CONGRUENCE SUBGROUPS AND TWISTED COHOMOLOGY OF SL_{n}(F [t])

KEVIN P. KNUDSON

In [7] we showed that if F is an infinite field, then the natural inclusion SL_{n}(F) → SL_{n}(F [t]) induces an isomorphism

H_{•}(SL_{n}(F), ℤ) ⟶ H_{•}(SL_{n}(F [t]), ℤ)

for all n ≥ 2. Here, we study the extent to which this isomorphism holds when the trivial ℤ coefficients are replaced by some rational representation of SL_{n}(F). The group SL_{n}(F [t]) acts on such a representation via the map

SL_{n}(F [t]) ⟶^{t=0} SL_{n}(F).

There are two approaches one might take. The first is to use the spectral sequence associated to the action of SL_{n}(F [t]) on a certain Bruhat–Tits building X. With this method, we obtain the following result. Suppose that F is a field of characteristic zero. Let V be an irreducible representation of SL_{n}(F). Theorem.(cf. Theorem 5.2) The group H^{1}(SL_{n}(F [t]), V) satisfies

H^{1}(SL_{n}(F [t]), V) = \begin{cases}H^{1}(SL_{n}(F), V) & V ≠ Ad \\ H^{1}(SL_{n}(F), V) ⊕ F^{∞} & V = Ad, n = 2 \\ H^{1}(SL_{n}(F), V) ⊕ F & V = Ad, n ≥ 3. \end{cases}

One expects that a similar result holds in positive characteristic as well. However, the author does not pretend to be an expert in representation theory, especially in positive characteristic; therefore, we restrict our attention to the characteristic zero case.

The homology of linear groups with twisted coefficients has been studied by Dwyer [2], van der Kallen [5], and others. Most known results concern what happens to these homology groups as n becomes large. In contrast, our results cover the unstable case.

The second approach to studying this question is to use the Hochschild– Serre spectral sequence associated to the extension

1 ⟶ K ⟶ SL_{n}(F [t]) ⟶^{t=0} SL_{n}(F) ⟶ 1

Date: October 7, 1997.

1991 Mathematics Subject Classification. 20G10.

Partially supported by an NSF Postdoctoral Fellowship.

where K denotes the subgroup of matrices which are congruent to the identity modulo t. A spectral sequence calculation shows that for any field F,

H^{1}(SL_{n}(F [t]), V) = H^{1}(SL_{n}(F), V) ⊕ Hom_{SL_{n}(F)}(H_{1}(K), V)

so that one need only compute the group H_{1}(K) explicitly. This seems to be rather difficult in general. When n = 2, we have a free product decomposition

K = ∗_{s∈SL_{2}(F)/B}sCs^{−1}

where C is the upper triangular subgroup of K and B is the upper triangular subgroup of SL_{2}(F). Hence we have

H_{1}(K) = ⨁_{s∈SL_{2}(F)/B} H_{1}(sCs^{−1})

and since C is an abelian group, this is easily calculated.

For n ≥ 3, however, we have no such free product decomposition. In fact, the natural map H_{1}(C) → H_{1}(K) is no longer injective (consider the matrix I + E_{12}(t^{2}) ∈ C; this gives a nontrivial element of H_{1}(C) which vanishes in H_{1}(K)). The group H_{1}(K) appears to be rather complicated in general. For any field F, H_{1}(K) surjects onto the adjoint representation 𝔰𝔩_{n}(F), and the kernel is nontrivial in general (see Section 7 below). We make the following conjecture.

Conjecture. If n ≥ 3, then H_{1}(K) = 𝔰𝔩_{n}(F) for any finite field F.

One might prove this by finding a fundamental domain for the action of K on the Bruhat–Tits building X and then utilizing the corresponding spectral sequence. We take the first steps toward this here. Unfortunately, the combinatorics involved are rather complicated. Still, we are able to prove the conjecture for n = 3, F = 𝔽_{2}, 𝔽_{3}.

The study of congruence subgroups has a long history, one which we will not try to reproduce here. The standard question one asks is whether a given normal subgroup of SL_{n}(R) is a congruence subgroup. In particular, one could ask this question for the subgroups of K which form its lower central series. While these groups are close to being congruence subgroups in a certain sense (see Section 7), they actually are not.

This paper consists of two parts. The first part deals with the characteristic zero case and is organized as follows. In Section 1, we prove the above theorem in the case n = 2 by considering the long exact sequence associated to Nagao’s amalgamated free product decomposition of SL_{2}(F [t]). In Section 2, we describe the spectral sequence used in the proof of Theorem 5.2. In Section 3, we reprove the theorem in the case n = 2 using the spectral sequence of Section 2. This helps guide the way to the general result. In Section 4, we describe some equivariant homomorphisms. In Section 5, we prove Theorem 5.2.

TWISTED COHOMOLOGY OF SL_{n}(F [t]) 3

The second part deals with the congruence subgroup K and its abelianization H_{1}(K). In Section 6, we consider the maximal algebraic quotient of H_{1}(K) in the characteristic zero case. In Section 7 we describe a descending central series in K. In Section 8 we consider the action of K on the building X and the resulting spectral sequence. In Section 9 we find a simpler subcomplex of X which suffices to compute H_{1}(K). Finally, in Section 10 we prove the above conjecture in certain cases.

This paper has its origins in discussions with Dick Hain, who wondered about the structure of H_{1}(K) and also whether unstable homotopy invariance holds with nontrivial coefficients. Also, I should point out that this work owes a great deal to C. Soulé’s paper [12]. I thank Eric Friedlander for many useful discussions. I am also grateful to the Mathematisches Forschungsinstitut Oberwolfach and l’Institut de Recherche Mathématique Avancée, Strasbourg (in particular to Jean-Louis Loday), for their hospitality during a visit in June 1996 when a preliminary version of this paper was written. Finally, I thank the referee who saved me from some embarrassing slips and pointed me to Krusemeyer’s paper [8].

Notation. In Sections 1 through 6, F is assumed to be of characteristic zero; thereafter, F is allowed to be any field unless otherwise specified. If G is a group acting on a set X, we denote the invariants of the G–action by X^{G}. The symbol P typically denotes a parabolic subgroup of SL_{n}(F) which contains the upper triangular subgroup B. We denote by Φ^{+} the set of positive roots of SL_{n}(F) determined by B; this set consists of roots {α_{ij} : i < j}. We denote by E_{ij}(a) the matrix having i, j entry a and zeros elsewhere. For a polynomial p(t), p^{(k)} denotes the coefficient of t^{k}. If R is a ring, we denote by R^{×} the group of units of R.

1. The SL_{2} Case, Part I

We single out the case n = 2 because we may use Nagao’s Theorem [10]

SL_{2}(F [t]) = SL_{2}(F) ∗_{B} B_{t}  (1)

to study H^{1}(SL_{2}(F [t]), V) (here, B_{t} denotes the group of upper triangular matrices over F [t]). This amalgamated free product decomposition yields a long exact sequence

⋯ → H^{k}(SL_{2}(F [t]), V) → H^{k}(SL_{2}(F), V)⊕H^{k}(B_{t}, V) → H^{k}(B, V) → ⋯

for computing the cohomology of SL_{2}(F [t]).

Proposition 1.1. We have a short exact sequence

0 → H^{1}(SL_{2}(F [t]), V) → H^{1}(SL_{2}(F), V) ⊕ H^{1}(B_{t}, V) → H^{1}(B, V) → 0. (2)

Proof. Since the map B_{t} ⟶^{t=0} B is split by the natural inclusion B ⟶ B_{t}, the induced map H^{k}(B_{t}, V) ⟶ H^{k}(B, V) is surjective.

We now study the relationship between H^{1}(B_{t}, V) and H^{1}(B, V). Consider the extension

1 ⟶ C ⟶ B_{t} ⟶^{t=0} B ⟶ 1  (3)

where

C = \{( \begin{matrix}1 & tp(t) \\ 0 & 1 \end{matrix} ) : p(t) ∈ F [t]\}.

The Hochschild–Serre spectral sequence associated to this satisfies

E_{2}^{p,q} = H^{p}(B, H^{q}(C, V)) ⟹ H^{p+q}(B_{t}, V).  (4)

Since the extension (3) is split, the map d_{2}^{0,1} : E_{2}^{0,1} → E_{2}^{2,0} vanishes. It follows that we have a short exact sequence

0 → H^{1}(B, H^{0}(C, V)) → H^{1}(B_{t}, V) → H^{0}(B, H^{1}(C, V)) → 0. (5)

Since C acts trivially on V, H^{0}(C, V) = V. Hence, the first term in (5) is simply H^{1}(B, V). Observe that H^{0}(B, H^{1}(C, V)) = Hom_{B}(H_{1}(C), V).

Proposition 1.2. The group Hom_{B}(H_{1}(C), V) satisfies

Hom_{B}(H_{1}(C), V) = \begin{cases}Hom_{F} (tF [t], F) & V = Ad \\ 0 & V ≠ Ad. \end{cases}

Proof. We may compute this group as follows. We have a split extension

1 ⟶ U ⟶ B ⟶ T ⟶ 1

where U is the subgroup of upper triangular unipotent matrices and T is the diagonal subgroup. The Hochschild–Serre spectral sequence implies that

H^{0}(B, H^{1}(C, V)) = H^{0}(T, H^{0}(U, H^{1}(C, V)));

that is,

Hom_{B}(H_{1}(C), V) = H^{0}(T, Hom_{U}(H_{1}(C), V)).

Observe that H_{1}(C) = C ≅ tF [t] and U acts trivially on C. It follows that

Hom_{U}(H_{1}(C), V) = Hom(C, V^{U}).

Now, V = Sym^{n}S, where S is the standard representation of SL_{2}(F) and n is some nonnegative integer (see e.g. [3, Ch. 11]). It follows that

V^{U} = V_{n}

where V_{n} denotes the highest weight space of V. Observe that T acts on C with weight 2 and on V_{n} with weight n. Note also that

H^{0}(T, Hom(C, V_{n})) = Hom_{T} (C, V_{n}).

We need the following result.

Lemma 1.3. Suppose f : C ⟶ V^{U} is a T -equivariant group homomorphism. Then f is F -linear.

Proof. We have V = Sym^{n}S and V^{U} = V_{n}. If a ∈ ℤ, denote by t_{a} the element of T with diagonal entries a, \frac{1}{a}. If w ∈ C, we have t_{a}.w = a^{2}w. Thus

a^{n}f(w) = t_{a}.f(w) = f(t_{a}.w) = f(a^{2}w) = a^{2}f(w)

(the last equality follows since f is ℤ-linear). It follows that f = 0 unless n = 2.

Consider the case n = 2 (i.e., V = Ad). Let α ∈ F and consider the element t_{α} = diag(α, \frac{1}{α}). Since f is T -equivariant, we have

α^{2}f(w) = t_{α}.f(w) = f(t_{α}.w) = f(α^{2}w)

for any w ∈ C. Note that since F has characteristic zero and f is ℤlinear, f is in fact ℚ-linear. Note also that for any α ∈ F, we have α = −\frac{1}{2} − \frac{1}{2}α^{2} + \frac{1}{2}(1 + α)^{2}. Thus,

f(αw) = f((−\frac{1}{2} − \frac{1}{2}α^{2} + \frac{1}{2}(1 + α)^{2})w)
= −\frac{1}{2}f(w) − \frac{1}{2}α^{2}f(w) + \frac{1}{2}(1 + α)^{2}f(w)
= αf(w).

Returning to the proof of Proposition 1.2, the proof of Lemma 1.3 shows that Hom_{T} (C, V_{n}) = 0 unless n = 2. It follows that

Hom_{B}(C, V) = \begin{cases}0 & V ≠ Ad \\ Hom_{F} (tF [t], F) & V = Ad. \end{cases}

Corollary 1.4. The group H^{1}(B_{t}, V) satisfies

H^{1}(B_{t}, V) = \begin{cases}H^{1}(B, V) & V ≠ Ad \\ H^{1}(B, V) ⊕ Hom_{F} (tF [t], F) & V = Ad. \end{cases}

Proof. This follows from the short exact sequence (5) and the fact that H^{1}(B, V) is a direct summand of H^{1}(B_{t}, V).

Theorem 1.5. The group H^{1}(SL_{2}(F [t]), V) satisfies

H^{1}(SL_{2}(F [t]), V) = \begin{cases}H^{1}(SL_{2}(F), V) & V ≠ Ad \\ H^{1}(SL_{2}(F), V) ⊕ Hom_{F} (tF [t], F) & V = Ad. \end{cases}

Proof. Consider the short exact sequence (2). By the corollary, the kernel of the map

H^{1}(SL_{2}(F), V) ⊕ H^{1}(B_{t}, V) ⟶ H^{1}(B, V)

is isomorphic to

\begin{matrix}H^{1}(SL_{2}(F), V) & V ≠ Ad \\ H^{1}(SL_{2}(F), V) ⊕ Hom_{F} (tF [t], F) & V = Ad. \end{matrix}

The direct sum decomposition follows because H^{1}(SL_{2}(F), V) is a direct summand of H^{1}(SL_{2}(F [t]), V).

We have seen that H^{1}(SL_{2}(F [t]), Ad) differs from H^{1}(SL_{2}(F), Ad) by the infinite dimensional F -vector space Hom_{B}(H_{1}(C), Ad). We describe an explicit basis for this space. For each k ≥ 1, denote by φ_{k} the map C → 𝔰𝔩_{2}(F) defined by

I + E_{12}(a_{12}^{(1)}t + a_{12}^{(2)}t^{2} + ⋯ + a_{12}^{(m)}t^{m}) ↦ E_{12}(a_{12}^{(k)}).

Then the set {φ_{k}} is a basis of Hom_{B}(H_{1}(C), Ad).

2. A Spectral Sequence

In general, we shall use the action of SL_{n}(F [t]) on a suitable simplicial complex to compute H^{1}(SL_{n}(F [t]), V).

Denote by X the Bruhat–Tits building associated to the vector space F (t)^{n}. Recall that the vertices of X are equivalence classes of O-lattices in F (t)^{n} (here, O consists of the set of a/b with deg b ≥ deg a), where two lattices L and L′ are equivalent if there is an x ∈ F^{×} with L′ = xL. A collection of vertices Λ_{0}, Λ_{1}, …, Λ_{m} forms an m-simplex if there are representatives L_{i} of the Λ_{i} with

t^{−1}L_{0} ⊂ L_{m} ⊂ L_{m−1} ⊂ ⋯ ⊂ L_{0}.

It is possible to put a metric on X so that each edge in X has length one. When we speak of the distance between vertices we implicitly use this metric. For a more complete description of X, see, for example, [7].

The group SL_{n}(F [t]) acts on X with fundamental domain an infinite wedge T , which is the subcomplex of X spanned by the vertices

[e_{1}t^{r_{1}}, e_{2}t^{r_{2}}, …, e_{n−1}t^{r_{n−1}}, e_{n}], r_{1} ≥ r_{2} ≥ ⋯ r_{n−1} ≥ 0

where e_{1}, e_{2}, …, e_{n} denotes the standard basis of F (t)^{n} (this is due to Serre [11] for n = 2 and Soulé [12] for n ≥ 3). See Figure 1 for the case n = 3.

Denote by v_{0} the vertex [e_{1}, …, e_{n}] and by v_{i} the vertex

[e_{1}t, e_{2}t, …, e_{i}t, e_{i+1}, …, e_{n}], i = 1, 2, …, n − 1.

For a k element subset I = {i_{1}, …, i_{k}} of {1, 2, …, n − 1}, define E_{I}^{(k)} to be the subcomplex of T which is the union of all rays with origin v_{0} passing through the (k − 1)–simplex ⟨v_{i_{1}}, …, v_{i_{k}}⟩. There are \binom{n−1}{k} such E_{I}^{(k)}. Observe that if I = {1, 2, …, n − 1}, then E_{I}^{(n−1)} = T . When we write E_{J}^{(l)}, the superscript l denotes the cardinality of the set J.

The structure of the various simplex stabilizers was described in [7]. If (x_{ij}(t)) ∈ SL_{n}(F [t]) stabilizes the vertex [e_{1}t^{r_{1}}, … e_{n−1}t^{r_{n−1}}, e_{n}], then we have

deg x_{ij}(t) ≤ r_{i} − r_{j}

(set r_{n} = 0). Note that since r_{1} ≥ r_{2} ≥ ⋯ ≥ r_{n−1}, some of the x_{ij}(t) with i > j may be 0. Denote the stabilizer of σ by Γ_{σ}. The group Γ_{σ} is the intersection of the stabilizers Γ_{v} where v ranges over the vertices of σ. In this case deg x_{ij}(t) ≤ min_{v∈σ}{r_{i}^{(v)} − r_{j}^{(v)}}. Observe that in any case, the group Γ_{σ} has a block form where the blocks along the diagonal are matrices with

Figure 1. The Fundamental Domain T for n = 3

entries in F, blocks below are zero, and blocks above contain polynomials of bounded degree. In the case n = 3, we have the following block forms:

\begin{aligned}v_{0} & : & Γ_{v_{0}} = SL_{3}(F) \\ σ ∈ E_{\{1\}}^{(1)} & : & Γ_{σ} = ( \begin{matrix}∗ & ∗ & ∗ \\ 0 & ∗ & ∗ \\ 0 & ∗ & ∗ \end{matrix} ) \end{aligned}

\begin{aligned}σ ∈ E_{\{2\}}^{(1)} & : & Γ_{σ} = ( \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \\ 0 & 0 & ∗ \end{matrix} ) \\ σ ∈ T − (E_{\{1\}}^{(1)} ∪ E_{\{2\}}^{(1)}) & : & Γ_{σ} = ( \begin{matrix}∗ & ∗ & ∗ \\ 0 & ∗ & ∗ \\ 0 & 0 & ∗ \end{matrix} ) \end{aligned}

We have a short exact sequence

1 ⟶ C_{σ} ⟶ Γ_{σ} ⟶^{t=0} P_{σ} ⟶ 1

where P_{σ} is a parabolic subgroup of SL_{n}(F). From the above description of Γ_{σ}, we see that the group C_{σ} has a block form where blocks along the diagonal are identity matrices, blocks below are zero, and blocks above contain polynomials of bounded degree which are divisible by t.

Filter the complex T by setting W^{(0)} = v_{0} and

W^{(l)} = ⋃_{|I|=l} E_{I}^{(l)}, 1 ≤ l ≤ n − 1

Observe that if σ and τ are simplices in the same component of W^{(i)}−W^{(i−1)}, then P_{σ} = P_{τ} since on such a component the relationships among the r_{i} defining the vertices do not vary from vertex to vertex (i.e., if r_{i} > r_{i+1} for one vertex in the component, then the same holds for every vertex in the component; since these relations determine which entries below the diagonal are zero, we see that the stabilizers of these vertices all have the same block form and hence so does the stabilizer of any simplex in the component).

The action of SL_{n}(F [t]) on X gives rise to a spectral sequence converging to H^{•}(SL_{n}(F [t]), V) with E_{1}–term

E_{1}^{p,q} = ∏_{dim σ=p} H^{q}(Γ_{σ}, V)  (6)

where Γ_{σ} denotes the stabilizer of the simplex σ ∈ T . We shall compute the terms E_{2}^{p,0} and E_{2}^{0,1} and thereby obtain the group H^{1}(SL_{n}(F [t]), V).

Proposition 2.1. The bottom row of the spectral sequence (6) satisfies

E_{2}^{p,0} = 0, p > 0.

Proof. We have a coefficient system H^{0} on T given by

H^{0}(σ) = H^{0}(Γ_{σ}, V).

Thus, the bottom row of (6) is simply the cochain complex C^{•}(T , H^{0}). The Hochschild–Serre spectral sequence associated to the extension Γ_{σ} → P_{σ} implies that

H^{0}(Γ_{σ}, V) = H^{0}(P_{σ}, V).

It follows that on each component of W^{(i)}−W^{(i−1)}, the system H^{0} is constant (see the remarks above). An easy modification of Lemma 5 of [13] shows that the inclusion v_{0} ⟶ T induces an isomorphism

H^{•}(T , H^{0}) ⟶^{≅} H^{•}(v_{0}, H^{0})

(see [7] for the corresponding statement for homology). It follows that

E_{2}^{p,0} = \begin{cases}H^{0}(SL_{n}(F), V) & p = 0 \\ 0 & p > 0. \end{cases}

It remains to compute the term

E_{2}^{0,1} = ker{d_{1} : E_{1}^{0,1} ⟶ E_{1}^{1,1}}

in the spectral sequence (6). That is, we must compute the group H^{0}(T , H^{1}), where H^{1} is the system which assigns the group H^{1}(Γ_{σ}, V) to σ.

3. The SL_{2} Case, Part II

Before computing the group H^{0}(T , H^{1}) in general, we first reconsider the case n = 2. This will help guide the way to the desired result. To this end, consider the fundamental domain T . In this case, T is an infinite path in the tree X. Label the vertices v_{0}, v_{1}, … and the edges e_{0}, e_{1}, . . .. Then we have

Γ_{v_{0}} = SL_{2}(F)
Γ_{v_{i}} = Γ_{e_{i}} = \{( \begin{matrix}a & p(t) \\ 0 & 1/a \end{matrix} ) : deg p(t) ≤ i\}, i ≥ 1.

For each i ≥ 1 we have a split extension

1 ⟶ C_{i} ⟶ Γ_{v_{i}} ⟶^{t=0} B ⟶ 1.

Arguing as in the proof of Proposition 1.2 and again using Lemma 1.3 we see that

H^{1}(Γ_{v_{i}}, V) = \begin{cases}H^{1}(B, V) & V ≠ Ad \\ H^{1}(B, V) ⊕ Hom_{B}(C_{i}, V) & V = Ad. \end{cases}

Theorem 3.1. The group H^{0}(T , H^{1}) satisfies

H^{0}(T , H^{1}) = \begin{cases}H^{1}(SL_{2}(F), V) & V ≠ Ad \\ H^{1}(SL_{2}(F), V) ⊕ Hom_{B}(C, V) & V = Ad. \end{cases}

Proof. If V is not the adjoint representation, we use the long exact sequence of the pair (T , v_{0}). Since H^{1}(Γ_{v_{i}}, V) = H^{1}(B, V) for each i ≥ 1, we see that H^{•}(T , v_{0}; H^{1}) = 0. It follows that

H^{0}(T , H^{1}) = H^{1}(SL_{2}(F), V).

In the case V = Ad, we have

C^{•}(T , H^{1}) = C^{•}(T , H^{1}(P, V)) ⊕ C^{•}(T , Hom(C, V))

where H^{1}(P, V) is the system

σ ↦ H^{1}(P_{σ}, V)

and Hom(C, V) is the system

σ ↦ Hom_{P_{σ}}(C_{σ}, V).

Again, we see that H^{0}(T , H^{1}(P, V)) = H^{1}(SL_{2}(F), V). It remains to compute H^{0}(T , Hom(C, V)). For each i, we have

Hom_{B}(C_{i}, V) = Hom_{T} ({a_{12}^{(1)}t + ⋯ + a_{12}^{(i)}t^{i}}, V)
= F {φ_{j}}_{j=1}^{i}

where φ_{j}(a_{12}^{(1)}t+⋯+a_{12}^{(i)}t^{i}) = E_{12}(a_{12}^{(j)}) (this follows from Lemma 1.3). Note that C_{v_{0}} = C_{e_{0}} = 1; i.e. Hom(C_{v_{0}}, V) = 0. It follows that

H^{0}(T , Hom(C, V)) = H^{0}(T′, Hom(C, V)),

where T′ is the complex obtained from deleting v_{0} and e_{0} (but not v_{1}) from T .

Consider the map d : C^{0}(T′, Hom(C, V)) → C^{1}(T′, Hom(C, V)). This map is given by

(x_{1}, x_{2}, x_{3}, …) ↦ (x_{2}|_{e_{1}} − x_{1}, x_{3}|_{e_{2}} − x_{2}, …)

where x_{k}|_{e_{k−1}} denotes the image of x_{k} under the restriction map

Hom_{B}(C_{v_{k}}, V) → Hom_{B}(C_{e_{k−1}}, V).

Note that d is F -linear since it is an alternating sum of restriction maps, each of which is clearly F -linear. The kernel of this map is spanned by the following elements:

(φ_{1}, φ_{1}, φ_{1}, …), (0, φ_{2}, φ_{2}, φ_{2}, …),
(0, 0, φ_{3}, φ_{3}, φ_{3}, …), …

If we identify the kth element of this list with the map φ_{k} : C → V, we see that

H^{0}(T′, Hom(C, V)) = Hom_{B}(C, V).

It follows that

H^{1}(SL_{2}(F [t]), Ad) = H^{1}(SL_{2}(F), Ad) ⊕ Hom_{B}(C, Ad)

which is the desired result.

4. P_{σ}–equivariant Homomorphisms

To compute the group H^{0}(T , H^{1}) in general, we will need to consider the groups Hom_{P_{σ}}(H_{1}(C_{σ}), V), where the groups H_{1}(C_{σ}) are more complicated than in the case n = 2. We begin by describing the structure of the H_{1}(C_{σ}).

Recall that the elements (x_{ij}(t)) of the subgroup C_{σ} have a block form where the blocks on the diagonal are identity matrices, blocks below are zero, and blocks above contain polynomials of bounded degree which are divisible by t.

Using elementary row operations, we may write for any (x_{ij}(t)) ∈ C_{σ},

(x_{ij}(t)) = ∏_{j=0}^{n−2} ∏_{i=1}^{n−j−1}(I + E_{i,n−j}(x_{i,n−j}(t))).  (7)

We also have

I + E_{ij}(x_{ij}(t)) = ∏_{l=1}^{m}(I + E_{ij}(x_{ij}^{(l)}t^{l})).  (8)

Lemma 4.1. If there exist distinct integers i, j, k and positive integers l, m with l + m = r such that I + E_{ik}(t^{l}), I + E_{kj}(t^{m}) ∈ C_{σ}, then the element I + E_{ij}(t^{r}) is zero in H_{1}(C_{σ}).

Proof. This follows from standard relations among commutators of elementary matrices.

We are now in a position to describe explicitly the structure of the groups H_{1}(C_{σ}).

Proposition 4.2. The group H_{1}(C_{σ}) admits a decomposition

H_{1}(C_{σ}) = ⨁_{α∈Φ^{+}} S_{α}

where S_{α} is a weight space (possibly 0) for the positive root α. Moreover, each S_{α_{ij}} is graded by powers of t:

S_{α_{ij}} = t^{l_{1}(α_{ij})}W_{α_{ij}} ⊕ t^{l_{2}(α_{ij})}W_{α_{ij}} ⊕ ⋯ ⊕ t^{l_{p}(α_{ij})}W_{α_{ij}}

where 0 ≤ l_{1}(α_{ij}) < l_{2}(α_{ij}) < ⋯ < l_{p}(α_{ij}) ≤ min_{v∈σ}{r_{i}^{(v)} − r_{j}^{(v)}} and W_{α_{ij}} is a

one-dimensional root space for α_{ij}. If S_{α_{ij}} ≠ 0, then l_{1}(α_{ij}) = 1.

Proof. Observe that H_{1}(C_{σ}) is a P_{σ}–module and as such admits a decomposition into weight spaces for the action of the diagonal subgroup T. In light of (7) and (8), we see that each element of H_{1}(C_{σ}) may be written as a product

∏_{i<j} I + E_{ij}(x_{ij}(t))

which implies the existence of the decomposition

H_{1}(C_{σ}) = ⨁_{α∈Φ^{+}} S_{α}

(the only weights which can occur are positive roots since C_{σ} is upper triangular and T acts on I + E_{ij}(x) with weight α_{ij}). The space S_{α_{ij}} is spanned by all elements of the form

I + E_{ij}(t^{r})

where 1 ≤ r ≤ min_{v∈σ}{r_{i}^{(v)} − r_{j}^{(v)}}. If min_{v∈σ}{r_{i}^{(v)} − r_{j}^{(v)}} = 0, then S_{α_{ij}} = 0. By

Lemma 4.1, if there is a k distinct from i and j with I +E_{ik}(t^{l}), I +E_{kj}(t^{m}) ∈ C_{σ} (l + m = r), then I + E_{ij}(t^{r}) = 0 in H_{1}(C_{σ}). Notice that the conditions of Lemma 4.1 cannot be met for r = 1 (since l, m ≥ 1) so that if S_{α_{ij}} ≠ 0 (which can only occur if min_{v∈σ}{r_{i}^{(v)} − r_{j}^{(v)}} ≥ 1), then I + E_{ij}(t) ∈ S_{α_{ij}}. It follows that

S_{α_{ij}} = tW_{α_{ij}} ⊕ t^{l_{2}(α_{ij})}W_{α_{ij}} ⊕ ⋯ ⊕ t^{l_{p}(α_{ij})}W_{α_{ij}}

where 1 < l_{2} < ⋯ < l_{p} ≤ min_{v∈σ}{r_{i}^{(v)} − r_{j}^{(v)}}.

Note that it is possible for powers of t to get skipped. For example, let n = 3 and consider the following group:

C_{σ} = ( \begin{matrix}1 & deg ≤ 1 & deg ≤ 3 \\ 0 & 1 & deg ≤ 1 \\ 0 & 0 & 1 \end{matrix} ) .

This is the C_{σ} of some edge in T . In H_{1}(C_{σ}), the element

( \begin{matrix}1 & 0 & t^{3} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

is nonzero, but the element

( \begin{matrix}1 & 0 & t^{2} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

is trivial. (In this case, H_{1}(C_{σ}) = tW_{α_{12}} ⊕ tW_{α_{23}} ⊕ tW_{α_{13}} ⊕ t^{3}W_{α_{13}}.) This problem arises because for each i, j we are only allowed polynomials of degree at most min_{v∈σ}{r_{i}^{(v)} − r_{j}^{(v)}}. This may allow certain elements to survive in H_{1}.

However, if v is a vertex, this phenomenon does not occur for the following reason. For each i, j, we have deg x_{ij}(t) ≤ r_{i} − r_{j}. Consider the elements I + E_{ij}(t^{r}), 1 ≤ r ≤ r_{i} − r_{j}, in C_{v}. Suppose there exists a k such that C_{v} contains nonzero elements of the form I +E_{ik}(p(t)), I +E_{kj}(q(t)). Note that since C_{v} is upper triangular, such a k satisfies i < k < j. Then C_{v} contains all the I + E_{ik}(t^{l}), I + E_{kj}(t^{m}), 1 ≤ l ≤ r_{i} − r_{k}, 1 ≤ m ≤ r_{k} − r_{j}. Since r_{i} ≥ r_{k} ≥ r_{j}, it follows that it is always possible to satisfy the conditions of Lemma 4.1 for each 1 < r ≤ r_{i} − r_{j}; i.e., S_{α_{ij}} = tW_{α_{ij}}. Note that such a pair i, j must satisfy j > i + 1 since each C_{v} is upper triangular.

We now determine the structure of the groups Hom_{P_{σ}}(H_{1}(C_{σ}), V).

Lemma 4.3. Suppose n ≥ 3 and f : H_{1}(C_{σ}) ⟶ V is a P_{σ}-equivariant group homomorphism. Then f(W_{α_{ij}}) ⊆ V_{α_{ij}}.

Proof. Write V = ⨁_{λ} V_{λ} and choose a basis {v_{λ}}_{λ} of V. Recall that each λ is a linear combination λ = ∑_{p=1}^{n} m_{p}(λ)L_{i} where L_{1}, …, L_{n} are the weights of the standard representation. Recall also that ∑_{i=1}^{n} L_{i} = 0. Fix i < j. For k ≠ i denote by t_{k} the element

t_{k} = diag(1, …, 1, b, 1, …, 1, 1/b, 1, …, 1)

where b is an integer greater than 1, b appears in the ith place and 1/b appears in the kth place. Let w ∈ W_{α_{ij}} and write f(w) = ∑_{λ} a_{λ}v_{λ}. Note that if k ≠ j, t_{k}.w = bw and t_{j}.w = b^{2}w. Now if k ≠ j,

t_{k}. ∑_{λ} a_{λ}v_{λ} = t_{k}.f(w)
= f(t_{k}.w)
= f(bw)
= bf(w)  (f additive)
= ∑_{λ} ba_{λ}v_{λ}

and similarly,

t_{j}. ∑_{λ} a_{λ}v_{λ} = ∑_{λ} b^{2}a_{λ}v_{λ}.

Now, if λ = ∑_{p=1}^{n} m_{p}(λ)L_{p}, then

t_{k}.v_{λ} = b^{m_{i}(λ)−m_{k}(λ)}v_{λ}

for all k ≠ i. Thus,

(b − b^{m_{i}(λ)−m_{k}(λ)})a_{λ} = 0, k ≠ j

and

(b^{2} − b^{m_{i}(λ)−m_{j}(λ)})a_{λ} = 0.

So, if some m_{i}(λ) − m_{k}(λ) ≠ 1 or m_{i}(λ) − m_{j}(λ) ≠ 2, we see that a_{λ} = 0 (since F has characteristic zero).

Suppose that for k ≠ j, m_{i}(λ) − m_{k}(λ) = 1 and m_{i}(λ) − m_{j}(λ) = 2. Then

λ = (m_{i}(λ) − 1)L_{1} + ⋯ + m_{i}(λ)L_{i} + ⋯ + (m_{i}(λ) − 2)L_{j} + ⋯
⋯ + (m_{i}(λ) − 1)L_{n}
= m_{i}(λ) ∑_{p=1}^{n} L_{p} − ∑_{p≠i} L_{p} − L_{j}

= m_{i}(λ) · 0 + L_{i} − ∑_{p=1}^{n} L_{p} − L_{j}
= 0 + L_{i} − 0 − L_{j}
= α_{ij}.

Thus, a_{λ} = 0 except for λ = α_{ij}; i.e., f(w) = a_{α_{ij}}v_{α_{ij}}.

Corollary 4.4. If n ≥ 3 and f : H_{1}(C_{σ}) ⟶ V is a P_{σ}-equivariant group homomorphism, then f is F -linear.

Proof. Let w ∈ W_{α_{ij}} and let α ∈ F. Let

t_{α} = diag(1, …, 1, α, 1, …, 1, 1/α, 1, …, 1)

where α appears in the ith position and 1/α appears in the kth position, where k ≠ j. Then t_{α}.w = αw. Hence,

t.f(w) = f(t.w) = f(αw).

Since f(W_{α_{ij}}) ⊆ V_{α_{ij}}, t.f(w) = αf(w). Thus, f(αw) = αf(w) for all α ∈ F. Since each element of H_{1}(C_{σ}) is an F -linear combination of elements of the various W_{α_{ij}}, we see that f is F -linear.

The preceding result allows us to assume that the homomorphisms under consideration are F -linear.

Proposition 4.5. If V is not the adjoint representation, then

Hom_{P_{σ}}(H_{1}(C_{σ}), V) = 0.

Proof. Suppose f is a nonzero element in Hom_{P_{σ}}(H_{1}(C_{σ}), V). Since V is irreducible, the image of f contains a highest weight vector whose weight γ must be a weight of H_{1}(C_{σ}), hence a root. On the other hand, γ must be a highest weight of V, so we must have V = Ad.

This argument also shows that if V = Ad, then the restriction map Hom_{P_{σ}}(H_{1}(C_{σ}), V) → Hom_{F} (H_{1}(C_{σ})_{α_{1n}}, V_{α_{1n}}) is injective. Observe that each map H_{1}(C_{σ})_{α_{1n}} → V_{α_{1n}} is a linear combination of maps of the form

I + E_{1n}(a_{1n}^{(1)}t + a_{1n}^{(2)}t^{2} + ⋯ + a_{1n}^{(l)}t^{l}) ↦ E_{1n}(a_{1n}^{(k)}).

It follows that any f in Hom_{P_{σ}}(H_{1}(C_{σ}), V) is a linear combination of maps φ_{k} : H_{1}(C_{σ}) → V defined by

φ_{k} : I + tX_{1} + ⋯ + t^{l}X_{l} ↦ X_{k}.

Let us consider what the proposition says about Hom_{B}(H_{1}(C), Ad). If v is a vertex in T − W^{(n−2)} (i.e., v is in the interior of T ), then

H_{1}(C_{v}) = ⨁_{i} W_{α_{i,i+1}}^{n_{α_{i,i+1}}} ⊕ ⨁_{j>i+1} tW_{α_{ij}}

where n_{α_{i,i+1}} ≥ 1. In this case, P_{v} = B and Hom_{P_{v}}(H_{1}(C_{v}), Ad) injects into Hom_{F} (H_{1}(C_{v})_{α_{1n}}, Ad_{α_{1n}}) = F (we assume that n ≥ 3). Hence, up to scalars any nonzero map f : H_{1}(C_{v}) → Ad is of the form

f(I + tX_{1} + t^{2}X_{2} + ⋯ + t^{k}X_{k}) = X_{1}.

Note that when n = 2 the situation is much different since W_{α_{12}}^{n_{α_{12}}} is such that n_{α_{12}} can be greater than 1.

Thus we arrive at the following description of Hom_{B}(H_{1}(C), Ad).

Corollary 4.6. The group Hom_{B}(H_{1}(C), Ad) satisfies

Hom_{B}(H_{1}(C), Ad) = \begin{cases}F \{φ_{k}\}_{k=1}^{∞} & n = 2 \\ F \{φ_{1}\} & n ≥ 3 \end{cases}

where φ_{k} is the map H_{1}(C) ⟶ Ad defined by

φ_{k}(I + tX_{1} + t^{2}X_{2} + ⋯ + t^{l}X_{l}) = X_{k}.
5. The Group H^{1}(SL_{n}(F [t]), V), n ≥ 3

Having dispensed with the case n = 2, we now turn our attention to the groups SL_{n}(F [t]) for n ≥ 3.

We begin by computing the groups H^{1}(Γ_{σ}, V), where σ is a simplex in T . We have a split short exact sequence

1 ⟶ C_{σ} ⟶ Γ_{σ} ⟶^{t=0} P_{σ} ⟶ 1.

By considering the Hochschild–Serre spectral sequence , we see that we have

H^{1}(Γ_{σ}, V) = H^{1}(P_{σ}, V) ⊕ Hom_{P_{σ}}(H_{1}(C_{σ}), V).

In light of Proposition 4.5, we have the following result.

Proposition 5.1. If V is not the adjoint representation, then

H^{1}(SL_{n}(F [t]), V) = H^{1}(SL_{n}(F), V).

Proof. One checks easily in this case that H^{0}(T , H^{1}) = H^{1}(SL_{n}(F), V) by considering the filtration W^{•} of T .

Now, if V is the adjoint representation we have already seen that the situation can be much different.

Theorem 5.2. If V is the adjoint representation, then

H^{1}(SL_{n}(F [t]), V) = H^{1}(SL_{n}(F), V) ⊕ Hom_{B}(H_{1}(C), V).

Proof. As before, one checks that

H^{0}(T , H^{1}) = H^{1}(SL_{n}(F), V) ⊕ H^{0}(T , Hom(C, V)).

It remains to compute the kernel of the map

d : C^{0}(T , Hom(C, V)) ⟶ C^{1}(T , Hom(C, V)).

Suppose that v is a vertex in T . The discussion following the proof of Proposition 4.5 shows that the group Hom_{P_{v}}(H_{1}(C_{v}), V) has basis the maps φ_{k} where

φ_{k}(I + tX_{1} + ⋯ + t^{l}X_{l}) = X_{k}

and k ≤ max{r_{i} − r_{j}} (here v = [e_{1}t^{r_{1}}, e_{2}t^{r_{2}}, …, e_{n−1}t^{r_{n−1}}, e_{n}]). We will show that only the map φ_{1} corresponds to an element in the kernel of d. Note that d is F -linear since it is a linear combination of restriction maps, each of which is F -linear.

Suppose that k ≥ 2 and consider the map φ_{k}. This map belongs to infinitely many Hom_{P_{v}}(H_{1}(C_{v}), V). However, given a vertex v, the map φ_{k} can only belong to Hom_{P_{v}}(H_{1}(C_{v}), V) if r_{i} = r_{i+1} for some i (i.e., v cannot lie in T −W^{(n−2)}). In this case, there exists a vertex v′ adjacent to v (where r_{i} > r_{i+1}) with φ_{k} ∉ Hom_{P_{v′}} (H_{1}(C_{v′}), V) and such that C_{e} = C_{v} ∩C_{v′} = C_{v}. The map d : C^{0}(T , Hom(C, V)) ⟶ C^{1}(T , Hom(C, V)) is then

d : (φ_{k})_{v} ↦ (φ_{k})_{e} + ∑_{e′}(φ_{k})_{e′}

where the sum is over the edges e′ incident with v. However, since φ_{k} ∉ Hom_{P_{v′}} (H_{1}(C_{v′}), V), the e component of d(φ_{k}) cannot be killed by a corresponding element of Hom_{P_{v′}} (H_{1}(C_{v′}), V), i.e., d(φ_{k}) ≠ 0.

However, the map φ_{1} does belong to each Hom_{P_{v}}(H_{1}(C_{v}), V) and to each Hom_{P_{e}}(H_{1}(C_{e}), V). Moreover, given an edge e with vertices v and v′, the restriction maps

Hom_{P_{v}}(H_{1}(C_{v}), V) ⟶ Hom_{P_{e}}(H_{1}(C_{e}), V)

and

Hom_{P_{v′}} (H_{1}(C_{v′}), V) ⟶ Hom_{P_{e}}(H_{1}(C_{e}), V)

map φ_{1} to the same element (with opposite sign). It follows that the kernel of d : C^{0} ⟶ C^{1} is spanned by the element x = (x_{v})_{v∈T} defined by

x_{v} = φ_{1} ∈ Hom_{P_{v}}(H_{1}(C_{v}), V).

If we identify this map with the map φ_{1} ∈ Hom_{B}(H_{1}(C), V), then we see that

H^{0}(T , Hom(C, V)) ≅ Hom_{B}(H_{1}(C), V).

This completes the proof of Theorem 5.2.

6. A Second Spectral Sequence

There is another spectral sequence which can be used to compute the group H^{1}(SL_{n}(F [t]), V); namely, the Hochschild–Serre spectral sequence associated to the split extension

1 ⟶ K ⟶ SL_{n}(F [t]) ⟶^{t=0} SL_{n}(F) ⟶ 1

where K denotes the subgroup of SL_{n}(F [t]) consisting of those matrices which are congruent to the identity modulo t. Since K acts trivially on V and the extension splits, we see that the map

d_{2}^{0,1} : E_{2}^{0,1} ⟶ E_{2}^{2,0}

vanishes and hence,

H^{1}(SL_{n}(F [t]), V) = E_{2}^{1,0} ⊕ E_{2}^{0,1}
= H^{1}(SL_{n}(F), V) ⊕ H^{0}(SL_{n}(F), H^{1}(K, V))
= H^{1}(SL_{n}(F), V) ⊕ Hom_{SL_{n}(F)}(H_{1}(K), V).

The results of the preceding section now imply the following result.

Corollary 6.1. If V is not the adjoint representation, then

Hom_{SL_{n}(F)}(H_{1}(K), V) = 0;

that is, there are no SL_{n}(F)-equivariant maps H_{1}(K) → V. If V is the adjoint representation, then

Hom_{SL_{n}(F)}(H_{1}(K), V) = Hom_{B}(H_{1}(C), V).

In other words, SL_{n}(F)-equivariant maps H_{1}(K) → V are in one-to-one correspondence with B-equivariant maps H_{1}(C) → V. Moreover, if we define H_{1}^{alg}(K) to be the maximal algebraic quotient of H_{1}(K) (i.e. the product of those SL_{n}(F) representations which admit an equivariant homomorphism H_{1}(K) → V) then we have

H_{1}^{alg}(K) = Ad.

The conclusion of Corollary 6.1 is clear in the case n = 2 since we have the free product decomposition (see e.g. [6])

K = ∗_{s∈ℙ^{1}(F)}sCs^{−1} (ℙ^{1}(F) = SL_{2}(F)/B)

which implies that

H_{1}(K) = Ind_{B}^{SL_{2}(F)}H_{1}(C).

By Shapiro’s Lemma, we have

H^{0}(B, H^{1}(C, V)) = H^{0}(SL_{2}(F), Coind_{B}^{SL_{2}(F)}H^{1}(C, V));

that is,

Hom_{B}(H_{1}(C), V) = ( ∏_{SL_{2}(F)/B} H^{1}(C, V))^{SL_{2}(F)}
= Hom_{SL_{2}(F)}( ⨁_{SL_{2}(F)/B} H_{1}(C), V)
= Hom_{SL_{2}(F)}(H_{1}(K), V).

Note that this argument works for any field F.

Proposition 6.2. If F is any field, then

H^{1}(SL_{2}(F [t], V) = H^{1}(SL_{2}(F), V) ⊕ Hom_{B}(H_{1}(C), V).

7. Central Series

In this section, we allow F to be any field of any characteristic.

For each i ≥ 1, denote by K^{i} the subgroup of K consisting of matrices congruent to the identity modulo t^{i}. Each K^{i} is a normal subgroup of SL_{n}(F [t]) as it is the kernel of the map

SL_{n}(F [t]) ⟶ SL_{n}(F [t]/(t^{i})).

Lemma 7.1. For each i, j, [K^{i}, K^{j}] ⊆ K^{i+j}.

Proof. If X = I + t^{i}X_{i} + ⋯ + t^{l}X_{l} and Y = I + t^{j}Y_{j} + ⋯ t^{m}Y_{m}, then

XY X^{−1}Y^{−1} = (I + t^{i}X_{i} + ⋯)(I + t^{j}Y_{j} + ⋯)
(I − t^{i}X_{i} + ⋯)(I − t^{j}Y_{j} + ⋯)
= (I + t^{i}X_{i} + t^{j}Y_{j} + ⋯)(I − t^{i}X_{i} − t^{j}Y_{j} + ⋯)
= I + t^{i+j}(−X_{i}Y_{j} − Y_{j}X_{i}) + ⋯
∈ K^{i+j}.

For each i define a map ρ_{i} : K^{i} ⟶ 𝔰𝔩_{n}(F) by

ρ_{i}(I + t^{i}X_{i} + ⋯ + t^{l}X_{l}) = X_{i}.

One checks easily that ρ_{i} is a surjective homomorphism (see Lee and Szczarba [9] for the ℤ case) with kernel K^{i+1}. Hence for each i ≥ 1, we have an isomorphism

K^{i}/K^{i+1} ≅ 𝔰𝔩_{n}(F).

For any group G, denote by Γ^{•} the lower central series of G.

Lemma 7.2. For each i, we have Γ^{i} ⊆ K^{i}.

Proof. Since K^{i}/K^{i+1} is abelian for each i, K^{•} is a central series in K and as such contains the lower central series.

Given a group G and a central series G = G^{1} ⊇ G^{2} ⊇ ⋯ we have the associated graded Lie algebra

Gr^{•}G = ⨁_{i≥1} G^{i}/G^{i+1}.

Lemma 7.3. The graded algebra Gr^{•}G is generated by Gr^{1}G if and only if G^{l} = G^{l+1}Γ^{l} for each l ≥ 1.

Proof. Note that Gr_{Γ}^{•}G = ⨁_{i≥1} Γ^{i}/Γ^{i+1} is generated by Gr_{Γ}^{1}G = Γ^{1}/Γ^{2} and that we have a surjective map

Γ^{1}/Γ^{2} ⟶ G^{1}/G^{2}.

For each l we have an exact sequence

0 → (Γ^{l} ∩ G^{l+1})/Γ^{l+1} → Γ^{l}/Γ^{l+1} → G^{l}/G^{l+1} → G^{l}/G^{l+1}Γ^{l} → 0.

We also have a commutative diagram

(Γ^{1}/Γ^{2})^{⊗l} ⟶ (G^{1}/G^{2})^{⊗l}
↓  ↓^{φ}
Γ^{l}/Γ^{l+1} ⟶^{ψ} G^{l}/G^{l+1}

Suppose that Gr^{•}G is generated by Gr^{1}G. Then φ is surjective. Commutativity of the diagram forces ψ to be surjective (since the top horizontal and left vertical maps are surjective). But then the exact sequence

Γ^{l}/Γ^{l+1} ⟶^{ψ} G^{l}/G^{l+1} ⟶ G^{l}/G^{l+1}Γ^{l} ⟶ 0

implies that G^{l}/G^{l+1}Γ^{l} = 0; that is, G^{l} = G^{l+1}Γ^{l}.

Conversely, if G^{l} = G^{l+1}Γ^{l} for each l, then ψ is surjective. But then φ is also surjective; that is, Gr^{•}G is generated by Gr^{1}G.

Now consider the algebra Gr^{•}K = ⨁_{i≥1} K^{i}/K^{i+1}. For each i, Gr^{i}K = 𝔰𝔩_{n}(F). Since 𝔰𝔩_{n}(F) = [𝔰𝔩_{n}(F), 𝔰𝔩_{n}(F)] (unless n = 2, char F = 2), Gr^{•}K is generated by Gr^{1}K. By the lemma, we have K^{l} = K^{l+1}Γ^{l} for each l. Note that this implies that K^{l} = K^{l+m}Γ^{l} for each m ≥ 1. Since ⋂_{m} K^{m} = {I}, we see that the filtrations K^{•} and Γ^{•} are close together in a certain sense. However, we will see that in general K^{l} ≠ Γ^{l}.

Remark. The fact that K^{l} = K^{l+m}Γ^{l} implies that in the completed group K̂ = lim_{←} K/K^{i}, we have K̂^{l} = Γ̂^{l} for each l.

Consider the short exact sequence

0 ⟶ K^{2}/Γ^{2} ⟶ H_{1}(K) ⟶^{ρ_{1}_{∗}} 𝔰𝔩_{n}(F) ⟶ 0.

The group K^{2} is often denoted by SL_{n}(F [t], (t^{2})) and we have

Γ^{2} = [SL_{n}(F [t], (t)), SL_{n}(F [t], (t))].

Lemma 7.4. If F is a field of characteristic not equal to 2 or 3 or if F is perfect, then Γ^{2} ⊂ E(F [t], (t^{2})).

Proof. According to Vaserstein [14], the stable commutator subgroup

[GL(R, I), GL(R, J)]

is generated by elements [E_{rs}(a), E_{sr}(b)] with (a, b) ∈ (I × J) ∪ (R × IJ) along with the E_{rs}(x) with x ∈ IJ (the latter elements are clearly in Γ^{2} and in E(F [t], (t^{2}))). The corresponding Mennicke symbol is \left[{a^{2}b \atop 1−ab}\right]. We show that these symbols vanish in SK_{1}(F [t], (t^{2})).

To do this, we make use of computations in Krusemeyer [8, Lemma 12.3]. The hypotheses on F are needed to show the relations \left[{t^{2} \atop 1−t^{2}k}\right] = 1 and

\left[{t^{3} \atop 1−t^{2}k}\right] = 1 for k ∈ F [t]. The hardest case is where (a, b) ∈ (t) × (t). Write a = tf, b = tg, g = g_{0} + g_{1}t + t^{2}h, where g_{i} ∈ F and f, h ∈ F [t]. Then

\left[{a^{2}b \atop 1 − ab}\right] = \left[{t^{3}f^{2}g \atop 1 − t^{2}fg}\right] = \left[{t^{3}f^{2}g \atop 1 − t^{2}fg}\right] \left[{t^{2} \atop 1 − t^{2}fg}\right]

= \left[{t^{5}f^{2}g \atop 1 − t^{2}fg}\right] = \left[{t^{3}f \atop 1 − t^{2}fg}\right] \left[{fg \atop 1 − t^{2}fg}\right]

= \left[{t^{3}f \atop 1 − t^{2}fg}\right] = \left[{t^{5}f \atop 1 − t^{2}fg}\right] = \left[{t^{2}f \atop 1 − t^{2}fg}\right]

= \left[{t^{2}f \atop 1 − t^{3}fg_{1}}\right] = \left[{t^{3}f \atop 1 − t^{3}fg_{1}}\right] = 1.

Similar computations show that \left[{a^{2}b \atop 1−ab}\right] = 1 for (a, b) ∈ F [t] × (t^{2}). This completes the proof.

The lemma implies that we have a surjective map

K^{2}/Γ^{2} ⟶ SK_{1}(F [t], (t^{2})).

The latter group was computed by Krusemeyer [8]; it equals the module of differentials Ω_{F}^{1} , if char F ≠ 2, 3 or if F is perfect. Observe that this is nonzero in general.

Proposition 7.5. Suppose char F ≠ 2, 3 or F is perfect. If Ω_{F}^{1} ≠ 0, then K^{l} ≠ Γ^{l} for all l ≥ 2.

Proof. Since K^{2}/Γ^{2} surjects onto Ω_{F}^{1} , we have K^{2} ≠ Γ^{2}. Suppose K^{l} = Γ^{l} for some l. Then K^{2} = K^{l}Γ^{2} = Γ^{l}Γ^{2} ⊆ Γ^{2}, a contradiction.

If Ω_{F}^{1} = 0, then we get no information about the structure of K^{2}/Γ^{2}. Note that if F is a finite field, then Ω_{F}^{1} = 0.

8. The Action of K on X

We now turn our attention to the computation of the group H_{1}(K) in the case where F is a finite field. While much of what we say below is true for any field, we restrict our attention to the finite case.

In [7, Prop. 4.1], we found a fundamental domain for the action of K on X. Denote by S a set of coset representatives for SL_{n}(F)/B, where B is the upper triangular subgroup. Define a subcomplex Z of X by

Z = ⋃_{s∈S} sT .

Then Z is a fundamental domain for the action of K on X.

If σ is a simplex in Z, denote by K_{σ} the stabilizer of σ in K. Since X is contractible, we have a spectral sequence converging to H_{•}(K) with E^{1}–term

E_{p,q}^{1} = ⨁_{dim σ=p} H_{q}(K_{σ})  (9)

where σ ranges over the simplices of Z.

Observe that the bottom row E_{∗,0}^{1} is simply the simplicial chain complex S_{•}(Z). Note that Z is contractible since the index set S is finite (F is a finite field) and each s ∈ S fixes the initial vertex v_{0}. Moreover, sT ∩ s′T is at most a codimension one face. It follows that the straight line contracting homotopies for each sT can be glued together to obtain a contracting homotopy of Z. It follows that

E_{p,0}^{2} = \begin{cases}ℤ & p = 0 \\ 0 & p > 0. \end{cases}

Thus, to compute H_{1}(K) we need only compute the group E_{0,1}^{2}.

9. Reduction

For each q define a coefficient system H_{q} on Z by setting H_{q}(σ) = H_{q}(K_{σ}). Then the qth row of the spectral sequence (9) is simply the chain complex C_{•}(Z, H_{q}), and the group E_{p,q}^{2} is the group H_{p}(Z, H_{q}).

In the previous section, we showed that H_{1}(K) = E_{0,1}^{2}. Thus we have

H_{1}(K) = H_{0}(Z, H_{1}).

To compute the latter group, we first define a sequence of subcomplexes of Z.

Denote by Z^{(1)} the 1-skeleton of Z. Clearly, H_{0}(Z, H_{1}) = H_{0}(Z^{(1)}, H_{1}). For each i ≥ 1, let Z_{i} be the subgraph of Z^{(1)} spanned by the vertices having distance at most i from v_{0}. Then

Z^{(1)} = ⋃_{i≥1} Z_{i}

and

H_{0}(Z^{(1)}, H_{1}) = \underrightarrow{lim} H_{0}(Z_{i}, H_{1}).

tW_{12}⊕
t^{2}W_{12}⊕
tW_{32}⊕
t^{2}W_{32}
↗  ↑  ↖
tW_{12}⊕  tW_{12}  tW_{12}⊕
tW_{32}⊕  ⊕  t^{2}W_{12}⊕
t^{2}W_{32}  tW_{32}  tW_{32}
↙  ↓  ↘
tW_{12}⊕  tW_{12}  tW_{12}⊕
tW_{31}⊕  ⊕  tW_{13}⊕
tW_{32}  tW_{32}  tW_{32}
Figure 2. The Vanishing of H_{0}(Z_{i+1}, Z_{i}; H_{1})

Proposition 9.1. The inclusion Z_{1} ⟶ Z^{(1)} induces a surjection

H_{0}(Z_{1}, H_{1}) ⟶ H_{0}(Z^{(1)}, H_{1}).

Proof. Consider the long exact sequence of the pair (Z_{i+1}, Z_{i}) for i ≥ 1:

H_{1}(Z_{i+1}, Z_{i}; H_{1}) → H_{0}(Z_{i}, H_{1}) → H_{0}(Z_{i+1}, H_{1}) → H_{0}(Z_{i+1}, Z_{i}; H_{1}) → 0.

Let v be a vertex in Z_{i+1}. Then there exists an s ∈ SL_{n}(F)/B and a vertex v′ ∈ T such that v = sv′. It follows that K_{v} = sK_{v′}s^{−1} and that H_{1}(K_{v}) admits a decomposition

H_{1}(K_{v}) = tW_{1} ⊕ t^{2}W_{2} ⊕ ⋯ ⊕ t^{i+1}W_{i+1}

where each W_{k} is an F –vector space spanned by certain elements of 𝔰𝔩_{n}(F) (this follows from the decomposition of H_{1}(K_{v′}) = H_{1}(C_{v′}) described in Section 4).

Let v ∈ Z_{i+1} − Z_{i}. Then there exist v_{1}, …, v_{l} ∈ Z_{i} adjacent to v with K_{v_{j}} ⊂ K_{v} for each j. If e_{j} denotes the edge connecting v_{j} with v, then K_{e_{j}} = K_{v_{j}}. Now, the boundary map

∂ : C_{1}(Z_{i+1}, Z_{i}; H_{1}) → C_{0}(Z_{i+1}, Z_{i}; H_{1})

maps ⨁_{j=1}^{l} H_{1}(K_{e_{j}}) surjectively onto the summand tW_{1} ⊕ ⋯ ⊕ t^{i}W_{i} of

H_{1}(K_{v}) = tW_{1} ⊕ ⋯ ⊕ t^{i}W_{i} ⊕ t^{i+1}W_{i+1}.

Figure 2 shows an example consisting of the vertices v = [t^{2}e_{1}, e_{2}, t^{2}e_{3}], v′ = [t^{2}e_{1}, e_{2}, te_{3}], v′′ = [te_{1}, e_{2}, t^{2}e_{3}], v_{1} = [te_{1}, e_{2}, te_{3}], where the top group is H_{1}(C_{v}), the three middle groups are H_{1}(C_{v′′,v}), H_{1}(C_{v_{1},v}), H_{1}(C_{v′,v}), and the bottom groups are H_{1}(C_{v′′}), H_{1}(C_{v_{1}}), H_{1}(C_{v′}). The space W_{ij} is a onedimensional weight space for the root α_{ij}. The vertices v, v′, v′′ lie in Z_{2} and v_{1} lies in Z_{1}.

It follows that the group H_{0}(Z_{i+1}, Z_{i}; H_{1}) can consist only of classes arising from the various t^{i+1}W_{i+1}. Moreover, such a class t^{i+1}w_{i+1} can arise only from vertices of the form v = s˙[e_{1}t^{r_{1}}, …, e_{n−1}t^{r_{n−1}}, e_{n}] = sv′ where r_{k} = r_{k+1} for some k = 1, 2, … n−1 (set r_{n} = 0). (Recall that those vertices for which the r_{k} are positive and distinct satisfy H_{1} = tW_{1} ⊕ ⋯ ⊕ t^{i}W_{i}— see the discussion following Proposition 4.2.) However, v is adjacent to a vertex w in Z_{i+1} − Z_{i} for which r_{k} > r_{k+1} and hence the corresponding class t^{i+1}w_{i+1} is trivial in H_{1}(K_{w}). Also, the class t^{i+1}w_{i+1} is nontrivial in H_{1}(K_{e}) where e is the edge joining v and w. It follows that the boundary map hits the class t^{i+1}w_{i+1} in H_{1}(K_{v}); that is, H_{0}(Z_{i+1}, Z_{i}; H_{1}) = 0. Figure 2 illustrates this for the classes t^{2}w_{12}, t^{2}w_{32} ∈ H_{1}(C_{v}). The class t^{2}w_{12} is hit by the class t^{2}w_{12} ∈ H_{1}(C_{v′′,v}) (notice that this class is trivial in H_{1}(C_{v′′})). Similarly, the class t^{2}w_{32} is hit by t^{2}w_{32} ∈ H_{1}(C_{v′,v}). Since H_{0}(Z^{(1)}, H_{1}) = \underrightarrow{lim} H_{0}(Z_{i}, H_{1}) and each map H_{0}(Z_{i}, H_{1}) → H_{0}(Z_{i+1}, H_{1}) is surjective, we have a surjection H_{0}(Z_{1}, H_{1}) → H_{0}(Z^{(1)}, H_{1}).

Remark. It is crucial that n ≥ 3 in the above proof. When n = 2, each vertex v_{i+1} in Z_{i+1} − Z_{i} is adjacent to a single vertex v_{i} in Z_{i}. The stabilizer C_{e_{i}} of the edge joining v_{i} to v_{i+1} maps surjectively onto the part of C_{v_{i+1}} consisting of polynomials of degree less than i + 1. However, v_{i+1} is not adjacent to any vertex in Z_{i+1} − Z_{i}. Hence,

H_{0}(Z_{i+1}, Z_{i}; H_{1}) = ⨁_{s∈SL_{2}(F)/B} F

with basis the various t^{i+1}w_{i+1}.

Corollary 9.2. The group H_{1}(K) is a quotient of H_{0}(Z_{1}, H_{1}).

If the field F is finite, then we see immediately that the graph Z_{1} is finite and hence H_{1}(K) is finite dimensional. This is in sharp contrast to the case n = 2.

10. Computation

We are now in a position to compute the group H_{1}(K) in certain cases.

Theorem 10.1. If n = 3 and F = 𝔽_{2}, then H_{1}(K) = 𝔰𝔩_{3}(𝔽_{2}).

Proof. By Proposition 9.1, we have a surjective map

H_{0}(Z_{1}, H_{1}) ⟶ H_{1}(K).

Since H_{1}(K) surjects onto 𝔰𝔩_{3}(F), we need only check that H_{0}(Z_{1}, H_{1}) is an 8-dimensional vector space.

The complex Z_{1} is the incidence geometry of 𝔽_{2}^{3}. It is a graph with 14 vertices and 21 edges. For each vertex v in Z_{1}, the group H_{1}(K_{v}) is twodimensional and for each edge e, the group H_{1}(K_{e}) is one-dimensional. It follows that

∂ : C_{1}(Z_{1}, H_{1}) ⟶ C_{0}(Z_{1}, H_{1})

is a map from a 21-dimensional vector space to a 28-dimensional vector space. Moreover, this map is clearly 𝔽_{2}-linear. Denote the generators of C_{0}(Z_{1}, H_{1}) by f_{1}, f_{2}, …, f_{28}. The relations imposed by ∂ are as follows:

\begin{matrix}f_{1} = f_{3} & f_{15} = f_{2} + f_{4} & f_{14} = f_{20} \\ f_{2} = f_{11} & f_{17} = f_{5} + f_{6} & f_{27} = f_{13} + f_{14} \\ f_{12} = f_{10} & f_{19} = f_{7} + f_{8} & f_{27} + f_{28} = f_{17} + f_{18} \\ f_{9} = f_{7} & f_{21} = f_{9} + f_{10} & f_{28} = f_{21} + f_{22} \\ f_{5} = f_{8} & f_{23} = f_{11} + f_{12} & f_{25} + f_{26} = f_{19} + f_{20} \\ f_{4} = f_{6} & f_{18} = f_{24} & f_{26} = f_{23} + f_{24} \\ f_{13} = f_{1} + f_{2} & f_{16} = f_{22} & f_{25} = f_{15} + f_{16} \end{matrix}

By considering the 28 × 21 matrix associated to ∂, we see that H_{0}(Z_{1}, H_{1}) is an 8-dimensional F -vector space (the rank of ∂ is 20).

Remarks. 1. A similar argument can be used to compute H_{1}(K) = 𝔰𝔩_{3}(𝔽_{3}) for F = 𝔽_{3}. In this case the complex Z_{1} has 25 vertices and 42 edges. The map ∂ is injective and hence H_{0}(Z_{1}, H_{1}) is 8-dimensional.

2. To prove that H_{1}(K) = 𝔰𝔩_{n}(F) for all n, it suffices to show that the vector space H_{0}(Z_{1}, H_{1}) has dimension n^{2} − 1. Since Z_{1} is a finite complex it should be possible to carry this out. However, the combinatorics seem to be rather complicated in general. The complexity increases as n does and for a given n, the computation becomes more difficult as the cardinality of F increases.

3. As a corollary, we see that the subgroup K^{2} of K consisting of matrices congruent to the identity modulo t^{2} is equal to the commutator subgroup of K for n = 3, F = 𝔽_{2}, 𝔽_{3}.

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Department of Mathematics, Northwestern University, Evanston, IL 60208

   E-mail address: knudson@math.nwu.edu