# Taipei Postdoc Seminar

## 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)

## Feb. 22, Simon-Raphaël Fischer (NCTS) Curved Yang-Mills gauge theories Mar. 8, Wille Shih-Wei Liu (AS) Sheaves on nilpotent cones and representation theory

### The Taipei Postdoc Seminar is a weekly seminar that is organized jointly by the Institute of Mathematics of Academia Sinica and the National Center for Theoretical Sciences during the regular academic year. The seminar is held in these two mathematical institutions, with each one of them hosting the event approximately once every two weeks. The National Taiwan University, Academia Sinica, and the NCTS, together employ a significant number of postdoctoral fellows in mathematics from all over the world every year. The diverse scientific and cultural backgrounds of these postdocs create a unique environment of learning, and the goal of the Taipei Postdoc Seminar is to take advantage of this opportunity by bringing together all these postdocs once in a week in an informal-style seminar to enhance discussions and exchange of ideas among the participants. It is our belief that not only will this seminar encourage and eventually lead to more collaboration among mathematicians in Taipei, but it will also give the participants a chance to broaden their own scientific viewpoints. It is our hope that this seminar will greatly enrich the academic experience of the postdoctoral fellows during their stay in Taipei. Organizers: Shih-Hsin Chen (shchen@ncts.ntu.edu.tw), Sz-Sheng Wang (sswangtw@gate.sinica.edu.tw)

Former organisers

## Yu-Yen Chien (NCTS):Sep. 2016~Jun. 2017 Jyun-Ao Lin (AS): Sep. 2016~May 2018 Yang-Kai Lue (NCTS): Sep. 2017~May 2018 Chien-Hsun Wang (NCTS): Sep. 2018~May 2019 Sheng-Fu Chiu (AS): Sep. 2018~June 2020 Jia-Yuan Dai (NCTS): Sep. 2019~June 2020 Wei-Bo Su (AS): Sep. 2020~June 2021 Peng-Jie Wong (NCTS): Sep. 2020~June 2021 Jun-Wen Peng (NCTS): Sep. 2021~June 2022 Yi-Sheng Wang (AS): Sep. 2021~June 2022

Group_Photo_20170628 HERE