Related papers: The stack of spherical Langlands parameters

For a split reductive group ${}^L G$ over a global field, we determine the spectrum of the spherical Hecke algebra coming from the unramified Eisenstein series for the minimal parabolic ${}^L B$. This is done using a certain decomposition…

We define Langlands parameters for connected reductive groups over finite fields and formulate the Langlands correspondence for finite fields using these parameters.

Let $F/\mathbb{Q}_p$ be finite and let $\mathfrak{X}_G$ be the moduli space of Langlands parameters valued in $G$, in characteristic distinct from $p$. First, we determine the irreducible components of $\mathfrak{X}_G$. Then, we determine…

We associate to every irreducible representation of a reductive group over a local field of equal characteristics a local Langlands parameter up to semisimplification and prove the compatibility with the global parameterization constructed…

Suppose $G$ is a tamely ramified $p$-adic reductive group. We construct a partial local Langlands correspondence between the set of irreducible smooth representations of $G$ having depth $r$ and a certain set of $G^\vee$-conjugacy classes…

For $E/F$ quadratic extension of local fields and $G$ a reductive algebraic group over $F$, the paper formulates a conjecture classifying irreducible admissible representations of $G(E)$ which carry a $G(F)$ invariant linear form, and the…

We prove that, over any elliptic global Langlands parameter $\sigma$, the cuspidal cohomology groups of moduli stacks of shtukas are given by a formula involving a finite dimensional representation of the centralizer of $\sigma$. It is a…

Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of…

Let $\pi$ be a simple supercuspidal representation of the split even special orthogonal group. We compute the Rankin-Selberg $\gamma$-factors for rank 1-twists of $\pi$ by quadratic tamely ramified characters of $F^*$. We then use our…

The geometric Langlands program is distinguished in assigning spectral decompositions to all representations, not only the irreducible ones. However, it is not even clear what is meant by a spectral decomposition when one works with…

Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks…

By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $\pi$ of…

Let G be a connected reductive group over a non-archimedean local field K, and assume that G splits over an unramified extension of K. We establish a local Langlands correspondence for irreducible unipotent representations of G. It comes as…

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in…

We will give an explicit construction of irreducible suparcuspidal representations of the special linear group over a non-archimedean local field and will speculate its Langlands parameter by means of verifying the Hiraga-Ichino-Ikeda…

Fix a smooth, complete algebraic curve $X$ over an algebraically closed field $k$ of characteristic zero. To a reductive group $G$ over $k$, we associate an algebraic stack $\operatorname{Par}_G$ of quantum parameters for the geometric…

We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility…

We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a…

We study the structure constants defining two related rings: the spherical Hecke algebra of a split connected reductive group over a non-Archimedean local field, and the representation ring of the Langlands dual group.

We define a new geometric object--the stack of local systems with restricted variation. We formulate a version of the categorical geometric Langlands conjecture that makes sense for any constructible sheaf theory (such as l-adic sheaves).…