Cohomology of Classifying Spaces of Central Quotients of Rank Two Kac-moody Groups

Abstract

This paper is devoted to the computation of the mod p cohomology of the classifying spaces of the quotients of (non-afine) Kac-Moody groups of rank two by finite central p-groups, as algebras over the Steenrod algebra and higher Bockstein operations. To this aim we enlarge the class of spaces by including some homotopy theoretic constructions in such a way that the new class of spaces is nicely parametrized by the integral p-adic representations of the infinite dihedral group in rank two. Furthermore, we show how these representations encode enough information in order to determine the mod p cohomology and Bockstein spectral sequence of this class of spaces. The goal of this paper is to compute the mod p cohomology algebra —including the action of the Steenrod algebra and the Bockstein spectral sequence— of the classifying spaces of the quotients of (non-afine) Kac-Moody groups of rank two by finite central p-groups. The wise have taught us (cf. Ithaca in [6]) that in the trips which are really worth doing what we see and learn along the trip always turns out to become more important than the final destination. We think that the present work might be an example of this. The rank two Kac-Moody groups have a large family of central subgroups which yield a rather complex series of interesting unstable algebras over the Steenrod algebra, and when we started computing these cohomology algebras we learned that in order to study them in a systematic way we needed to relate them to representation theory and to invariant theory. In this context, we believe than the relationship between rank two Kac-Moody groups, representations of the infinite dihedral group, invariant theory of pseudoreflection groups and cohomology algebras that we display here is more interesting than the particular values of each cohomology algebra. Although our starting point is to compute the cohomology of the spaces B(K/F ), for K a rank two Kac-Moody group and F a central subgroup (we will be more precise after this introduction), we soon realize that we should better compute the cohomology of a larger family of spaces —the family which we call S— which can be viewed as a homotopy theoretic generalization (and also a p-adic completion) of the spaces B(K/F ). The spaces in the set S can be parametrized by matrices in GL2(Ẑp) plus some extra data. A better parametrization of S ∗ is given by a set called R ∗ whose elements are faithful representations of the infinite dihedral group D∞ in GL2(Ẑp) plus some obstruction classes, modulo the action of the outer automorphisms of D∞. Hence, our results depend on the integral p-adic representation theory of the infinite dihedral group. This theory has been developed in [2] and we would like to J. A., C. B. and L. S. acknowledge support from MCYT grant BFM2001-2035. 1 2 JAUME AGUADÉ, CARLES BROTO, NITU KITCHLOO, AND LAIA SAUMELL point out that the work in [2] was motivated by the present paper and, moreover, the cohomological results in the present paper helped us in shaping the results in [2]. We want to mention also that this work is part of a more general project which, roughly speaking, aims to investigate the homotopy theory of the Kac-Moody groups with the tools that have lead to the development of the homotopy theory of compact Lie groups (see, for instance, the surveys [8] and [13]). We would like to point out the papers [5] and [1] as further examples of how some of the homotopy theoretical results and techniques of compact Lie groups can be extended, with some appropriate reformulation, to Kac-Moody groups. This paper generated also our motivation for the work in [3]. The present paper is organized as follows. In section 1 we recollect the notation on Kac-Moody groups and their central quotients that we use later. In section 2 we introduce the colimit decomposition of B(K/F ) that we will use to compute the cohomology and we define the family of spaces S which contains all spaces B(K/F ). In section 3 we relate the spaces in S to representations of D∞ and we introduce a set R ∗ of representation data with obstruction classes which parametrizes S. Moreover, we review the results on representations ofD∞ which we proved in [2], in a form which is more appropriate to our needs. Then, in the next three sections, we compute, for each data element of R, the mod p cohomology of the space in S corresponding to this data. The first of these three sections gives some generic information and the two other sections consider the case of the prime two and the case of the odd primes, respectively. We use the invariant theory of finite reflection groups as developed in [14]. In the final section, we return to Kac-Moody groups and we compute, for each quotient K/F , the representation data in R which gives B(K/F ) and we have in this way enough information to compute the mod p cohomology algebras, the Steenrod algebra actions and the Bockstein spectral sequences of all spaces, of the form B(K/F ), but a few cases. We are grateful to the Centre de Recerca Matemàtica for making possible several meetings of the authors of this paper. J. Aguadé wants to thank the Department of Mathematics of the University of Wisconsin-Madison for its hospitality during the final stages of the preparation of this paper. 1. Rank two Kac-Moody groups and central quotients We choose positive integers a, b such that ab > 4. Along this paper K will always denote the unitary form of the Kac-Moody group associated to the generalized Cartan matrix ( 2 −a −b 2 ) . Sometimes we write K(a, b) instead of K when we want to make explicit the values of a and b used to construct K. The integers a and b can be interchanged, since the group associated to (a, b) is isomorphic to the group associated to (b, a). The case ab < 4 gives rise to compact Lie groups while the case ab = 4 is called the affine case and will be left aside. These infinite dimensional topological groups and their classifying spaces BK have been studied from a homotopical point of view in several works, like [10], [9], [11], [12], [5], [1]. We recall here some properties of K and BK CENTRAL QUOTIENTS OF RANK TWO KAC-MOODY GROUPS 3 which we will use along this work and which can be found in the references that we have just mentioned. By construction, K comes with a standard maximal torus of rank two TK which is a maximal connected abelian subgroup of K. Any two such subgroups are conjugated. The Weyl group W of K is an infinite dihedral group acting on the Lie algebra of TK through reflections ω1 and ω2 given, in the standard basis, by the integral matrices: w1 = ( −1 b 0 1 ) , w2 = ( 1 0 a −1 ) . The matrices of determinant +1 in W form a subgroup W of index two which is infinite cyclic generated by ω1ω2. The cohomology of K and BK was computed by Kitchloo ([12]) using, among other tools, the existence of a Schubert calculus for the homogeneous space K/TK . The center of K is also well understood ([10]): ZK = { 2Z/(ab− 4)× Z/2, a ≡ b ≡ 0 (mod 2) Z/(ab− 4), otherwise. Hence, we have a family of central p-subgroups F of K and the purpose of this paper is to study the spaces B(K/F ). Let us fix now the notation that we will use in this paper to refer to the various quotients of K by central subgroups. We denote by νp the p-adic valuation. • If p is an odd prime or p = 2 and a or b is odd then there is a unique central subgroup F of K of order p for any 0 ≤ m ≤ νp(ab − 4). We denote K/F by PpmK. • If p = 2 and a and b are both even then the 2-primary part of the center of K is non-cyclic of the form Z/2 × Z/2, t = ν2((ab − 4)/2). There are several quotient groups. We denote by P 2 K the quotient of K by the right subgroup of the center of order two. We denote by P 2mK, 0 ≤ m ≤ t the quotient of K by the left subgroup of the center of order 2. We denote by P 2mK, 0 < m ≤ t the quotient of K by the diagonal subgroup of the center of order 2. Finally, we denote by P 2m+1K, 0 < m ≤ t the quotient of K by the non-cyclic subgroup of the center of order 2. 2. Colimit decompositions of BK and B(K/F ) and the spaces in S A fundamental result in the homotopy theory of the classifying spaces of KacMoody groups is the following (cf. [12]). If L is any Kac-Moody group with infinite Weyl group and {PI} are the parabolic subgroups of L indexed by proper subsets I of {1, . . . , rank(L)} then there is a homotopy equivalence BL ≃ hocolim I BPI . Let us give a more precise description of this homotopy colimit in the case of the rank two group K = K(a, b). 4 JAUME AGUADÉ, CARLES BROTO, NITU KITCHLOO, AND LAIA SAUMELL There are group homomorphisms φi : SU(2) → K, i = 1, 2, such that the images of φ1 and φ2 generate K. If D is the unit disc in C and we write zi(u) = φi ( u (1− ||u||2)1/2 −(1− ||u||2)1/2 ū ) then K has a presentation with generators {zi(u) | u ∈ D, i = 1, 2} and relations (i) zi(u)zi(v) = zi(uv) if u, v ∈ S. (ii) zi(u)zi(−ū) = zi(−1) if u ∈ D\S1. (iii) zi(u)zi(v) = zi(u )zi(v ) if u, v ∈ D\S1, u 6= v, for some unique u ∈ D\S1 and v ∈ S. (iv) zi(u)zj(v)zi(u) −1 = zj(u ijv)zj(u −aij ) if u ∈ S, v ∈ D and (aij) is the Cartan matrix of K. Then BK is a homotopy push out

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@inproceedings{AGUAD2007CohomologyOC, title={Cohomology of Classifying Spaces of Central Quotients of Rank Two Kac-moody Groups}, author={JAUME AGUAD{\'E} and CARLES BROTO and LAIA SAUMELL}, year={2007} }