The based ring of the lowest two-sided cell of an affine Weyl group
Let G be a simply connected, almost simple complex algebraic group and T a maximal torus of G. For the root lattice \(P\subset X=Hom(T,{\mathbb{C}}^*)\) and the Weyl group \(W_ 0=N_ G(T)/T\) of G, the semidirect product \(W=W_ 0\ltimes P\) is the affine Weyl group associated to G. Let \(J_ 0\) be the based ring associated to the lowest two-sided cell of W (see [\textit{G. Lusztig}, Proc. Symp. Pure Math. 47, 235-262 (1987; Zbl 0657.20037)] for the definition of a based ring). For a G-set Y, let \(K_ G(Y\times Y)\) be the Grothendieck group of the category of G-equivariant complex vector bundles on the G-set \(Y\times Y\). \(K_ G(Y\times Y)\) has a based ring structure if we define the multiplication * by setting \[ (V*U)_{(x,y)}=\oplus_{z\in Y}V_{(x,z)}\otimes U_{(z,y)} \] for any \(V,U\in K_ G(Y\times Y)\) and define \(\tilde V_{(x,y)}=V^*_{(y,x)}\) (the complex dual of \(V_{(y,x)})\). Then the main result of this paper is to assert that \(J_ 0\) is isomorphic as a based ring to \(K_ G(Y\times Y)\) for a certain G-set Y with \(| Y| =| W_ 0|\). This result supports a conjecture of \textit{G. Lusztig} [see J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No.2, 297-328 (1989; Zbl 0688.20020)].
simply connected, almost simple complex algebraic group
category of G-equivariant complex vector bundles
scientific article; zbMATH DE number 4165168