ncts

For a reductive group G defined and split over $\mathbb{Z}$ , let B_G be the ring of functions of the affine scheme (T//W)^F, where T is a split maximal torus of G, W is the Weyl group of (G,T), and F is the q-power endomorphism on T with q a power of a prime number. Our interest in the ring B_G comes from the following result: upon denoting by G^* the dual group of G and by $\Gamma_{G^\ast}$ a Gelfand-Graev representation of the finite group $G^\ast(\mathbb{F}_q)$ , the ring B_G offers a combinatorial description of the endomorphism algebra of $\Gamma_{G^\ast}$ when the derived subgroup D(G) of G is simply-connected (see [1, Thm.10.1] for the case of G=GL(n), and [2][3] for general G with mild assumptions on the coefficients of $\Gamma_{G^\ast}$ ). On the other hand, from an algebro-geometric point of view, it is also natural to study B_G itself without reference to Gelfand-Graev representations; for example, it is known that B_G is a reduced ring (that is, (T//W)^F is a reduced scheme) when D(G) is simply-connected, but at the moment, except for a few special cases, we don't know whether B_G remains reduced beyond the case of simply-connected D(G). In this talk, we shall try to elaborate the above aspects on B_G, and examples will be given to illustrate the general theory.