Equivariant \(K\)-theory of compact Lie group actions with maximal rank isotropy (Q2892847)

In the paper under review, calculations of equivariant \(K\)-groups are given for certain \(G\)-spaces \(X\). The group \(G\) is a compact, connected Lie group with torsion-free fundamental group. The \(G\)-space \(X\) is assumed to be compact and to satisfy the connected maximal rank property. That means that the isotropy groups of the action are connected and each one contains a maximal torus. Classical examples of such actions are the action of \(G\) on itself by conjugation as well as the path of the \(n\)-tuple consisting of the identity elements in the space of the \(n\)-tuples of commuting elements in \(G\). Those conditions guarantee that \(G\)-CW structures on \(X\), with specified isotropy groups that satisfy the connected maximal rank condition, are in 1-1 correspondence with \(W\)-CW (where \(W\) is the Weyl group of a maximal torus \(T\)) structures on \(X^T\) with isotropy groups the corresponding Weyl groups.NEWLINENEWLINEWith the above restrictions, the rational equivariant \(K\)-theory of \(X\) is shown to be a free module over the rational representation ring and its rank is calculated. For the integral case, a similar result is proved under certain conditions on the fixed-point sets \(X^T\). More precisely, it is assumed that \(X^T\) contains a \(W\)-subcomplex whose isotropy groups satisfy the coset intersection property. The proofs relay on the calculation of the equivariant \(K\)-theory of \(X\) using a spectral sequence that is derived by the skeletal filtration of \(X\). The \(E^2\)-term of the sequence are Bredon cohomology groups. The authors provide a very detailed calculation of the homology groups. The proof is completed by proving that the spectral sequence collapses.NEWLINENEWLINEAs applications the authors calculate the groups \(K_G(G)\) where \(G\) acts on itself by conjugation. Also, for the action of \(G\) on its Lie algebra \(\mathfrak{g}\), they calculate the \(K_G\)-theory of the one-point compactification of \(\mathfrak{g}\) as well as of the unit sphere of \(\mathfrak{g}\). Furthermore, the \(K_G\)-groups of the one-point compactification of the variety of the \(n\)-commuting tuples in \(\mathfrak{g}\) are calculated. Two more examples include the calculation of \(K_G\) of the union of the orbits, under the adjoint action, of the space of the complement of a \(W\)-hyperplane arrangement in \(\mathfrak{t}\) (the Lie algebra of \(T\)). In all cases the (rational or integral) results are given either explicitly or as free modules over the appropriate representation ring.NEWLINENEWLINEFinally, if \(G\) is the product of groups \(SU(r)\), \(U(q)\) or \(Sp(k)\), then the methods provide the calculation of the \(K_G\)-groups of the space of commuting of \(n\)-tuples in \(G\). The calculations allow the authors to show that the inclusion map from the space of \(n\)-commuting tuples to all \(n\)-tuples induces an injection in \(K_G\)-groups.