Bernstein center for the category of smooth representations of p-adic groups. In a joint work with Tsao-Hsien Chen (A description of the integral depth-r Bernstein center), we give a description of the integral depth center. A rational depth analogue was treated in our work A description of the depth-r Bernstein center for rational depths, which generalizes several results from the previous paper.
Local Langlands parameters and topics around the local Langlands correspondence. In a joint work with Tsao-Hsien Chen (A description of the depth-r Bernstein center for rational depths), we attach restricted Langlands parameters to smooth irreducible representations, and decompose the category of smooth representations using them. These parameters are conjecturally arise as restrictions of Langlands parameters, and are treated by Chen-Debacker-Tsai in Langlands parameters for Moy-Prasad types.
Geometrization / categorification of the center along the lines of Bezrukavnikov-Kazhdan-Varshavsky.
Categorification of the Bernstein center (with Tsao-Hsien Chen)(in preparation)
Let G be a semisimple simply connected group over an algebraically closed field. We study stable infinity categories of sheaves on the loop group LG and other ind-schemes and ind-stacks, along the lines of Bezrukavnikov-Kazhdan-Varshavsky and define a categorical version of the center and the depth-r center. We prove that the categorical integral depth center is equivalent to a limit of certain parahoric Hecke categories, giving a categorification of the results studied in our previous works. One of the major goals is to develop the geometric results and use affine Springer theory to prove the stability of the elements conjectured to be stable in our earlier works.
Let G be a split connected reductive algebraic group over a non-archimedean local field k . In this paper we give a description of the depth-r Bernstein center of G(k) for rational depths as a limit of depth-r standard parahoric Hecke algebras, extending our previous work in the integral depths case (A description of the integral depth-r Bernstein center). Using this description, we construct maps from the space of stable functions on depth-r Moy-Prasad quotients to the depth-r center, and attach depth-r Deligne-Lusztig parameters to smooth irreducible representations, with the assignment of parameters to irreducible representations shown to be consistent with restricted Langlands parameters for Moy-Prasad types described by Chen-Debacker-Tsai (Langlands parameters for Moy-Prasad types). As an application, we give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth-r Langlands parameters.
We give a description of the depth-r Bernstein center for non-negative integers r for a connected reductive group G over a non-archimedean local field, as a limit of depth-r standard parahoric Hecke algebras. Using this description, we produce elements of the depth-r center by constructing a map to the depth- r center from a certain subalgebra (which we call stable functions) of the algebra of invariant functions on the r-th Moy Prasad filtration quotient of hyperspecial parahorics. We use these elements to attach an invariant to each depth-r irreducible representation, and show that this is equal to the semi-simple part of minimal K-types contained in the irreducible representation.
Statutory warning : Most of these are rough handwritten notes and might contain mistakes. Please email me if you encounter them
Notes from a talk on the étale topology given in a reading seminar on étale cohomology.
Notes on the proof of Springer correspondence and some examples.
Springer correspondence for SP(4).
Buildings and Moy-Prasad filtrations of split padic groups
Constructible derived categories - notes for a talk at a learning seminar on Perverse sheaves
A very short introduction to local class field theory, which I wrote during my Bachelor's thesis reading project under the supervision of Dr. Radhika Ganapathy at the Indian Institute of Science.