PHD THESIS: HOMOTOPY REPRESENTATIONS OF SIMPLY CONNECTED p-compact GROUPS OF RANK 1 OR 2

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1 PHD THESIS: HOMOTOPY REPRESENTATIONS OF SIMPLY CONNECTED p-compact GROUPS OF RANK 1 OR 2 TOKE NØRGÅRD-SØRENSEN ADVISOR: JESPER GRODAL Abstract. In this thesis we show that the restriction map R p (X) R p (T ) W is an isomorphism for all simply connected p-compact groups of rank at most 2. Here T is a maximal torus of X with Weyl group W, R p (X) is the complex homotopy representation ring of X and R p (T ) W is the invariants under W of the representation ring of T. (In Danish:) I denne afhandling viser vi at restriktionsafbildningen R p (X) R p (T ) W er en isomorfi for all enkeltsammenhængende p-kompakte grupper af rang højst 2. Her er T en maksimal torus for X med Weyl-gruppe W, R p (X) er den komplekse homotopi-repræsentationsring for X og R p (T ) W er invarianterne under W af repræsentationsringen for T. Date: 13/5/2013. Submitted at the Department of Mathematics, Faculty of Science, University of Copenhagen. 1

2 2 TOKE NØRGÅRD-SØRENSEN Contents 1. Introduction 3 2. Notation 4 3. Background for compact Lie groups Factorisation of Rep p (G) Rep p (T ) W Describing lim Rp(G) Rep n p(p ) Obstruction theory for Φ Describing Rep p (T ) 8 4. Background for p-compact groups 8 5. General results When p does not divide the order of the Weyl group Condition for fusion invariance Injectivity of Φ Bound on non-zero obstruction groups Using unstable Adams operations Proof of case: Sp(1) Proof of case: SU(3) Proof of case: Sp(1) Sp(1) at p = Representations of P 1,, P Determining fusion invariance Injectivity of Gr(Φ 1 ): R 2 (G) Gr(lim Rep 2 (P )) Proof of case: G 2 at p = radical subgroups of G Surjectivity of Φ 1 : Rep 2 (G 2 ) lim Rep 2 (P ) Representations of P 1,, P Determining fusion invariance Example: The adjoint representation Injectivity of Gr(Φ 1 ): R 2 (G) Gr(lim Rep 2 (P )) Surjectivity of Gr(Φ 2 ): Gr(lim Rep 2 (P )) R 2 (T ) W Proof of case: Sp(2) at p = Representations of P 1,..., P Relating R 2 (G) to orbit categories of finite groups Fix point functors and spectral sequences Injectivity of Gr(Φ 1 ): R 2 (G) Gr(lim Rep 2 (P )) Proof of case: G 2 at p = Representations of N Fusion invariance Surjectivity of Gr(Φ 2 ) Proof of case: DI(2) Fusion invariance Surjectivity of Gr(Φ 2 ) 43 References 43

3 HOMOTOPY REPRESENTATIONS 3 1. Introduction The goal of this thesis is to understand the complex homotopy representation ring (henceforth just called the representation ring) of simply connected p-compact groups of small rank. The p-completion of any connected compact Lie group is a p-compact group and the major part of the thesis can be understood without knowing about p-compact groups. The representation ring R p (X) of a p-compact group X is defined as R p (X) := Gr( n 0[BX, BU(n)ˆp]) where Gr is the Grothendieck group/group completion and BU(n)ˆp is the p-completion of BU(n) as defined in [5]. Our main theorem is: Theorem 1.1. Let X be a simply connected p-compact group of rank 1 or 2. Let T be a maximal torus of X with Weyl group W. Then the restriction map R p (X) R p (T ) W is an isomorphism. Here R p (T ) W representation ring of T. is the invariants under W of the The above theorem is analogous the following classical result: Let G be a connected compact Lie group and let T G be a maximal torus of G with Weyl group W = W G (T ). Then the restriction map R(G) R(T ) is an isomorphism onto the invariant subring R(T ) W [1, theorem 6.20]. Here R(G) is the classical representation ring of G. We also have the following integral result, shown in [21]: Gr([BG, n 0 BU(n)]) = R(G) This result motivates the definition of the homotopy representations [BG, n 0 BU(n)ˆp] of a compact Lie group G at a prime p or more generally the definition of R p (X). In lemma 5.1 our main theorem is shown for all connected p-compact groups X where p does not divide the order of the Weyl group. Since connected p-compact groups have been classified (see [3] and [4]) we can list all the remaining p-compact groups of rank 1 or 2 and we show our main theorem case by case for all these. By factoring the map Φ: R p (X) R p (T ) W into two maps Φ = (R P (G) Gr(Φ 1) Gr(lim Rep p (P )) Gr(Φ 2) R p (T ) W ) (with definitions and notation to be given later) the proof for each case is divided into two parts: To show that Gr(Φ 1 ) is an isomorphism we use a certain obstruction theory whereas showing that Gr(Φ 2 ) is an isomorphism is more combinatorial. For both parts we need to understand lim Rep p (P ), but luckily this limit can be described purely

4 4 TOKE NØRGÅRD-SØRENSEN algebraically and we can use classical representation theory (that is character theory) to understand it. In section 3 and 4 we survey the necessary background material needed for our proofs of the main theorem, first for compact Lie groups and then for general p-compact groups. In section 5 we give general proofs not related to one specific p-compact group. The remainder of the thesis consists of all the case by case proofs of our main theorem. We have tried to order the proofs in terms of similarity: For example the proof for G 2 at p = 2 has similarities with the proof for Sp(1) Sp(1) at p = 2 and the proof for DI(2) has similarities with the proof for G 2 at p = 3. With regards to proving that Gr(Φ 1 ) is an isomorphism it is best to read the proof for Sp(1) Sp(1) first; this is the simplest non-trivial proof, and it has been written in much greater detail than the later proofs. 2. Notation Let X be a p-compact group. We denote the (complex) n-dimensional homotopy representations of X by and define Rep n p(x) := [BX, BU(n)ˆp] Rep p (X) := n 0 Rep n p(x) which is a semiring with addition and product induced by direct sum and tensor product in n 0 U(n). So we can write R p (X) := Gr(Rep p (X)) Let D 1 : G U(n 1 ) and D 2 : H U(n 2 ) be representations of the groups G and H. Then D 1 D 2 : G H U(n 1 n 2 ) is the outer tensor product of D 1 and D 2. That is (D 1 D 2 )(g, h) = D 1 (g) D 2 (h). W will always denote the Weyl group of the particular p-compact group we are discussing, that is the Weyl group of the chosen maximal torus. 3. Background for compact Lie groups Let G be a connected compact Lie group with maximal torus T and Weyl group W Factorisation of Rep p (G) Rep p (T ) W. A p-toral subgroup P G is a closed subgroup such that its one-component P 1 is a torus, and π 0 (P ) is a finite p-group. A p-toral subgroup P is called a p-radical subgroup if and only if its Weyl group W (P ) := N G (P )/P is a finite group and O p (W (P )) = 1. Let O p (G) be the category with objects G/P for P a p-toral subgroup of G and with Mor(G/P, G/Q) being the G-equivariant maps. Here G/P = {P g g G} on which G acts

5 HOMOTOPY REPRESENTATIONS 5 on the right. Let R p (G) be the full subcategory of O p (G) with objects G/P for P p-radical. In [17] it is shown that hocolim G/P Rp(G) EG G G/P BG is an F p homology isomorphism, so that Rep k p(g) [hocolim G/P Rp(G) EG G G/P, BU(k)ˆp] is a bijection (since BG is p-good and BU(k)ˆp is p-complete, cf. [5]). We have a map [hocolim G/P Rp(G) EG/P, BU(k)ˆp] lim G/P Rp(G)[EG G G/P, BU(k)] = lim G/P Rp(G) Rep k p(p ). So we get a factorization Rep p (G) Φ 1 lim Rep p(p ) Φ 2 Rep p (T ) W G/P R p(g) For descriptions of R p (G) for the classical compact simple Lie groups, see [23] Describing lim Rp(G) Rep n p(p ). Let N p G be a maximal p-toral subgroup of G. One such can be constructed by taking the preimage of a Sylow-p-subgroup of W in N G (T ). Then lim Rep p (P ) Rep p (N p ) R p(g) is injective, since any p-toral subgroup of G conjugates into N p (see [18, lemma A.1]). We have an isomorphism lim Op(G) Rep p (P ) = lim Rp(G) Rep p (P ) induced by the natural inclusion (see [17]). Let φ: G/P G/Q and let g G be such that φ(p x) = P gx. Let c g : P Q be conjugation from the left. Then the following square commutes up to homotopy: BP Bc g BQ EG G G/P 1 φ EG G G/Q Because of this square we use the following language: Definition 3.1. An element of Rep p (N p ) is called fusion invariant if it comes from lim Rp(G) Rep p (P ) = lim Op(G) Rep p (P ). Definition 3.2. Let P be p-toral. P P is a p-discrete approximation of P if P is dense in P and P 1 := P P 1 = {x P 1 x pr = 1 for some r N} A p-discrete approximation of P always exists. And P 1 is unique up to conjugation with an element of P 1 (see [20, theorem 1.1]). P should be regarded as a discrete group. P is called a p-discrete toral group.

6 6 TOKE NØRGÅRD-SØRENSEN For a group K define Rep(K, U(k)) := Hom(K, U(k))/ Inn(U(k)). We have isomorphisms Rep( P, U(k)) B [B P, BU(k)ˆp] Rep k p(p ) For the proof of this see [20, theorem 1.1]: The proof is based on [10] which shows it for finite p-groups, and on the fact that B P BP is a mod p equivalence. Thus lim Rp(G) Rep n p(p ) can be given a quite algebraic description. A p-discrete toral group is a special type of a countable locally finite group. That G is countable locally finite means that there exists an ascending sequence 1 = G 0 G 1 G 2 G of finite groups such that G = n Gn. The representation theory of a countable locally finite group G is very similar to the representation theory of finite groups, as explained in [29, Appendix B]: Among other things, any representation of G splits uniquely (up to permutation) into a finite sum of irreducible representations, and any representation of G i determined by its character. Also Schur s lemma holds Obstruction theory for Φ 1. Let ρ be a k-dimensional fusion invariant representation of N p (or equivalently let ρ lim Rp(G) Rep k p(p )). For i 1 define Π ρ i : R p(g) op Grp as Π ρ i (G/P ) := π i(map(eg G G/P, BU(k)ˆp) Bρ If H i+1 (R p (G); Π ρ i ) = 0 for all i 1 then Φ 1 hits ρ. And if H i (R p (G); Π ρ i ) = 0 for all i 1 then the element hitting ρ is unique (see [27]). We have a natural weak equivalence BC U(k) (ρ( P ))ˆp Map(EG G G/P, BU(k)ˆp) Bρ This is shown in [20, theorem 1.1] and the proof is based on [10]. Say ρ P = ρ k 1 1 ρ kr r where the ρ i s are non-isomorphic irreducible representations. Then by Schur s lemma C U(k) (ρ( P )) = U(k 1 ) U(k r ) In particular, since π 0 (U(l)) = π 2 (U(l)) = 0 for all l 0 we get that Π ρ 1 = Π ρ 3 = Understanding Π ρ 2. Let k be the dimension of ρ. We have natural isomorphisms Π ρ 2(G/P ) = π 2 (Map(EG G G/P, BU(k)ˆp) Bρ ) = π2 (BC U(k) (ρ( P ))) Z p = π1 (C U(k) (ρ( P ))) Z p

7 HOMOTOPY REPRESENTATIONS 7 Write ρ P = ρ a 1 1 ρ ar r where the ρ i s are pairwise non-isomorphic irreducible representations of P. Say ρ i has dimension k i. Then C U(k) (ρ( P )) = U(a 1 ) I k1 U(a r ) I kr where I ki is the identity matrix of rank k i. Since π 1 (U(l)) = Z for all l 1 we get Π ρ 2(G/P ) = Z p {ρ 1,..., ρ r } This is a W (P ) op -permutation representation of rank r. Now assume Q P. For simplicity say ρ P = ρ a 1 1. Assume ρ 1 Q = σ b 1 1 σs bs, where the σ i s are non-isomorphic irreducible representations, so that ρ Q = (σ b 1 1 σs bs ) a 1. Say σ i has dimension l i. We then want to calculate the map Π ρ 2(P ) Π ρ 2(Q) as a map Z p {ρ 1 } Z p {σ 1,..., σ s } The element ρ 1 Z p {ρ 1 } corresponds to the element in π 1 (C U(k) (ρ( P ))) with representative f : S 1 U(a 1 ) I k1 where f(z) = diag(z I k1, I k1,..., I k1 ). Postcomposing with the inclusion C U(k) (ρ( P )) C U(k) (ρ( Q)) we get a map S 1 U(a 1 ) (U(b 1 ) I l1 U(b s ) I ls ) z diag(z I k1, I k1,, I k1 ) This map represents the element b 1 σ b s σ s. So we get ρ 1 b 1 σ b s σ s Spectral sequence for H (R p (G); F ). Fix a height function ht: Ob(R p (G)) Z 0 satisfying G/P = G/Q ht(g/p ) = ht(g/q) and (G/P = G/Q and Mor(G/P, G/Q) 0) ht(g/p ) > ht(g/q). Theorem 3.3. [13, theorem 1.3] There is a cohomological spectral sequence converging to H (R p (G); F ) with E 1 page given by E s,t 1 = ht(g/p )=s Λ s+t (W (P ); F (G/P )) Here Λ (Γ, M) := H (R p (Γ); F M ) where F M (Γ/1) = M (M is a Γ op - module) and F M (Γ/P ) = 0 for P 1. See [13] and [18] for methods for calculating the Λ groups. The height ht(c) of a category C is the maximal length of a chain of inclusions in the category.

8 8 TOKE NØRGÅRD-SØRENSEN 3.4. Describing Rep p (T ). Say T = (Z/p ) r U(1) r. Since T is abelian all irreducible representations of T are 1-dimensional (see [24, exercise 3.1]). And Rep( T, U(1)) = Hom( T, U(1)) = Hom(Z/p, U(1)) r = Hom(colim Z/p n, U(1)) r = (lim Hom(Z/p n, U(1))) r = (lim Z/p n ) r = Z r p An element (α 1,..., α r ) Z r p corresponds to the map (t 1,..., t r ) t α 1 mod p n 1 t αr mod pn r t i Z/p n U(1) The element (α 1,..., α r ) is called a weight. We will write Rep p (T ) = Z 0 [x 1,..., x r ] (called the character lattice) where an exponent of x i is in Z p. As an element of Rep p (T ) the weight (α 1,..., α r ) is written as x α 1 1 x αr r. Given a N p representation ρ, its restriction ρ T, as an element of Z 0 [x 1,..., x r ], is called the Lie character of ρ (not to be confused with the character χ of ρ, that is χ = trace ρ). We will define [x α 1 1 x αr r ] to be the orbit sum of x α 1 1 x αr r under the action of the Weyl group W, that is [x α 1 1 x αr r ] = y W.(x α 1 1 xαr r ) 4. Background for p-compact groups A p-compact group (where p is a prime) is a triple (X, BX, e) where BX is a connected pointed p-complete space, X is a space with finitedimensional F p -homology and e: X ΩBX is a homotopy equivalence. p-compact groups were first defined in [11]. See [11] and [12] for basic definitions and facts about p-compact groups, which will not be repeated here. Recall though that P is called a p-compact toral group if BP is the total space of a fibration with fiber the p-completed classifying space of a torus and with base the classifying space of a finite p-group. Now assume X is a p-compact group. Define the category O(X) as follows: The objects are all pairs (P, i P ) where P is a p-compact toral group and i P : P X is a monomorphism (in the sense of p-compact groups). A morphism α: (P, i p ) (Q, i Q ) is a free (i.e. non-pointed) homotopy class of maps Bα: BP BQ such that Bi Q Bα is freely homotopic to Bi P. Define the category R(X) to be the full subcategory y

9 HOMOTOPY REPRESENTATIONS 9 of O(X) with objects (P, i P ) where P is p-radical and centric. Here p- radical is defined in terms of the Weyl group W (P ) := Aut O(X) (P ) in the same way as for compact Lie groups and centric means that the natural map BZ(P ) BC X (P ) is a weak equivalence (here the centralizers are to be understood in the sense of p-compact groups). In [8] it is shown that there exists a functor Φ: R(X) Top such that Φ(P, i P ) BP for all (P, i P ) R(X) and such that there there exists a natural F p -homology equivalence hocolim R(X) Φ BX Let G be a connected compact Lie group. In [8, Appendix B] it is furthermore shown that the natural map R p (G) R(Gˆp) is a equivalence of categories. And it is shown that via this equivalence the above homology decomposition is equivalent to the homology decomposition hocolim G/P Rp(G) EG G G/P BG of the previous section up to p-completion. As in the previous section the above homology decomposition gives a factorization Rep p (X) Φ 1 lim Rep p(p ) Φ 2 Rep p (T ) W (P,i P ) R(X) where T is a maximal torus of X with Weyl group W. The rest of the previous section more or less generalizes to this context: Any p-compact toral group has a discrete approximation, and any homomorphism between p-compact toral groups lifts uniquely to a homomorphism of the chosen discrete approximation (see [12, proposition 3.2]). So we get the same algebraic description for lim (P,iP ) R(X) Rep p (P ) as we did for compact Lie groups. The obstruction theory for Φ 1, including the spectral sequence for calculating the obstruction groups, is the same as for compact Lie groups. Choose a maximal p-compact toral subgroup N p of X (say the p- normalizer of the chosen maximal torus T ). Then lim (P,iP ) R(X) Rep p (P ) Rep p (N p ) is injective: This is because any morphism i P : P X in R(X) lifts to a morphism P N p by [12, proposition 2.14] where we use the fact that p does not divide the Euler characteristic χ(x/n p ) (see [12, proposition 2.10]). Now we again say that an element of Rep p (N p ) is fusion invariant if and only if it lies in lim R(X) Rep p (P ) = lim O(X) Rep p (P ) (cf. lemma 5.2). Connected p-compact groups have been completely classified in [3] and [4]. If X is a connected 2-compact group not isomorphic to the 2-completion of compact Lie group then X contains DI(4) as factor by [4, theorem 1.1] and since DI(4) has rank 3 X has rank at least 3. In the case of odd primes p the classification says that there is a one to one

10 10 TOKE NØRGÅRD-SØRENSEN correspondence between isomorphism classes of simply connected p- compact groups and isomorphism classes of finite Q p -reflection groups (see [3, theorem 1.1] and [4, theorem 8.13(2)]). Now if X is a connected p-compact group with p not dividing the order of the Weyl group, the main theorem (theorem 1.1) is true for X by lemma 5.1. By inspecting a table of the irreducible Q p -reflection groups (see [14, page 4]) we see that the remaining cases we need to prove the main theorem for are Sp(1) at p = 2, Sp(1) Sp(1) at p = 2, SU(3) at p = 2, 3, Sp(2) at p = 2, G 2 at p = 2, 3 and DI(2). Here DI(2) is a 3-compact group the only simple exotic p-compact group of rank at most 2 where p divides the order of the Weyl group. 5. General results 5.1. When p does not divide the order of the Weyl group. Lemma 5.1. Let X be a connected compact Lie group with maximal torus T and Weyl group W. Let p be a prime such that p W. Then is an isomorphism. Rep p (X) = Rep p (T ) W Proof. In this case T is a p-radical subgroup and it is a maximal one since the p-normalizer of T is T itself. Choose any P O(X) not isomorphic to T. Then we have a monomorphism P T and P has rank strictly less that T. On the other hand C X (P ) has the same rank as T by [12, proposition 4.3]. It follows that P is not centric, so P R(X). From this calculation of R(X) we get that lim R(X) Rep p (P ) = Rep p (T ) W. Since R p (T ) has height 0, we also get that all obstructions for the map Φ 1 : Rep p (X) lim R(X) Rep p (P ) vanish by lemma 5.5, so that Φ 1 is an isomorphism Condition for fusion invariance. Lemma 5.2. Let X be a p-compact group and let ρ Rep p (N p ) where N p is a maximal p-compact toral subgroup of X. Then ρ is fusion invariant if and only if α (ρ P ) = ρ P for all α W (P ) for all (P, i P ) R(X). Proof. This is an application of Alperin s Fusion Theorem (AFT): AFT is proven in [6, theorem 3.6] for p-local compact groups, and thus also holds for O(X) since any p-compact group is a p-local compact group by [6, chapter 10]. Let α: P P be a morphism in O(X). We can assume that α is an isomorphism, since any morphism factors as an isomorphism followed by an inclusion. By AFT we have objects P = P 0,..., P k = P in O(X), objects Q 1,..., Q k in R(X) and morphisms α i W (Q i ) such that P i 1, P i

11 HOMOTOPY REPRESENTATIONS 11 Q i, α i : P i 1 P i is an isomorphism and α = α k α 1. Now αi (ρ P i ) = ρ P i 1 for all i implying that α (ρ P ) = ρ P. So ρ lies in lim O(X) Rep p (P ) and hence in lim R(X) Rep p (P ), that is ρ is fusion invariant Injectivity of Φ 2. Lemma 5.3. Let X be a connected p-compact group with maximal torus T. Then Φ 2 : lim Rp(G) Rep n p(p ) Rep n p(t ) is injective. Proof. Let N p N p be a p-discrete approximation of a maximal p- toral subgroup and let T be a p-discrete approximation of T. Let ρ 1 and ρ 2 be k-dimensional fusion invariant N p -representations. Assume ρ 1 T ρ 2 T. Remember that ρ i is determined by its character χ i. So let n N p ; we then have to show that χ 1 (n) = χ 2 (n). The map B n BN p X is a monomorphism and there exists a map Bφ: B n BT in O(X): This follows by a proof almost identical to the proof of [11, proposition 8.11] except that one uses theorem 4.6 instead of theorem 4.7 in the proof. Now since ρ i is fusion invariant we have that χ i (n) = χ i (φ(n)). So we get χ 1 (n) = χ 1 (φ(n)) = χ 2 (φ(n)) = χ 2 (n) Corollary 5.4. Gr(Φ 2 ) is injective. Proof. This follows from the previous lemma and the fact that R p (T ) satisfies additive cancellation Bound on non-zero obstruction groups. Lemma 5.5. Let X be p-compact group. Then H n (R(X); F ) = 0 for n > ht(r(x)) for all functors F. Proof. We want to use proposition in [22]. First we note that a skeleton of R(X) is a finite EI-category. Let M : R(X) op Z p - mod be the constant functor M(P ) = Z p. Obviously M(P ) is projective over Z p. Now by [13, theorem 1.1] Λ n (W (P ), F (P )) = 0 for n > ht(r p (W (P ))) so by the spectral sequence converging to H (R(X); F ) we have that there exists an N such that Ext n Z op(m, F ) = pr(x) Hn (R(X); F ) = 0 for n > N, and this N is independent of F. This implies that M has a finite projective resolution. Now by [22, proposition 17.31] the projective dimension of M is less than or equal to ht(r(x)). This implies that H n (R(X); F ) = Ext n Z pr(x) op(m, F ) = 0 for n > ht(r(x)).

12 12 TOKE NØRGÅRD-SØRENSEN 5.5. Using unstable Adams operations. Lemma 5.6. Let G be a connected compact Lie group with maximal torus T and Weyl group W. Let R p (T ) = Z[x 1,..., x r ] be any isomorphism, and let α Z p. Then [x α i ] R P (T ) W is hit by Φ: R p (G) R p (T ) W. Proof. Write α = kp i with i 0 and k Z p. Since we have an isomorphism R(G) R(T ) W the integral orbit sum [x pi i ] is hit by Φ. Furthermore for all k Z p there exists an unstable Adams operation ψ k : BGˆp BGˆp and precomposing a representation with ψ k corresponds on the character lattice to multiplying each exponential by k. So by precomposing with ψ k we see that also [x α i ] is hit by Φ. Remark 5.7. Notice that by tensoring virtual representations we see that Φ also hits products of the above orbit sums; for example [x α 1 ] [x β 2] for α, β Z p is also hit. 6. Proof of case: Sp(1) Let G = Sp(1). Here R 2 (T ) = Z[x 1 ] and the Weyl group Σ 2 acts by x α 1 x α 1. By lemma 5.6 any orbit sum [x α 1 ] is hit by R 2 (G) R 2 (T ) W. A skeleton of R 2 (G) is Q N. So R 2 (G) has height 1, so Φ 1 is an isomorphism. So is an isomorphism. R 2 (Sp(1)) = R 2 (T ) W 7. Proof of case: SU(3) Here we can use lemma 5.6 to show surjectivity. We have R p (T ) = Z[x 1, x 2, x 3 ]/(x α 1 x α 2 x α 3, α Z p ) and the Weyl group Σ 3 acts by permuting x 1,..., x 3. We already know that any orbit sum of the form [x α 1 ] is hit. Then, for α, β 0 and α β, [x α 1 ] [x β 1] = [x α+β 1 ] + [x α 1 x β 2] so also [x α 1 x β 2] is hit. Both R 2 (SU(3)) and R 3 (SU(3)) have height 1 (see [23]), so Φ 1 is always an isomorphism. So is an isomorphism for all primes p. R p (SU(3)) = R p (T ) W 8. Proof of case: Sp(1) Sp(1) at p = 2 Let G = Sp(1) Sp(1). By using lemma 5.6 and its following remark it is easy to see that R 2 (G) R 2 (T ) W is surjective.

13 HOMOTOPY REPRESENTATIONS 13 Representatives for the conjugacy classes of groups in R 2 (G) = R 2 (Sp(1)) R 2 (Sp(1)) are given by the following table: P W (P ) ht(p ) P 1 = N N = N 2 (T ) 1 0 P 2 = N Q 1 Σ 3 1 P 3 = Q N Σ P 4 = Q Q Σ 3 Σ 3 2 Here N = U(1), j and Q = i, j, the quaternion group. N N N Q Q N Q Q The morphisms between the P i s are generated by the automorphisms and the inclusions: This follows by the following lemma by noting that R 2 (G) = R 2 (Sp(1)) R 2 (Sp(1)): Lemma 8.1. Let H N, H = Q (for example H = x Q for x Sp(1)). Then there exists n N such that H = n Q. Proof. Assume x, y = H N is isomorphic to the quaternion group via x i and y j. Since not all elements of order 4 in H can lie in j U(1) we must have one of the elements equal to i, say x = i. Then y j U(1). Write y = y j for y U(1), choose n U(1) such that (n ) 2 = y and put n = n j. Then n i = i and n j = (n ) 2 j = y, so n Q = H. Let N = Ŭ(1), j Sp(1) (a p-discrete approximation of N). Choose discrete approximations P 1,..., P 4 of P 1,..., P 4 by replacing any factor N by N. As R 2 (G) has height 2 there is just one potential obstruction group to deal with, namely H 2 (R 2 (G), Π ρ 2) Representations of P 1,, P 4. Lemma 8.2. The irreducible representations of N are given as follows: Two 1-dimensional representations φ ɛ for ɛ {1, 1} given by φ ɛ (Ŭ(1)) = 1 and φ ɛ(j) = ɛ. For all α Z 2 {0} a representation ψ α = Ind NŬ(1) (α). Then ( ) t α 0 ψ α (t) = 0 t α

14 14 TOKE NØRGÅRD-SØRENSEN for all t Ŭ(1) and ( ) 0 ( 1) α ψ α (j) = 1 0 Proof. Any irreducible representation of N is contained in a representation induced from Ŭ(1) by [24, exercise 3.4]. Let α Z 2 be an irreducible representation of Ŭ(1). Then Ind NŬ(1) (α) is irreducible if and only if the action of N/ Ŭ(1) on α has trivial stabilizer (see [15, problem 6.1] which can be proven using theorem 6.11), that is if and only if α j α. Since α j = α, this is if and only if α 0. And Ind NŬ(1) (0) = φ 1 φ 1. The irreducible representations of P 1 are exactly the products of an irreducible representation of N with an irreducible representation of N (see [29, Appendix B]). So they are given as follows: 1-dimensional representations τ ɛ1,ɛ 2 = φ ɛ1 φ ɛ2, ɛ i {±1}. 2-dimensional representations θα,ɛ 1 = ψ α φ ɛ with character x ±α 1 and θβ,ɛ 2 = φ ɛ ψ β with character x ±β 2. Here α, β 0 and ɛ {±1}. 4-dimensional representations ρ α,β = ψ α ψ β with character x ±α 1 x ±β 2. Here α, β 0. Q has five irreducible representations. Four 1-dimensional representations χ ɛ1,ɛ 1, ɛ i {1, 1} given by χ ɛ1,ɛ 2 (i) = ɛ 1 χ ɛ1,ɛ 2 (j) = ɛ 2 and one 2-dimensional representation ζ. Then P 4 = Q Q has 25 irreducible representations given by products of these. To determine how representations of P 1 restrict to P 2,..., P 4 we use the fact that, for ρ 1, ρ 2 representations of R, S, the character of ρ 1 ρ 2 (a representation of R S) can be calculated as χ ρ1 ρ 2 (x, y) = χ ρ1 (x) χ ρ2 (y) And then we use the following table Representation of N ψ α, α 1 (2) Restriction to Q ζ ψ α, α 0 (2) χ δ,1 χ δ, 1 δ = φ ɛ χ 1,ɛ So for example ρ 1,2 Q Q = ζ χ 1,1 ζ χ 1, 1. { 1 α 0 (4) 1 α 2 (4)

15 HOMOTOPY REPRESENTATIONS Determining fusion invariance. We will now determine fusion invariance (see definition 3.1 and theorem 5.2). Lemma 8.3. A representation of the discrete 2-normalizer P 1 is fusion invariant if and only if its character χ satisfies χ(x, i) χ(x, j) = 0 and χ(i, x) χ(j, x) = 0 for all x N. Proof. An P 1 -representation with character χ is fusion invariant if and only its restriction to P i is invariant under the action of W (P i ) for i = 1,..., 4. It is invariant under W (P 2 ) if and only if χ(x, i) = χ(x, j) = χ(x, k) and invariant under W (P 3 ) if and only if χ(i, x) = χ(j, x) = χ(k, x) for all x N. Since χ(x, j) = χ(x, k) and χ(j, x) = χ(k, x) for all representations of P 1 this is equivalent to χ(x, i) χ(x, j) = 0 and χ(i, x) χ(j, x) = 0. By straightforward calculation one sees the representation is also invariant under W (P 4 ) if these two equations are satisfied. For fusion invariance the values of χ for the irreducible representations are (where t Ŭ(1) and congruences are module 4): Rep. Value χ(t, i) χ(t, j) τ ±, 2 θα, 1 2t ±α θβ,± 2 β 2: 2, β 0: 2 ρ α,β β 2: 2t ±α, β 0: 2t ±α χ(jt, i) χ(jt, j) τ +, 2 τ, 2 θβ,+ 2 β 2: 2, β 0: 2 θβ, 2 β 2: 2, β 0: 2 χ(i, t) χ(j, t) τ,± 2 θβ, 2 2t ±β θα,± 1 α 2: 2, α 0: 2 ρ α,β α 2: 2t ±β, α 0: 2t ±β χ(i, jt) χ(j, jt) τ,+ 2 τ, 2 θα,+ 1 α 2: 2, α 0: 2 θα, 1 α 2: 2, α 0: 2 For representations not listed the values are 0. Here for example τ ±, means either τ +, or τ,. And 2t ±α = 2t α + 2t α

16 16 TOKE NØRGÅRD-SØRENSEN Studying the above table we see that the representation is fusion invariant if and only if the following equations are all satisfied: Equation no. Equation (1a) For each α Z 2 {0}: ρ α,β + #θα, 1 = # # β 0 (1b) #τ ±, + # β 0 θ 2 β,± = # ρ α,β β 2 θβ,± 2 β 2 (2) #τ +, + # θβ,+ 2 + # β 0 β 2 (3a) For each β Z 2 {0}: # α 0 ρ α,β + #θ 2 β, = # (3b) #τ,± + # α 0 (4) #τ,+ + # α 0 θ 1 α,± = # θβ, 2 = #τ, + # θβ,+ 2 + # ρ α,β α 2 θα,± 1 α 2 β 2 θα,+ 1 + # θα, 1 = #τ, + # θα,+ 1 + # α 2 α 2 θβ, 2 β 0 θα, 1 α 0 Here, for example, # ρ α,β means the number of irreducible summands β 0 in the representation of the form ρ α,β with α = α and β 0 (4). For example the representation ρ 2,2 ρ 2,4 ρ 4,2 ρ 4,4 is fusion invariant Injectivity of Gr(Φ 1 ): R 2 (G) Gr(lim Rep 2 (P )). Injectivity is governed by the uniqueness obstruction group H 2 (R 2 (G); Π ρ 2). We will construct a specific fusion invariant representation ρ such that this group is 0. For this particular representation the following holds: For any other fusion invariant representation Ṽ also H2 (R 2 (G); ΠṼ ρ 2 ) = 0 (see below). Injectivity will then follow by the following lemma (noting that Φ 1 is surjective, since all existence obstruction groups are 0, since the height of R 2 (G) is 2) Lemma 8.4. Let p be a prime and let G be a connected compact Lie group. Assume ht(r p (G)) 3, assume Φ 1 is surjective and assume that for all Ṽ lim Rep p(p ) there exists a representation X lim Rep p (P ) such that H 2 (R p (G); ΠṼ X 2 ) = 0. Then Gr(Φ 1 ) is injective. Proof. As ht(r p (G)) 3 all uniqueness obstruction groups except H 2 vanish (remember Π ρ 1 = Π ρ 3 = 0). Assume Gr(Φ 1 )([V 1 V 2 ]) = 0 that is Φ 1 (V 1 ) W = Φ 1 (V 2 ) W for some W. Then by assumption there exists X such that H 2 (R p (G); Π Φ 1(V 1 ) W X 2 ) = 0. Choose W and X such that W = Φ 1 (W ) and X = Φ 1 (X). Since the obstruction group vanishes for Φ 1 (V 1 W X) = Φ 1 (V 2 W X) we get V 1 W X = V 2 W X. So [V 1 V 2 ] = 0. We will show that H 2 (R 2 (G); Π ρ 2) = 0 by showing that the differential in the spectral sequence Λ 1 (1 Σ 3 ; Π ρ 2(P 2 )) Λ 1 (Σ 3 1; Π ρ 2(P 3 )) Λ 2 (Σ 3 Σ 3 ; Π ρ 2(P 4 ))

17 HOMOTOPY REPRESENTATIONS 17 is surjective. Let Z 2 R 2 (G) op Z 2 W (P 4 ) op be the functor T T (P 4 ). This functor has a right adjoint, a right Kan extension, which we will call Ran. Let M = Π ρ 2(P 4 ) and put F = Ran(M). The unit of the adjunction gives a natural transformation Π ρ 2 F. This induces a natural transformation of spectral sequences giving a commutative square (8.1) Λ 1 (1 Σ 3 ; Π ρ 2(P 2 )) Λ 1 (Σ 3 1; Π ρ 2(P 3 )) Λ 2 (Σ 3 Σ 3 ; Π ρ 2(P 4 )) Λ 1 (1 Σ 3 ; F (P 2 )) Λ 1 (Σ 3 1; F (P 3 )) Λ 2 (Σ 3 Σ 3 ; M) giving a factorization of. First we will show that is surjective. To do this we need to understand the category R 2 (G) and the functor F a little better: Q < N are 2-toral groups, so Q < N N (Q) by [18, lemma A.2]. Since Out(Q) = Σ 3 (identifying i with 1, j with 2 and k with 3) we must have N N (Q)/Q = C 2. In fact N N (Q)/Q = (1+i) 2 : Putting x = (1+i) 2 we see x i = i, x j = k and x k = j. So N N (Q)/Q = τ Σ 3 where τ = (2 3). Let C 2 = τ denote this particular subgroup of Σ 3. Let O denote the following full subcategory of Σ 3 Σ 3 = W (P 4 ): N N N (Q Q)/(Q Q) N N Q (Q Q)/(Q Q) N Q N (Q Q)/(Q Q) 1 that is the subcategory C 2 C 2 C C 2 1 Notice that this is a skeleton of O 2 (Σ 3 Σ 3 ). Let R G denote the full subcategory (skeleton) of R 2 (G) with objects P 1,..., P 4. We want to show that the obvious map O R G on objects gives an isomorphism of categories: First notice that the Weyl groups are isomorphic. To determine Mor R2 (G)(G/P 4, G/P 2 ) = N(P 4, P 2 )/P 2 we use that any such map is given by an automorphism of G/P 4 followed by the projection G/P 4

18 18 TOKE NØRGÅRD-SØRENSEN G/P 2 followed by an automorphism of G/P 2. And by the description of the Weyl groups of P 4 and P 2 we see that any composition of the projection with an automorphism of G/P 2 is equal to an automorphism of G/P 4 followed by the projection. This implies that W (P 4 ) N(P 4, P 2 )/P 2 (mapping x to x) is surjective. So N(P 4, P 2 )/P 2 = W (P 4 )/(N P2 (P 4 )/P 4 ) = Σ 3 Σ 3 /C 2 1 By similar calculations we get N(P 4, P 3 )/P 3 = Σ 3 Σ 3 /1 C 2 N(P 4, P 1 )/P 1 = Σ 3 Σ 3 /C 2 C 2 N(P 2, P 1 )/P 1 = 1 Σ 3 /1 C 2 N(P 3, P 1 )/P 1 = Σ 3 1/C 2 1 This shows that O = R G. Now, F (P 2 ) = = N(P 4,P 2 )/P 2 M Σ 3 Σ 3 /C 2 1 M W (P 4 ) Σ 3 Σ 3 = Hom Σ3 Σ 3 (Z 2, Ind Σ 3 Σ 3 C 2 1 (M)) = Hom C2 1(Z 2, M) = M C 2 1 where we in the first equality write up a concrete expression for F. Similarly we calculate F (P 3 ) and F (P 1 ). All in all F (P 1 ) = M C 2 C 2 F (P 2 ) = M C 2 1 F (P 3 ) = M 1 C 2 F (P 4 ) = M Restricting F to a functor O Z 2 -mod we see that F is a fixed point functor. Such functors are known to be acyclic by [18, proposition 5.2] (the proof is to show that F is a proto-mackey functor and then use [16, proposition 5.14]). Looking at the spectral sequence converging to H (O; F ) we see that the map Λ 1 (1 Σ 3 ; F (P 2 )) Λ 1 (Σ 3 1; F (P 3 )) Λ 2 (Σ 3 Σ 3 ; M)

19 HOMOTOPY REPRESENTATIONS 19 is surjective. Returning to the square 8.1 this implies that is surjective if the left vertical map is surjective. We now want to find a representation ρ where this is the case: Put ρ =(ρ 2,2 ρ 2,4 ρ 4,2 ρ 4,4 ) (ρ 1,2 ρ 1,4 ) (ρ 2,1 ρ 4,1 ) ρ 1,1 (τ,+ θ 1 2,+) (τ +, θ 2 2,+) θ 1 1,+ θ 2 1,+ Then ρ is fusion invariant (check that the equations in section 8.2 are satisfied). For this ρ we have that ρ P 4 contains all irreducible representations of P 4, so M has as basis all the irreducible representations. Let M 2 = Π ρ 2(P 2 ). Then M 2 has basis {ψ 2 χ ɛ1,ɛ 2 ɛ i {±1}} {ψ 4 χ ɛ1,ɛ 2 ɛ i {±1}} {ψ 1 χ ɛ1,ɛ 2 ɛ i {±1}} {ψ 2 ζ, ψ 4 ζ, ψ 1 ζ} {φ 1 χ 1,1, φ 1 χ 1, 1 } {φ 1 χ 1,ɛ ɛ {±1} {φ 1 ζ} The map M 2 M induced by the inclusion P 4 P 2 is easily determined from the table above detailing how representations of P2 restrict to P 4. We will now show that Λ 1 (1 Σ 3 ; Π ρ 2(P 2 )) Λ 1 (1 Σ 3 ; F (P 2 )) is surjective. In general Λ 1 (1 Σ 3 ; L) = L 1 C 2 /L 1 Σ 3. So it is enough to show that Π ρ 2(P 2 ) 1 C 2 F (P 2 ) 1 C 2 is surjective. That is, that M 1 C 2 2 M C 2 C 2 is surjective. M 1 C 2 2 has basis {ψ 2 χ 1,ɛ2, ψ 4 χ 1,ɛ2, ψ 1 χ 1,ɛ2 ɛ 2 {±1}} {ψ 2 χ 1,±1, ψ 4 χ 1,±1 ψ 1 χ 1,±1 } {ψ 2 ζ, ψ 4 ζ, ψ 1 ζ} {φ 1 χ 1,1, φ 1 χ 1, 1, φ 1 χ 1,±1 } {φ 1 ζ} Here we are using the summing convention that for example ψ 2 χ 1,±1 = ψ 2 χ 1,1 + ψ 2 χ 1, 1. M C 2 C 2 has basis {χ 1,ɛ2 χ 1,ɛ4 ɛ i {±1}} {χ 1,ɛ2 χ 1,±1 ɛ 2 {±1}} {χ 1,±1 χ 1,ɛ4 ɛ 4 {±1}} {χ 1,±1 χ 1,±1 } {ζ χ 1,ɛ, χ 1,ɛ ζ ɛ {±1}} {ζ χ 1,±1, χ 1,±1 ζ} {ζ ζ}

20 20 TOKE NØRGÅRD-SØRENSEN We can now calculate that Γ: M 1 C 2 2 M C 2 C 2 is surjective. Explicitly: χ 1,1 χ 1,1 = Γ(ψ 4 χ 1,1 φ 1 χ 1,1 ) χ 1,1 χ 1, 1 = Γ(φ 1 χ 1, 1 ) χ 1, 1 χ 1,1 = Γ(φ 1 χ 1,1 ) χ 1, 1 χ 1, 1 = Γ(ψ 4 χ 1, 1 φ 1 χ 1, 1 ) χ 1,1 χ 1,±1 = Γ(φ 1 χ 1,±1 ) χ 1, 1 χ 1,±1 = Γ(ψ 4 χ 1,±1 φ 1 χ 1,±1 ) χ 1,±1 χ 1,ɛ = Γ(ψ 2 χ 1,ɛ ) χ 1,±1 χ 1,±1 = Γ(ψ 2 χ 1,±1 ) ζ χ 1,ɛ = Γ(ψ 1 χ 1,ɛ ) ζ χ 1,±1 = Γ(ψ 1 χ 1,±1 ) χ 1,1 ζ = Γ(φ 1 ζ) χ 1, 1 ζ = Γ(ψ 4 ζ φ 1 ζ) χ 1,±1 ζ = Γ(ψ 2 ζ) ζ ζ = Γ(ψ 1 ζ) By the symmetry in the definition of ρ we get by symmetrical calculations that also Λ 1 (Σ 3 1; Π ρ 2(P 3 )) Λ 1 (Σ 3 1; F (P 3 )) is surjective. So we conclude, for this particular ρ, that the left vertical map in diagram 8.1 is surjective. To finish the argument we note that this vertical map is surjective for the representation Ṽ ρ for any other fusion invariant representation Ṽ. This follows because the basis of M consists of all irreducible representations of P 4. We conclude that H 2 (R 2 (G); ΠṼ ρ 2 ) = 0. In conclusion is an isomorphism. R 2 (Sp(1) Sp(1)) = R 2 (T ) W 9. Proof of case: G 2 at p = radical subgroups of G 2. Let G = G 2. Following [19] let z G be an element of order 2 (all these are conjugate in G). Then C G (z) is isomorphic to Sp(1) C2 Sp(1) where C 2 = ( 1, 1). Then T = U(1) C2 U(1) Sp(1) C2 Sp(1) is a maximal torus of G and T = Z/2 C2 Z/2 T is a 2-discrete approximation. The weight lattice of T is {(α, β) Z 2 Z 2 α + β 0 (2)}. The Weyl group of G acts on the weight lattice by the two generating matrices D = 1 ( )

21 HOMOTOPY REPRESENTATIONS 21 (rotation) and S = ( ) (reflection). Each conjugacy class of 2-radical subgroups of G has a representative in Sp(1) C2 Sp(1). The representatives are given in the following list (copied from [19]), where N = U(1), j and Q = i, j : P W (P ) ht(p ) P 1 = N C2 N = N 2 (T ) 1 0 P 2 = N C2 Q 1 Σ 3 1 P 3 = Q C2 N Σ P 4 = Q C2 Q Σ 3 Σ 3 2 P 5 = T, (j, j) Σ 3 1 P 6 = (i, i), (j, j), (1, 1) GL 3 (F 2 ) 3 The morphisms between the P i s are generated by the automorphisms and the inclusions (see [19]) P 1 6 P 2 P 3 P P 4 P Surjectivity of Φ 1 : Rep 2 (G 2 ) lim Rep 2 (P ). Let ρ be a fusion invariant representation of P 1. Surjectivity follows if we show that H 3 (R 2 (G); Π ρ 2) = 0. By the spectral sequence this follows if we show that Λ 3 (GL 3 (F 2 ); M) = 0 where M = Π ρ 2(P 6 ). Since GL 3 (F 2 ) is a finite group of Lie type we have that Λ 3 (GL 3 (F 2 ); M) = Hom GL3 (F 2 )(St GL3 (F 2 ) Z 2, M) (see [13]). Here St GL3 (F 2 ) is the Steinberg Z GL 3 (F 2 ) op -module, a module which is free over Z of rank 8. Now M is a permutation module on the isomorphism classes of irreducible summands of ρ P 6. We have P 6 = C 3 2 and the irreducible representations can also be identified with the elements of C 3 2. Assume ρ P 6 contains all irreducible representations of P 6 as summand. Then M = Z C3 2 p where GL 3 (F 2 ) acts on the basis C 3 2 in the canonical way (that is, C 3 2 is identified with the vector space F 3 2).

22 22 TOKE NØRGÅRD-SØRENSEN St GL3 (F 2 ) F 2 belongs to a block with trivial defect group (see [9, remark 67.13]). By standard modular representations theory this implies that St GL3 (F 2 ) Q 2 is simple, and that its restriction to a Sylow- 2-subgroup S = is isomorphic to the regular module Q 2 S. But this is not true for the restriction of M to S: The value on its character on is 4 whereas the value of the regular representation s character on this element is 0. Since St GL3 (F 2 ) Q 2 = St GL3 (F 2 ) Z 2 Q is simple this implies that Hom GL3 (F 2 )(St GL3 (F 2 ) Q 2, M Q) = 0. Now by the following diagram St GL3 (F 2 ) Z 2 M St GL3 (F 2 ) Q 2 0 M Q where the horizontal maps are inclusions, since St GL3 (F 2 ) Z 2 and M are free Z 2 -modules, we see that also Hom GL3 (F 2 )(St GL3 (F 2 ) Z 2, M) = 0 If ρ P 6 does not contain all irreducible representations of P 6 as summand then M has rank strictly less that 8, and again, by similar arguments as above, we get Hom GL3 (F 2 )(St GL3 (F 2 ) Z 2, M) = Representations of P 1,, P 4. The description for Sp(1) Sp(1) above gives the necessary information for this case as well. The irreducible representations of P1 are the irreducible representations of N N that factor through C 2. That is: For ρ α,β we require that α β (2) and for θα,ɛ i we require that α 0 (2). And for P 4 there are 17 irreducible representations: The 16 1-dimensional representations χ ɛ1,ɛ 2 χ ɛ3,ɛ 4, ɛ i {±1} and the 4-dimensional representation ζ ζ Determining fusion invariance. Lemma 9.1. A representation of the discrete 2-normalizer P 1 is fusion invariant if and only if (1) it is invariant under the action of the Weyl group W and (2) its character χ satisfies χ(x, i) χ(x, j) = 0 and χ(i, x) χ(j, x) = 0 for all x N.

23 HOMOTOPY REPRESENTATIONS 23 Proof. The condition on χ comes from being invariant under W (P 2 ) and W (P 3 ). See the proof for Sp(1) Sp(1) above. If this condition is satisfied then the representation is also invariant under W (P 4 ) and W (P 6 ). Regarding invariance under W (P 5 ): In [19] it is shown that W (P 5 ) = W/ [(j, j)], so invariance at P 5 implies invariance at T (that is invariance under W ). So we just have to worry about how W (P 5 ) acts on (j, j). A representative of an element of W (P 5 ) maps (j, j) to (j, j)t for some t T. Now similar to the proof of lemma 8.2 one can show that any irreducible representation (with character χ ) of P 5 is either induced from T or is trivial on T. In both cases χ (j, j) = χ ((j, j)t). So invariance under W also implies invariance at P 5. The tables for the case Sp(1) Sp(1) used to determine when the condition on χ is satisfied (see above) are the same for this case Example: The adjoint representation. The adjoint representation of G is a 14-dimensional representation with character 2 + [x 2 1] + [x 2 2]. It restricts to the P 1 -representation τ +, τ,+ θ 1 2,+ θ 2 2,+ ρ 1,1 ρ 3,1 This can be seen by noting, that this is the only way to make a fusion invariant representation with the given character. The adjoint representation splits as a sum of 2 fusion invariant representations, namely and τ,+ θ 1 2,+ ρ 1,1 τ +, θ 2 2,+ ρ 3, Injectivity of Gr(Φ 1 ): R 2 (G) Gr(lim Rep 2 (P )). The proof is basically the same as for the case Sp(1) Sp(1). Hence we will not repeat all the arguments. Only the fusion invariant P 1 -representation used to stabilize with has to be changed: Again we define F = Ran(M), M = Π ρ 2(P 4 ), where Ran is the same right Kan extension as for Sp(1) Sp(1). And again we get a natural transformation of spectral sequence. Then we note that the full subcategory of R 2 (G) with objects P 1,..., P 4 is isomorphic to the skeleton O of O(Σ 3 Σ 3 ). Then we note that the restriction F : O Z 2 -mod is a fixpoint functor and hence acyclic. Now put ρ =(ρ 2,2 ρ 2,4 ρ 4,2 ρ 4,4 ) (τ,+ θ 1 2,+) (τ +, θ 2 2,+) ρ 1,1 ρ is invariant at P 1,..., P 4 and P 6 but is not invariant under the action of the Weyl group. We note that ρ P 4 contains all irreducible representations of P 4, which is exactly what we need to be able to use ρ for stabilizing.

24 24 TOKE NØRGÅRD-SØRENSEN Let ρ = ρ θ 1 4,+ (ρ D (2,4) ρ D2 (2,4)) (ρ D (4,2) ρ D2 (4,2)) θ 1 8,+ ρ 3,1 2(θ 1 2,+ ρ 1,1 ) See above for the definition of the matrix D. Then ρ is fusion invariant. One can ignore the extra representations of ρ not in ρ as adding extra representations does not hurt the argument. Now one can check that Π ρ 2(P 2 ) 1 C 2 M C 2 C 2 is surjective, so that Λ 1 (1 Σ 3 ; Π ρ 2(P 2 )) Λ 1 (1 Σ 3 ; F (P 2 )) is surjective. And by the symmetry in the definition of ρ also Λ 1 (Σ 3 1; Π ρ 2(P 3 )) Λ 1 (Σ 3 1; F (P 3 )) is surjective. The rest of the proof of injectivity of Gr(Φ 1 ) is the same as for Sp(1) Sp(1) Surjectivity of Gr(Φ 2 ): Gr(lim Rep 2 (P )) R 2 (T ) W. I write R 2 (T ) W = Z[x 1, x 2 ] W similarly to how I wrote characters above. Given a free orbit sum x ±α 1 1 x ±β 1 2 +x ±α 2 1 x ±β 2 2 +x ±α 3 1 x ±β 3 2 then, calculating modulo 4, (α 1, β 1 ), (α 2, β 2 ), (α 3, β 3 ) equals one of (after possibly changing sign on α i or β i and permuting the i s) (0, 0), (0, 0), (0, 0) (1) (0, 0), (2, 2), (2, 2) (2) (0, 2), (1, 1), (1, 1) (3) (2, 0), (1, 1), (1, 1) (4) Let (α, β) be a weight with α, β 0. I will define the following families of P 1 -representations: For α, β 2 (4): Ψ α,β = ρ α,β (θ 1 α, ρ α/2,α/2 ) (θ 2 β, ρ β/2,β/2 ) τ, This is almost fusion invariant, except for missing the rest of the orbit of (α, β). The parentheses indicate which subrepresentations make up a Weyl group invariant orbit sum. For α, β 0 (4) let Ψ α,β = ρ α,β (ρ α,2 ρ D (α,2) ρ D2 (α,2)) (ρ 2,β ρ D (2,β) ρ D2 (2,β)) (ρ 2,2 θ 1 4,+) (θ 1 2,+ ρ 1,1 ) Here D is the matrix defined above. This is almost fusion invariant, except for missing the rest of the orbit of (α, β). In this regard notice that (α, 2) and (2, β) always generate free orbits.

25 HOMOTOPY REPRESENTATIONS 25 Now assume that (α, β) generates a free orbit. For α 0 (4) and β 2 (4) let Ψ α,β = (ρ α,β ρ D (α,β) ρ D2 (α,β)) (ρ β,β θ 1 2β,+) (θ 1 α, Ψ α/2,α/2 ) (θ 1 β, ρ β/2,β/2 ) (θ 1 2,+ ρ 1,1 ) This is fusion invariant. For α 2 (4) and β 0 (4) let Ψ α,β = (ρ α,β ρ D (α,β) ρ D2 (α,β)) (ρ α,α θ 1 2α,+) This is fusion invariant. (θ 2 α, ρ 3α/2,α/2 ) (θ 2 β, Ψ 3β/2,β/2 ) (θ 1 2,+ ρ 1,1 ) Lemma 9.2. Gr(Φ 2 ) is surjective. Proof. We have to show that all orbit sums [x α 1 x β 2] in R 2 (T ) W are hit by Gr(Φ 2 ). (1) All non-free orbit sums are hit. This follows by lemma 5.6. (2) Assume ν(α) = 2 and ν(β) = 1 (here ν is the valuation of the 2-adic number). The orbit sum [x α 1 x β 2] is hit since all the other orbit sums in Ψ α,β are hit (check the definition of Ψ α,β and Ψ α/2,α/2 ). Similarly for the case ν(α) = 1 and ν(β) = 2. (3) Assume ν(α) > 2 and ν(β) = 1. Here we use that by induction we can assume that the orbit sums [x α/2 1 x 2 2] and [x 2 1x α/2 2 ] are hit. Similarly for ν(α) = 1 and ν(β) > 2. So all orbit sums of type (3) or (4) above are hit. (4) Assume [x α 1 x β 2] is a free orbit sum of type (1) or type (2) above. Then Ψ α,β Ψ D (α,β) Ψ D2 (α,β) only consists of the orbit sum [x α 1 x β 2] plus orbit sums of type (3) and (4) plus non-free orbit sums. So also the orbit sum [x α 1 x β 2] is hit. We conclude that is an isomorphism. R 2 (G 2 ) = R 2 (T ) W 10. Proof of case: Sp(2) at p = 2 Let G = Sp(2). We have R 2 (T ) = Z[x 1, x 2 ] and the Weyl group, the dihedral group of order 8, acts by transposing x 1 and x 2 and by changing the sign of the exponent on x 1 and on x 2. Now by lemma 5.6 the map Φ: R 2 (G) R 2 (T ) W hits any non-free orbit sum (that is [x α 1 ] or [(x 1 x 2 ) α ]). Let α, β Z 2 {0} such that α ±β. Then [x α 1 ] [x β 1] = [x α 1 x β 2] + [x α+β 1 ] + [x α β 1 ] showing that Φ also hits any free orbit sum [x α 1 x β 2]. So Φ is surjective.

26 26 TOKE NØRGÅRD-SØRENSEN So we just have to show that Gr(Φ 1 ): R 2 (G) Gr(lim Rep 2 (P )) is injective. We will use the same general method as we did for Sp(1) Sp(1). Representatives for the conjugacy classes of groups in R 2 (G) are given by the following table (see [23]): P W (P ) ht(p ) P 1 = N C P 2 = N Q 1 Σ 3 1 P 3 = Q C 2 Σ 3 1 P 4 = Γ 2 Σ 3 1 P 5 = Q Q Σ 3 C 2 2 P 6 = Γ 2 O4 (F 2 ) 2 Here N C 2 N Q Q C 2 Γ Q Q Γ 2 N = U(1), j Sp(1) Q = i, j Sp(1) Γ 2 = N I, A, B Γ 2 = Q I, A, B where I is the identity matrix and ( ) 1 0 A = 0 1 ( ) 0 1 B = 1 0 The morphisms between the P i s are generated by the automorphisms and the inclusions: This follows by the following lemma: Lemma Let P j < P i, P i, P j {P 1,..., P 6 }. Let A Sp(2) such that A P j P i. Then there exists n P i such that A P j = n P j. To show this lemma we need the following lemma: Lemma Let K = Sp(1). Let A Sp(2) such that A (Q Q) K C 2. Then A (Q Q) K K.

27 HOMOTOPY REPRESENTATIONS 27 ( ) ( ) a b x 0 Proof. Write A = and let X Q Q, X =. Notice c d 0 y that if X is a square in Q Q then A X is diagonal. We calculate ( ) axa + byb A X = cxc + dyd ( ) 0 K By proof of contradiction assume A X is in. Then all of a,..., d K 0 ( ) 0 K must be non-zero. Also ( A X) is in so we get K 0 a(x + x)a + b(y + y)b = 0 Now assume x {±1} and y {±i, ±j, ±k}. Then we see that a = 0, a contradiction. Similarly if y {±1} and x {±i, ±j, ±k}. We conclude that A X is diagonal if x {±1} or y {±1}. And if neither of x and y is in {±1} then is also diagonal. A X = A ( x ) ) A( y Proof of lemma Case Q Q < Q C 2 : This follows immediately by lemma Case Q Q < N C 2. By lemma 10.2 A (Q Q) N N. And it is easy to see that if φ: Q Q N N is a monomorphism then φ(q Q) = H 1 H 2 with H i N, H i = Q by using that φ(x, 1) commutes with φ(1, y) for all x, y Q. Now use lemma 8.1. Case Q Q < N Q: Here A (Q Q) = H Q with H N, H = Q. Now use lemma 8.1. Case N Q < N C 2 : Since A (Q Q) N N also A (N Q) N N (since any element of N can be written as a linear combination of elements of Q). So A (N Q) = N H or A (N Q) = H N with H N, H = Q. Now use lemma 8.1. Case Q C 2 < N C 2 : Let φ: Q C 2 N C 2 be a monomorphism. We have φ(q Q) = H 1 H 2 with H i N, H i ( ) ( ) = Q like in the case n Q Q < N C 2. Now φ = since the image has to be antidiagonal and have order ( 2. ) Now by replacing φ with ( φ postcomposed ) ( ) 1 0 n 0 n with conjugation by we can assume that φ = Then we must have H 1 = H 2 so that φ(q C 2 ) = H C 2 for H N, H = Q. Now use lemma 8.1. Case Γ 2 < Q C 2 : Let φ: Γ 2 N C 2 be a monomorphism. By checking the possible subgroups of Q C 2 isomorphic to D 4 = A, B (first list all elements of order 2 and then find all pairs whose product has order 4) we see that by precomposing φ with an automorphism of Γ 2

28 28 TOKE NØRGÅRD-SØRENSEN (look closely at O 4 (F 2 )) we can assume that φ(a) = A and φ(b) = B. Then for x Q we must have φ ( ) ( ) x 0 α(x) 0 = 0 x 0 α(x) for α: Q N a monomorphism: The two diagonal entries of the image have to( be equal, ) since ( the element ) commutes ( with ) B. Also we cannot x 0 0 α(x) x 0 have φ = since φ A has order 4. Now 0 x α(x) 0 0 x use lemma 8.1. Case Γ 2 < N C 2 : Consider Sp(2) modulo its center that is Sp(2)/{±I} = SO(5). A 2-normalizer of the standard maximal torus in SO(5) is O(2) C 2 O(4) SO(5) and Γ 2 /{±I} = C2. 4 We will show that all elementary abelian subgroups of rank 4 in O(2) C 2 are conjugate via an element of O(2) C 2. Then by lifting such an element to Sp(2) we see that any subgroup of N C 2 isomorphic to Γ 2 is conjugate to Γ 2 via an element in N C 2. Let L N C 2, L = C2. 4 If L O(2) O(2) then we are done, since all elementary abelian subgroups of rank 2 in O(2) are conjugate by an element of O(2). By ( proof of) contradiction 0 N assume L O(2) O(2), say X L, X = N 1 M 0 2 (O(2)) (X ( must have ) this form since X has( order) 2). Then by conjugating L N by we can assume that L. But since all other elements in L have to commute with this element, we see that L can have rank at most 3 (since we can think of L as lying in O(2) C 2 ), a contradiction. Case Γ 2 < Γ 2 : As in the case of Γ 2 < N C 2 we reduce this to a question in SO(5), namely: Are all elementary abelian subgroups of rank 4 in Γ 2 /{±I} = O(2) C2 2 conjugate by an element of O(2) C2? 2 And the answer to this question is yes. Case Γ 2 < N C 2 : Similar to the case Γ 2 < N C 2 : Here we use that all subgroups of O(2) C 2 isomorphic to O(2) C2 2 are conjugate by an element of O(2) C 2 : Let φ: O(2) C2 2 O(2) C 2 be a monomorphism. By changing φ by an automorphism of O(2) C2 2 and by conjugation ( with ) an element of O(2) C 2 we can assume φ(n, x, y) = x 0 (N, α(n) ) O(2) O(2) where α: O(2) O(2) is a homomorphism. Since α(n) has to commute with we see that the 0 y ( ) image of α(n) has to be a diagonal matrix. Now use{( that O(2) )} is generated by its elements of order 2 to see that Im α. ±1 0 0 ±1