Let G be a p-adic connected reductive group with Lie algebra g. For a parabolic subgroup P ⊂ G and a finite-dimensional locally analytic representation V of a Levi subgroup of P, we study the induced locally analytic G-representation W = IndP (V ). Our result is the following criterion concerning the topological irreducibility of W : If the Verma module U(g) U(p) V ′ associated to the dual repr...

An example of a strong invariant of an admissible representation (by which I mean an invariant capable of splitting points in Ĝadm) is the distributional character of a representation. Let G be a reductive Lie group, (π,H) an irreducible admissible representation of G and HK is associated Harish-Chandra (g,K)-module, and let Θπ be its distributional character Θπ (f) ≡ TrH (∫ f (g)π (g) dg ) , ∀...

We study the principal series representations of central extensions of a split reductive algebraic group by a cyclic group of order n. We compute the Plancherel measure of the representation using Eisenstein series and a comparison method. In addition, we construct genuine central characters of the metaplectic torus in the simply-laced case.

The character of an irreducible admissible representation of a p-adic reductive group is known to be a constant function in some neighborhood of any regular semisimple element γ in the group. Under certain mild restrictions on γ, we give an explicit description of a neighborhood of γ on which the character is constant.

— We develop the relative theory of reductive group schemes, using dynamic techniques and algebraic spaces to streamline the original de-

Conjecture 1 (Langlands). To each reductive group G over a number field K, each automorphic (complex) representation π of G, and each finite dimensional representation r of the (complex) group G, there is defined an automorphic L-function L(s, π, r), which enjoys an analytic continuation and functional equation generalizing the Riemann zeta function πΓ(s/2)ζ(s) (or Artin’s L-function L(s, σ), w...

Let G be a connected reductive linear algebraic group over C and let (ρ, V ) be a regular representation of G . There is a locally finite representation (ρ̂,C[V ]) on the affine algebra C[V ] of V defined by ρ̂(g)f(v) = f(g−1v) for f ∈ C[V ] . Since G is reductive, (ρ̂,C[V ]) decomposes as a direct sum of irreducible regular representations of G . The representation (ρ, V ) is said to be multiplic...

Let G be a reductive Lie group and O the coadjoint orbit of a hyperbolic element of g. By π is denoted the unitary irreducible representation of G associated with O by the orbit method. We give geometric interpretations in terms of concepts related to O of the constant π(g), for g ∈ Z(G). In the spirit of the orbit method we interpret geometrically the infinitesimal character of the differentia...