Yiannis Fam (London School of Geometry and Number Theory)
Rigidity of depth-zero automorphic data
Abstract:
Over a function field, one can sometimes construct unique automorphic representations with prescribed local factors. A famous example is the Kloosterman automorphic data found by Gross and Heinloth—Ngô--Yun, for which the corresponding automorphic representation has a simple supercuspidal at one place. The latter paper constructed the corresponding Galois representation using techniques from the geometric Langlands correspondence. In this talk I will describe another instance of such rigidity where the simple supercuspidal is replaced by a depth-zero supercuspidal.
Yuta Takaya (University of Tokyo)
Geometry around unramified CM points
Abstract:
Recently, cohomological realization of Yu's construction has been actively studied via positive-depth Deligne-Lusztig varieties. The main cohomological contribution comes from certain subquotients, which we call Heisenberg Deligne-Lusztig varieties. Meanwhile, local Shimura varieties are the objects usually used in the geometric approach to the local Langlands correspondence. In this talk, I will discuss the relationship between these two geometric objects via special affinoid neighborhoods of unramified CM points. Our approach extends the work of Yoshida and Boyarchenko-Weinstein to general local Shimura varieties.
Wille Liu (Academia Sinica)
Fourier transform of invariant functions on a finite reductive Lie algebra
Abstract:
Given a connected reductive group G defined over a finite field Fq of big enough characteristic, we are interested in the Fourier transform of G(Fq)-invariant functions on the rational points of its Lie algebra g, defined as certain trigonometric sums.
In a joint work with Cheng-Chiang Tsai and Wei-Hsuan Hsin, we show that if an invariant function on g and its Fourier transform are both nilpotently supported, then it is an eigenfunction for the Fourier transform with an explicit eigenvalue given by a quadratic Gauss sum.
I'll explain the main ingredients of the proof, namely, lifting the problem to p-adic Lie algebras and reducing the problem to Lie algebras of smaller dimension via endoscopy.
Noriyuki Abe (University of Tokyo)
Supercuspidal blocks and the local Langlands correspondence
Abstract:
The category of smooth representations of a p-adic reductive group decomposes into Bernstein blocks. Each block is equivalent to the category of modules of a certain algebra, which is usually realized as a Hecke algebra. In this talk, I focus on a supercuspidal block. In this case, the relevant algebra is a twisted group algebra. I explain how to capture the Morita equivalence class of this algebra using the theory of Azumaya algebras. I propose a description of this equivalence class via the L-parameter. This description holds for Kaletha's local Langlands correspondence for non-singular supercuspidal representations. This is a joint work with Kazuma Ohara.
Masao Oi (NTU)
On Iwahori fundamental lemma
Abstract:
For a p-adic reductive group G and its twisted endoscopic group H, it is known that test functions on these groups are related through identities between orbital integrals (geometric transfer of Langlands-Shelstad-Kottwitz).
An important example is provided by the celebrated "fundamental lemma", established through the work of Waldspurger and Ngô: the characteristic functions of hyperspecial subgroups of G and H are matching functions. Although the existence of transfer is known in general, explicit descriptions of matching functions (like the fundamental lemma) are only known in very few cases.
In this talk, I will explain that the normalized characteristic function of the Iwahori subgroup of GL(2n) transfers to that of SO(2n+1).