Related papers: Shtukas for reductive groups and Langlands corresp…
Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic. We study a generalization of the notion of G-complete reducibility in the context of Steinberg endomorphisms of G. Our…
Let $G/H$ be a reductive symmetric space of split rank $1$ and let $K$ be a maximal compact subgroup of $G$. In a previous article the first two authors introduced a notion of cusp forms for $G/H$. We show that the space of cusp forms…
We give cases in which nearby cycles commutes with pushforward from sheaves on the moduli stack of shtukas to a product of curves over a finite field. The proof systematically uses the property that taking nearby cycles of Satake sheaves on…
These notes present an application of the geometric Satake equivalence to the description of characters of indecomposable tilting modules for reductive algebraic groups over fields of positive characteristic, obtained in joint work with G.…
The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant…
We provide a motivated introduction to the theory of categorical actions of groups and the local geometric Langlands program. Along the way we emphasize applications, old and new, to the usual representation theory of reductive and affine…
Motivated by the Langlands' beyond endoscopy proposal for establishing functoriality, we study the representation $\otimes^3$ in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic…
Every Lie group $G$ carries a distinguished algebra of particularly well-behaved real-analytic mappings: The entire functions $\mathcal{E}(G)$. They were introduced for the purposes of strict deformation quantization. This paper establishes…
We classify the linearly reductive finite subgroup schemes $G$ of $SL_2=SL(V)$ over an algebraically closed field $k$ of positive characteristic, up to conjugation. As a corollary, we prove that such $G$ is in one-to-one correspondence with…
We define a new cohomology set for an affine algebraic group G and a multiplicative finite central subgroup Z, both defined over a local field of characteristic zero, which is an enlargement of the usual first Galois cohomology set of G. We…
On a suitable category of formal schemes equipped with codimension functions we construct a canonical pseudofunctor (-)^# taking values in the corresponding categories of Cousin complexes. Cousin complexes on such a formal scheme X…
We study the endomorphism algebras attached to Bernstein components of reductive $p$-adic groups and construct a local Langlands correspondence with the appropriate set of enhanced $L$-parameters, using certain "desiderata" properties for…
We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the…
We study the local Langlands functoriality transfer from $\text{SO}(5, F)$ to $\text{GL}(4, F)$ for arbitrary twists of several families of irreducible supercuspidal representations of $\text{GL}(4, F)$, where $F$ is a non-archimedean local…
The Langlands functoriality conjecture envisaged in the bisemialgebra framework is proved to correspond to the nonorthogonal completely reducible cuspidal representations of the bilinear algebraic semigroups.
We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the…
This article deals with the tamely ramified geometric Langlands correspondence for GL_2 on $\mathbf{P}_{\mathbf{F}_q}^1$, where $q$ is a prime power, with tame ramification at four distinct points $D = \{\infty, 0,1, t\} \subset…
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…
This paper studies spherical Whittaker functions for central extensions of reductive groups over local fields. We follow the development of Chinta and Offen to produce a metaplectic Casselman-Shalika formula for tame covers of all…
We develop the analog of crystalline Dieudonn\'e theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonn\'e modules are replaced by…