Tag - Representation theory
Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to use techniques from local harmonic analysis to compute the asymptotics of a certain twisted orbital integral which counts the number of 𝔽q-points on the ADLV as q goes to infinity. This is joint work with Yihang Zhu.
The seminal work of Deligne and Lusztig on the representations of finite reductive groups has influenced an industry studying parallel constructions in the same theme. In this talk, we will discuss recent progress on studying analogues of Deligne-Lusztig varieties attached to p-adic groups.
A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.
We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic representations. Consequently, any Galois representation attached to these coherent cohomology should be automorphic. Our proof is based on Franke’s work on singular cohomology of locally symmteric spaces and via Faltings' BGG spectral sequence we’ve also strengthened Franke’s result in the Shimura variety case.
We show that the reduction mod p of an orthogonal linear representation is orthogonal.
I will describe applications of a 6-dimensional string theory to the Geometric Langlands Programme and
to the Knot Categorification Programme.
Mumford's geometric invariant theory (GIT) provides a method for constructing quotient varieties for linear actions of reductive groups on projective varieties. The GIT quotient depends on the choice of linearization for the group action, and this dependence was described using 'variation of GIT' (VGIT) by Thaddeus and Dolgachev & Hu in the 1990s. GIT has been extended to non-reductive actions; many of the nice features of classical GIT fail in general, but are satisfied given the extra data of a graded linearization for an action of a linear algebraic group with graded unipotent radical. The aim of this talk is to describe this picture and a version of VGIT which applies to it.
In a recent preprint with Sug Woo Shin I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.
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