#### ** Upcoming Talk**

**Mar. 29 (Wed.)**** 14:00 - 15:00**
李自然 Tzu-Jan Li (本所 Academia Sinica)

Venue: Room 515, Cosmology Building (NTU Campus)

Title: *On the reducedness of a ring from the invariant theory*

Abstract:

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)