For learning about the Langlands program, knowledge of Lie-group structure theory, algebraic number theory, algebraic geometry, modular forms, and infinite-dimensional representation theory is appropriate. This article describes how one can get a glimpse of the program with less background than this.

This article is an introduction to automorphic forms on the adeles of a linear reductive group over a number field. The first half is a summary of aspects of local and global class field theory, with emphasis on the local Weil group, the L functions of Artin and Hecke, and the role of Artin reciprocity in relating the two kinds of L functions. The first half serves as background for the second ...

Suppose that G is a connected reductive group over a p-adic field F , that K is a hyperspecial maximal compact subgroup of G(F ), and that V is an irreducible representation of K over the algebraic closure of the residue field of F . We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported, Kbiequivariant functions f : G(F ) EndV . These Hecke algebras wer...

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = SO2m, G = Sp2n over X with H split. Write BunG and BunH for the stacks of G-torsors and H-torsors on X . The theta-kernel AutG,H on BunG ×BunH yields the theta-lifting functors FG : D(BunH) → D(BunG) and FH : D(BunG) → D(BunH) between the corresponding derived categories. Assu...

We prove the compatibility at places dividing l of the local and global Langlands correspondences for the l-adic Galois representations associated to regular algebraic essentially (conjugate) self-dual cuspidal automorphic representations of GLn over an imaginary CM or totally real field. We prove this compatibility up to semisimplification in all cases, and up to Frobenius semisimplification i...

Let F be a p-adic field of characteristic 0, and let A be an F -central division algebra of dimension dA over F . In the paper we first develop the representation theory of GL(m,A), assuming that holds the conjecture which claims that unitary parabolic induction is irreducible for GL(m,A)’s. Among others, we obtain the formula for characters of irreducible unitary representations of GL(m,A) in ...

In the setting of geometric Langlands conjecture, we argue that phenomenon divergence at infinity on Bun_G (that is, difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by locus semisimple $\check{G}$-local systems. To see this, first rephrase question in terms Deligne-Lusztig duality then study functor DL_G^\spec acting spectral DG category IndCoh_N(LS_G). We...

We analyze the geometry of the tower of LubinTate deformation spaces, which parametrize deformations of a onedimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their l-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working...

An algebraic extended bilinear Hilbert semispace H∓ is proposed as being the natural representation space for the algebras of von Neumann. This bilinear Hilbert semispace has a well defined structure given by the representation space Repsp(GLn(Lv ×Lv)) of an algebraic general bilinear semigroup GLn(Lv ×Lv) over the product of sets of completions characterized by increasing ranks. This represent...