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 $G^\ast$ 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.

References:
[1] D. Helm, Curtis homomorphisms and the integral Bernstein center for $GL_n$, Algebra & Number Theory, Vol.14, No.10 (2020)
[2] T.-J. Li, On endomorphism algebras of Gelfand-Graev representations, preprint (2021)
[3] T.-J. Li and J. Shotton, On endomorphism algebras of Gelfand-Graev representations II, preprint (2022)