\(K\)-theory of twisted differential operators on flag varieties
Let \({\mathfrak g}\) be a semisimple Lie algebra over \(k\), an algebraically closed field of characteristic zero, and let \({\mathfrak h} \subset {\mathfrak b}\) be a Cartan subalgebra inside a Borel subalgebra of \({\mathfrak g}\). Let \({\mathcal U}\) be the enveloping algebra of \({\mathfrak g}\). For \(\mu \in {\mathfrak h}^*\) let \(M (\mu)\) denote the corresponding Verma module and let \({\mathcal U}_\mu = {\mathcal U}/ \text{Ann} M (\mu)\). Let \(W\) be the Weyl group and let \(W^0_\mu\) be the stabiliser of \(\mu\) in \(W\). We prove the following theorem, which affirms a conjecture of T. J. Hodges. Theorem. Let \(\mu \in {\mathfrak h}^*\). Then \(K_0 ({\mathcal U}_\mu)\) is free of \(\text{rank } |W/ W^0_\mu|\). -- In fact, this theorem is a special case of our results which compute \(K_0 ({\mathcal U} /{\mathcal J})\) for certain induced primitive ideals \({\mathcal J}\).
scientific article; zbMATH DE number 856596