Locally symmetric spaces and \(K\)-theory of number fields
Let \(G/K\) be a symmetric space of noncompact type and dimension \(2n-1\), and let \(\rho: G\to\text{GL}(N,{\mathbb C})\) be a representation such that \(\rho^*b_{2n-1}\neq0\) for the Borel class \(b_{2n-1}\in H^{2n-1}(\text{U}(N);{\mathbb R})\). Among other things, the author proves in the paper under review that there exists a constant \(c_\rho\neq0\) such that for every locally symmetric space \(M=\Gamma\setminus G/K\) which is orientable, has finite volume, and either is compact or \(\text{rk}(G/K)=1\), there exists \(\gamma(M)\in K_{2n-1}(\overline{\mathbb Q})\otimes{\mathbb Q}\) such that \(\langle b_{2n-1},\gamma(M)\rangle=c_\rho\text{vol}(M)\). He also determines the irreducible symmetric spaces \(G/K\) and the fundamental representations satisfying the above conditions.
scientific article; zbMATH DE number 6035474
scientific article