Borel-Moore homology and \(K\)-theory on the Steinberg variety
For a simply connected complex semisimple Lie group \(G\), let \(\mathfrak g\) be its Lie algebra, \(\mathcal B\) the flag variety of \(G,\) \(\mathcal N\) the nilpotent cone in \(\mathfrak g\), \(\widetilde{\mathcal N} =\{(x,{\mathfrak b}) \in {\mathcal N}\times {\mathcal B}\mid x\in {\mathfrak b}\}\) the Springer resolution of \(\mathcal N,\) and \(Z={\widetilde{\mathcal N}}{\times}_{{\mathcal N}}{\widetilde{\mathcal N}}=\{(x,{\mathfrak b}, {\mathfrak b}') \in {\mathcal N}\times {\mathcal B}\times {\mathcal B}\mid x\in {\mathfrak b}\cap {\mathfrak b}'\}\) the Steinberg variety. In the paper the author studies the \(\mathbb C\)-algebra structure of \(H_*(Z, {\mathbb C})\). Recall that a product on the complex graded Borel Moore homology is given by the convolution product. The study is done by examining multiplications on other objects such as the homology of a flag variety, the Grothendieck group of a graph variety and explicitly constructed by crossed product algebra. This is possible since the author proves the compatibility of the corresponding maps with the products. The main theorem of the paper is the computation of the \(\mathbb C\)-algebra structure of \(H_*(Z, {\mathbb C})\). A similar result was proven independently by \textit{J. M. Douglass} and \textit{G. Röhrle} [J. Algebra 321, No. 11, 3158--3196 (2009; Zbl 1183.20044), Doc. Math., J. DMV 14, 339--357 (2009; Zbl 1209.14020)].
scientific article; zbMATH DE number 5665444
scientific article