Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

Let G be a reductive p-adic group and let Rep(G)s Bernstein block in the category of smooth complex G-representations. We investigate structure Rep(G)s, by analysing algebra G-endomorphisms progenerator Π that category. show is “almost” Morita equivalent with (twisted) affine Hecke algebra. This statement made precise several ways, most importantly family graded algebras. It entails that, as fa...

We define a new notion of cuspidality for representations of GLn over a finite quotient ok of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups Gλ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of ...

Let F be a local/global field. Let E and K be quadratic semisimple F–algebras. Let φ and λ be characters/grössencharacters of E× and K×. We define a local/global symbol ( φ λ ) to be essentially the root number of the representation BCK/F (AIE/F (φ))⊗ λ−1. In the spirit of quadratic reciprocity, we prove that ( φ λ )( λ φ ) = φ(−1)λ(−1). We then derive some consequences of this reciprocity for ...

We define a new notion of cuspidality for representations of GLn over a finite quotient ok of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups Gλ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of ...

We construct a class of ℓ $\ell$ -adic local systems on A 1 $\mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These are finite field analogues generalizations classical equation y ′ ( z ) = $y^{\prime \prime }(z)=zy(z)$ . employ geometric Langlands correspondence sought-after as eigenvalues certain rigid Hecke eigensheaves, following methods developed Hei...

We prove that every irreducible, admissible representation π of GSp(4, F ), where F is a non-archimedean local field of characteristic zero, admits a Bessel functional, provided π is not onedimensional. If π is not supercuspidal, we explicitly determine the set of all Bessel functionals admitted by π, and prove that Bessel functionals of a fixed type are unique. If π is supercuspidal, we do the...