dec. ⊕ r2 ISL(C)
/dec. ⊕ r3 ISL(H)
/dec.
= r1 ISL(R)
The summands are computed in [3, 10.6]. We get one R-summand for each
factor for n ≡ 1 mod 4, n > 1, and one for each complex factor for n ≡ 3
mod 4.
It remains to compare πn (BSL(Z[G])+ ) with πn (BGL(Z[G])+ ). Let A =
NQ[G] (GLq (Z[G])) ⊂ C(Q[G]). Then there is a fibration (up to homotopy)
BSL(Z[G]) → BGL(Z[G]) → BA .
∼ A is abelian, so the plus construction on this fibration gives a
But π1 BA =
new fibration
BSL(Z[G])+ → BGL(Z[G])+ → BA
6
(See [2]). Therefore πn (BSL(Z[G])+ ) ≈ πn (BGL(Z[G])+ ) = Kn (Z[G]) for
n ≥ 2.
Thus we have proved:
Theorem 2.2. Let G be a finite group with r irreducible real representations,
c of them of complex type. For n > 1 we have


r if n ≡ 1 mod 4
rank Kn (Z[G]) = c if n ≡ 3 mod 4


0 if n is even .
Remark 2.3. (i) By a theorem of Bass [1], rank K1 (Z[G]) = r − q, where q
is the number of rational irreducible representations of G. In the present
context this result can be interpreted as follows:
Rank π1 (BSL(Z[G])+ ) = 0, so rank K1 (Z[G]) = rank A where A =
N (GL(Z[G])) is the group of units in a lattice in C(Q[G]).
C(Q[G]) is a product of q number fields k1 , . . . , kq , and A is commensurable with the products of the units in the integers Aki of the ki ’s. By the
Dirichlet unit theorem, A∗ki has rank r1i + r2i − 1, where r1i is the number of
real, 2r2i the number of complex embeddings of ki . It also means that the
Wedderburn decomposition of ki ⊗Q R is Rr1i × Cr1i . Therefore
rank A =
q
X
i=1
rank Aki =
q
X
(r1i + r2i − 1) =
i=1
q
X
(r1i + r2i ) − q .
i=1
But now
C(R[G]) ≈ C(Q[G]) ⊗Q R ≈
P
P
Y
(ki ⊗Q R) ≈ R i r1i × C i r2i ,
i
P
which means that qi=1 (r1i + r2i ) is equal to the number of simple components in the Wedderburn decomposition of C(R[G]) (as an algebra over R),
which is equal to the number r of real representations.
(ii) The result of Theorem 2.2 is true more generally for any order in the
semi–simple Q–algebra Q[G].
3
Remarks on functoriality
For the identification of the involutions in the next sections, we need to
study the effect on Borel’s constructions of automorphisms of SLq (Z[G]).
7
Let K be a maximal compact subgroup of SLq (R[G]). Then all maximal
subgroups are conjugate to K, and g 7→ g −1 Kg induces an identification of
the set X of maximal compact subgroups with K\SLq (R[G]). Then Borel’s
theorem is a connectivity result for the homomorphism
SLq (Z[G])
ISLq (R[G]) = H ∗ (ISLq (R[G]) ) → H ∗ (ΩX
SLq (R[G])
induced by the inclusion ISLq (R[G]) = ΩX
) ≈ H ∗ (SLq (Z[G]))
SLq (Z[G])
⊂ ΩX
.
Now let α be an automorphism of SLq (R[G]) which preserves SLq (Z[G]).
Then α takes maximal subgroups to maximal subgroups — hence induces a
self–map αX of X, which can be described as follows:
Let α(K) = γ −1 Kγ, γ ∈ SLq (R[G]). Then
α(g −1 Kg) = (α(g))−1 α(K)α(g) = (γα(g))−1 Kγα(g) ,
which corresponds to the coset Kγα(g) ∈ K\SLq (Z[G]). Therefore we have
αX (Kg) = Kγα(g) .
(9)
If Rh stands for right multiplication by h (on X), we can write αX ◦Rh =
∗ restricts to isomorphisms of both I
Rα(h) ◦ αX . It follows that αX
SLq (R[G])
SLq (Z[G])
and ΩX
, and we have a commutative diagram
≈
SL (Z[G])
−−−−→ H ∗ (SLq (Z[G]))

α∗
y X
SLq (Z[G])
−−−−→ H ∗ (SLq (Z[G]))
ISLq (R[G]) −−−−→ ΩX q


α∗
α∗
y X
y X
ISLq (R[G]) −−−−→ ΩX
≈
In the examples to be studied below, α will also preserve the factors
or pairs of factors in the decomposition (8). The discussion above then
applies to each factor as well, so if we can analyze these, we get the result
for SLq (R[G]) from the isomorphism ISLq (R[G]) ≈ ⊗i ISLqd (Di ) .
i
(Note that the product of maximal compact subgroups of the factors
corresponds to a maximal compact subgroup of SLq (R[G])).
Example 3.1. There are two cases that will be of particular interest to us:
i. If α(K) = K, we may chose γ = 1. Via the isomorphism ISLn (D) =
∗ is induced by the differential of α on
Λ∗ (sln (D)/k)k , we see that αX
the Lie algebra sln (D).
8
ii. If α is conjugation by an element β ∈ SLn (D), we may chose γ =
β in (9), and we have α(Kg) = Kgβ on K\SLn (D). But β may
∗ must be the identity on
be deformed to the identity matrix, so αX
cohomology.
4
Involutions. The untwisted case
Note first that formula (1) defines an involution also on the real group ring
R[G]. We start byQ
examining the effect on the decomposition (6), which we
now write R[G] ≈ ρ Mdρ (Dρ ), where Dρ = R, C or H, and the index ρ runs
over all (isomorphism classes of) real irreducible representations of G. The
factors in this decomposition are uniquely determined summands of R[G],
namely the simple 2-sided ideals in R[G]. (However, the representation of
the factors as matrix algebras obviously depends on choice of basis for the
irreducible representation spaces.) But then τ must respect the decomposition, in the sense that it either leaves a component invariant or interchanges
them in pairs.
Now we restrict to the case ω = 1 (the untwisted or ‘orientable’ case).
The compositions
Y
ρ : G → R[G] ≈
Mdρ (Dρ ) → Mdρ (Dρ )
ρ
are the irreducible representations. We can choose orthogonal bases with
respect to G-invariant inner products in the three settings (R, C, or H).
Then ρ(G) ⊂ Odρ , Udρ , or Spdρ and hence ρ(g)−1 = (ρ(g))t , where ‘bar’
means the usual conjugation (of each matrix entry) in C and H, and the
identity on R. Thus the diagram
⊂
G −−−−→ R[G] −−−−→ Mdρ (Dρ )




τ

−1
g7
→
g
y
y
yA7→(A)t
⊂
G −−−−→ R[G] −−−−→ Mdρ (Dρ )
commutes. Since this is true for all ρ, it follows that (with
suitable choices
Q
of bases) τ on R[G] corresponds to the involution on ρ Mdρ (Dρ ) which is
A 7→ (A)t on each factor (and that in this case all the factors are preserved).
Alternatively, it is easy to see that τ is a positive involution in the sense
of [12, section 7]. Therefore, by the arguments of that paper, it follows that
9
τ preserves each component, and by the uniqueness of positive involutions,
t
t
∗
τ is equivalent to any other —
Q e. g. A 7→ (A) . Let us denote (A) by A .
Passing to SLq (R[G]) ≈ ρ SLqdρ (Dρ ), it is now not hard to see that
the involution here is given by A 7→ (A∗ )−1 on each factor. The effect of this
operation on the space ISLk (D) of invariant forms on the symmetric space
X = K\SLk (D), where K is the maximal compact subgroup SOk , SUk or
Spk , can be analyzed as in section 3. In fact, we are now in the situation of
Example (3.1.i), so we may pass to the induced operation on Lie algebras.
In the two cases D = R and C, slk (D) is the set of k × k matrices with
trace zero. This is also true for D = H, provided that we use the reduced
trace. This is the trace we get after the standard
Mk (H) →
embedding
A −B̄
M2k (C). (A matrix in the image has the form
and then the
B Ā
reduced trace is tr(A + Ā). This is clearly real, and hence slk (H) has real
dimension 4k 2 − 1.)
In all three cases, the involution A 7→ (A∗ )−1 on the group induces the
involution A 7→ −A∗ on the Lie algebra, and this involution fixes precisely
the sub-Lie algebra K of the maximal compact subgroup. Hence it induces
multiplication by −1 on slk (D)/K.
Since ISLk (D) ≈ H ∗ (slk (D), k) ≈ Λ∗ (slk (D)/k)k , it follows that the involution induces multiplication by (−1)j in degree j. But since all the
cohomology lies in odd degrees, we get
Theorem 4.1. The untwisted involution τ induces multiplication by −1 on
K∗ (Z[G]) ⊗ R for ∗ > 1.
Example 4.2. In [6] the following was proved:
Let M 2n+1 be an orientable spherical space form with fundamental group
n
G. For 0 < i < − 4 we then have
3
−
es
πi (Diff M 2n+1 ) ⊗ Q ≈ Ki+2
(Z[G]) ⊕ L
(Z[G])
⊗ Q ⊕ Qi ,
2n+i+3
es
where L
2n+i+3 (Z[G]) are the reduced Wall surgery groups and Qi = Q if
i = 4k − 1 and 0 otherwise.
All such groups G are of course finite, and since M is orientable, Theorem
4.1 gives K − (Z[G]) ⊗ Q = K(Z[G]) ⊗ Q. Hence we get (in the stable range)
the largest possible contribution to the rank of πi (Diff M 2n+1 ) for i odd,
whereas the rank of π2i (Diff M 2n+1 ) is 0 .
Note that rk Ls2k (Z[G]) is r − c if k is even and 2c if k is odd, given by
the multisignature [7].
10
5
Involutions. The twisted case
P
P
Consider now a general involution of the form τω ( ng g) =
ω(g)ng g −1 ,
where ω : G → {±1} is a homomorphism. Then τω = α◦τ
P = τ ◦α,Pwhere τ is
the involution already considered in section 4, and α( ng g) = ω(g)ng g.
(α depends on ω, and if we want to emphasize this, we write αω .) α is a ring
homomorphism satisfying α2 = id. It induces an involution on K∗ (Z[G]),
and given the results of section 4, it then suffices to analyze the cohomology
automorphism induced by α. (Clearly α is defined both on SLq (R[G]) and
SLq (Z[G]).)
First we observe that (like τ ) α either preserves a simple factor, or it
interchanges it with another. Observe also that ω may be thought of as a
(irreducible) one-dimensional representation of G (hence occurs as one of
the factors in the decomposition (6). The composition
α
G → R[G] → Mdρ (Dρ ) −→ α(Mdρ (Dρ ))
takes g ∈ G to ω(g)ρ(g) = (ω ⊗ ρ)(g), thus α(Mdρ (Dρ )) = Mdω⊗ρ (Dω⊗ρ ).
Therefore α preserves Mdρ (Dρ ) if and only if ω ⊗ ρ ∼
= ρ.
Example 5.1. A simple example to keep in mind is when G is the symmetric
group Σ3 , and ω the parity representation. Then R[Σ3 ] ≈ R × R × M2 (R),
and α interchanges the two one-dimensional representations and preserves
the two-dimensional representation.
In the case where α interchanges two factors, we are free to chose coordinates such that on the product of these factors α(A, B) = (B, A). Then
the induced homomorphism just interchanges the corresponding two factors
in the decomposition (8) — hence it will also interchange the contributions
to K-theory from these factors.
The other case is more subtle. Let ρ be an irreducible representation
such that ω ⊗ ρ ≈ ρ, such that we have α : Md (D) → Md (D). By the
Skolem–Noether theorem, the crucial thing is what happens to the center.
In the real or quaternionic case α must restrict to the identity on the center
(= RI), but in the complex case, it could also be complex conjugation. But
then α ◦ c is an automorphism which is the identity on the center, where
c : Md (D) → Md (D) is A 7→ A, i. e. complex conjugation on each matrix
element. (This is an algebra automorphism over R.)
The Skolem–Noether theorem then says that there exists an invertible
matrix B ∈ Md (D) such that α(A) = B −1 AB, or possibly α(A) = B −1 AB
in the complex case. The relevant factor in (8) is SLqd (D), and here
11
the induced map will be A 7→ Bq−1 ABq (or A 7→ Bq−1 ABq ), where Bq =


B
..

.
.
B
If D = C or H, we may actually assume that B ∈ SLd (D), hence also
Bq ∈ SLqd (D). If D = R, we can only assume that det B = ±1, but if q is
even (which we may assume, since we are only interested in stable results),
Bq will still be in SLqd (D). In all cases we then are in the situation of
Example (3.1.ii), and we can conclude that conjugation by Bq induces the
identity on cohomology. This takes care of the real and quaternionic cases,
and “half” of the complex cases.
It only remains to consider the case when α(A) = B −1 AB. By the preceeding discussion such α induces the same homomorphism in cohomology
as α(A) = A, so we assume that α has this simple form.
hen D = C, K = SUn , α(K) = K, and by Example (3.1.i) the homomorphism on ISLn (C) = Λ∗ (sln (C)/sun )sun is also induced by complex
conjugation on matrices.
On sln (C), the canonical splitting sln (C) = sln (R) + i sln (R) is the splitting into the +1 and −1 eigenspaces of complex conjugation. But sln (R)
also has a canonical splitting sln (R) = son +pn , where pn is the space of real,
trace zero symmetric matrices. Thus we have the canonical decomposition
sln (C) = son + pn + i son + i pn .
In this sum sun = son + i pn , so we have
sln (C)/sun = pn + i son ,
which also is the splitting into +1 and −1 eigenspaces.
The computation of Λ∗ (sln (C)/sun )sun now proceeds as follows: First
observe that multiplication by i gives an isomorphism of pn + i son with
son + i pn = sun as sun -modules. Complex conjugation on pn + i son then
corresponds to minus complex conjugation on sun , with son as the −1
eigenspace and i pn as the +1 eigenspace.
Therefore we have
∗
Λ∗ (sln (C)/sun )sun ≈ Λ∗ (sun )sun ≈ HDR
(SUn ) ,
∗ (SU )/dec. has rank
and the computation of this is well known. In fact, HDR
1 in every odd degree > 1, and zero otherwise.
To see what the involution does on the generators, we use the fibration
(up to homotopy) SO → SU → SU/SO and the fact that generators in
12
degrees 4j + 1 come from SU/SO and the generators in degrees 4j + 3
restricts to generators in SO. This is so because the generators for the
indecomposable part in cohomology are dual to homology generators that
come from homotopy groups, and there it follows from the Bott periodicity
computations.
The inclusion of SOn in SUn now induces a homomorphism Λ∗ (sun ) →
∗
Λ (son ), which commutes with involutions if we give son the involution A 7→
−A. This induces multiplication by (−1)m in degree m, hence multiplication
∗ (SO ). Hence
by −1 on the indecomposable generators of H ∗ (son )) ≈ HDR
n
the involution multiplies the indecomposable generators of ISLn (C) in degrees
4j +3 by −1. For the generators in degrees 4j +1, we use a similar argument
with the projection SUn → SUn /SOn , which induces an inclusion
Λ∗ (sun /son )son ≈ Λ∗ (i pn )son ⊂ Λ∗ (sun ) .
(10)
This will clearly commute with involutions if we give sun /son the trivial involution. Since the map in (10) induces the natural homomorphism
∗ (SU /SO ) → H ∗ (SU ) in cohomology, it follows that the indecomHDR
n
n
n
DR
∗ (SU ) — hence also of I
posable generators of HDR
n
SLn (C) in degrees 4k + 1
— are invariant under the involution.
In order to collect all this into a theorem, we introduce some terminology:
Call a representation ρ ω-invariant if ρ ≈ ρ ⊗ ω. Otherwise we call the
unordered pair {ρ, ρ ⊗ ω} ω-noninvariant.
If ρ is ω-invariant and of complex type, we say that ρ is of C-linear
(resp. C-conjugate) type if αω : Md (C) → Md (C) is complex linear (resp.
conjugate linear).
Now we can formulate the results as follows:
Theorem 5.2. Let ω : G → {±1} be a surjective homomorphism, and let
K∗± be the ±1 eigenspace of the associated involution τω on K∗ (Z[G]) ⊗ R.
i. In degrees 4j + 1 > 1 the dimension of K∗+ is equal to the number of
pairs of ω-noninvariant real representations.
ii. In degrees 4j+3 the dimension of K∗+ is equal to the number of pairs as
in (a) of complex type, plus the number of ω-invariant representations
of C-conjugate type.
Proof. From Theorem 4.1 and the formula τω = αω ◦ τ , it follows that K +
is equal to the −1 eigenspace of αω∗ . In degrees 4j + 1 > 1, K∗ (Z[G]) ⊗ R
has one R-summand for each real representation. Those associated to a ωnoninvariant pair are interchanged by α∗ — hence have both +1 and −1
13
eigenspaces of dimension one. The summands associated to the ω-invariant
representations are all, as we have seen, invariant under αω∗ .
In degrees 4j + 3 we only get R-summands associated with the representations of complex type, but now also the ω-invariant representations of
conjugate type give contributions to the −1 eigenspace of αω∗ .
Theorem 4.1 says that K∗+ is trivial in the orientable case. In contrast,
we now have
Corollary 5.3. In the nonorientable case (ω 6= 0), both K∗+ and K∗− are
always nontrivial in degrees 4j + 1 > 1.
Proof. {ε, ω} is a ω-noninvariant pair, where ε is the trivial representation.
Example 5.4. We illustrate these results with two rather simple examples.
i. Take G = Σ3 as in Example (5.1). Then we have three irreducible representations, all of them of real type. Therefore rank K4j+1 = 3, j > 1
and rank Kj = 0 for j 6≡ 1 mod 4. (In fact, all three representations
are rational, so rank K1 = 0.)
From the remarks in example 5.1 we then see that if ω is the parity
representation (which is the nontrivial representation of degree one),
+
we get rank K4j+1
= 1.
ii. For the simplest example with ω 6= 0 and c 6= 0, take G = Z/4 and let
ω be reduction mod 2.
Then R[Z/4] ≈ R × R × C . αω interchanges the two one-dimensional
factors, and it is easy to see that it is conjugate linear on the twodimensional factor. Therefore we read off:
+
– rank K4j+1 = 3, rank K4j+1
= 1. (And again rank K1 = 0.)
+
– rank K4j+3 = rank K4j+3
= 1.
6
A different interpretation of αω and Theorem 5.3
Let ω : G → {±1} be a surjective homomorphism as above, and let H =
ker ω. In this section we shall consider the results in section 5 from the point
of view of the pair (G, H), and obtain a formulation of Theorem 5.2 in terms
of more familiar concepts from group representation theory.
14
The inclusion ι : H ⊂ G gives rise to two maps in K-theory, the induced
map ι∗ : Kn (Z[H]) → Kn (Z[G]) and the transfer ι∗ : Kn (Z[G]) → Kn (Z[H]).
On the category level, these are induced from P 7→ Z[G] ⊗Z[H] P where P
is a projective Z[H]-module (‘induction’), and the forgetful map taking a
Z[G]–module to its underlying Z[H]–module (‘restriction’). Let γ ∈ G be
an element not in H. Then conjugation by γ induces a ring homomorphism
on Z[H] and hence a map c = cγ on K–theory.
Proposition 6.1. The compositions of ι∗ and ι∗ on rational K–groups are
given by
i. ι∗ ◦ ι∗ = id + c
ii. ι∗ ◦ ι∗ = id + αω
Moreover, c = cγ is independent of γ.
Proof. i. follows immediately from a well known fact in group representation
theory. To prove the independence of γ, just observe that if γ 0 is an other
choice, then γ 0 = hγ for some h ∈ H. But then cγ 0 = ch ◦cγ , and conjugation
by an element in H induces the identity on K∗ (Z[H]).
ii. If P is a Z[G]–module, let µ : Z[G] ⊗Z[H] P → P be induced by the
Z[G]–action. If C(P ) = ker µ, there is an exact sequence of functors of P :
0 → C(P ) → Z[G] ⊗Z[H] P → P → 0
preserving the category of projective Z[G]–modules. Let γ ∈
/ H as above,
and consider the map φ : P → C(P ) given by φ(p) = 1 ⊗ p − γ ⊗ γ −1 p. Then
it is an easy exercise to show that
• φ is independent of γ
• φ is an additive isomorphism
• φ(gp) = αω (g)φ(p)
This means that φ is an isomorphism between C(P ) and P with the Z[G]–
structure defined by composition with αω . Then (ii) follows from the additivity theorem in K–theory.
Example 6.2. (G, H) = (S3 , A3 ). Then K4j+3 (Z[H]) ⊗ Q ≈ Q, and c must
be multiplication by −1, since K4j+3 (Z[G]) ⊗ Q = 0.
In view of Example (5.4) this is a spesial case of the following result.
15
Theorem 6.3. The +1 eigenspaces for the involutions αω on Kn (Z[G]) ⊗ Q
and c on Kn (Z[H]) ⊗ Q have the same dimensions.
Proof. Note that αω and c are involutions on K∗ (Z[G]). Hence the homomorphisms 12 (id + αω ) = 12 ι∗ ◦ ι∗ and 21 (id + c) = 12 ι∗ ◦ ι∗ (defined over Q)
are both projection maps. Consequently,
rank (ι∗ ◦ ι∗ ) = rank (ι∗ ◦ ι∗ ◦ ι∗ ◦ ι∗ ) ≤ rank (ι∗ ◦ ι∗ )
Similarly, rank (ι∗ ◦ ι∗ ) ≤ rank (ι∗ ◦ ι∗ ). Thus rank (id + αω ) = rank (id + c),
which is equivalent to the statement of the theorem.
To compute the ±1 eigenspaces of c, we observe that the discussion in
section 5 applies equally well in this case; the only necessary modification
is the replacement of every occurrence of ‘ρ ⊗ ω’ with ‘γργ −1 ’. Call two H–
representations ρ and ρ0 G–equivalent if either ρ ≈ ρ0 or ρ ≈ γρ0 γ −1 . (This
is clearly an equivalence relation, independent of γ.) The fixpoints are the
G–invariant representations. Then we have a new formulation of Theorem
5.2, which in many cases is even simpler to use:
Theorem 6.4. Let ω : G → {±1} be a surjective homomorphism with kernel
H, and let K∗± (Z[G]) be the ±1 eigenspace of the associated involution τω
on K∗ (Z[G]) ⊗ R. Then
i. In degrees 4j +1 > 1 the dimension of K∗− (Z[G]) is equal to the number
of G–equivalence classes of real representations of H.
ii. In degrees 4j + 3 the dimension of K∗− (Z[G]) is equal to the number
of G–equivalence classes of real representations of H of complex type,
minus the number of G–invariant representations of C-conjugate type.
Proof. By Theorems 6.3 and 4.1 we need to compute the dimension of the
+1 eigenspace of c on K∗ (Z[H]). The argument is similar to the proof of
Theorem 5.2.
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