Equivariant K-theory of real vector spaces and real projective spaces
Let \(G\) be a finite group and \(V\) a real vector space provided with a linear action of \(G\). When \(V=\mathbb{R}^{8n}\) the equivariant \(K\)-theory \(K^*_G(V)\) can be described in terms of representations of \(G\) under some spinoriality conditions. One purpose of the author is to fill the gap in this result due to \textit{M. F. Atiyah} [Q. J. Math., Oxf., II. Ser. 19, 113-140 (1968; Zbl 0159.53501)], and another one is to study the problems which interest the author in such a computation of \(K^*_G(V)\). The paper consists of fifteen theorems, four corollaries and also five remarks. In this review we consider only complex \(K\)-theory and confine ourself to the case when \(G=S_n\) the symmetric group and \(V=\mathbb{R}^n\) on which \(G\) acts naturally. Then one of the main results is for example: (Corollary 1.9) \(K^0_G(V)\cong \mathbb{Z}^{p_n}\) and \(K^1_G(V) \cong\mathbb{Z}^{i_n}\) where \(p_n\) and \(i_n\) denote the number of partitions of \(n\) of the type \(n=\lambda_1+ \cdots+\lambda_{2k}\) with \(1\leq \lambda_1< \cdots< \lambda_{2k}\) and of the type \(n=\lambda_1+ \cdots+ \lambda_{2 k+1}\) with \(1\leq\lambda_1 <\cdots< \lambda_{2k+1}\), respectively. The key to the author's approach is using the \(K\)-theories of Banach categories built up in his book [\textit{M. Karoubi}, ``\(K\)-theory'', Grundlehren 226 (1978; Zbl 0382.55002)] for instance and the method developed by himself as well. In fact the proof of the freeness of the group \(K^*_G(V)\) above is referred to the author's earlier results [\textit{H. Karoubi}, Lect. Notes Math. 136, 187-253 (1970; Zbl 0197.19902)]. On the one hand, the computation of its rank is done using an isomorphism \(K^*_G(V) \otimes\mathbb{C} \cong\oplus H^*_c(V^{g_i} /C_{g_i}; \mathbb{C})\) induced by the equivariant Chern character introduced by \textit{J. Slominska} et al. Here \(g_i\)'s denote representatives of the conjugacy classes of elements of \(G\) and \(C_{g_i}\) the centralizer of \(g_i\). Let \(P(V)\) denote the real projective space of \(V\). Then a second main concern of this paper is to determine the rank of the groups \(K^0_G(P(V))\) and \(K_G^1(P(V))\). They are \(2P(n)-p_n-i_n\) and \(p_n\), respectively, where \(P(n)\) denotes the total number of partitions of \(n\) (Theorem 3.8). This is proved also in the same way as in the above case. It is seen that the rank of such groups in general can be derived from the numbers of conjugacy classes of \(G\) and \(\mathbb{Z}/2 \ltimes G\) which are decomposed in their associated Schur groups, respectively. The proof of this fact seems to be the essence of this work. Finally we note that the \(R(G)\)-module structure of \(K_G^*(V)\) is also completely determined (Theorem 4.8).
scientific article; zbMATH DE number 1787583
scientific article