On unitary deformations of smooth modular representations

Abstract

Abstract. Let G be a locally Qp-analytic group and K a finite extension of Qp with residue field k. Adapting a strategy of B. Mazur (cf.[Maz89]) we use deformation theory to study the possible liftings of a given smooth G-representation ρ over k to unitary G-Banach space representations over K. The main result proves the existence of a universal deformation space in case ρ admits only scalar endomorphisms. As an application we let G = GL 2 (Qp) and compute the fibers of the reduction map in principal series representations.

Deformation rings and parabolic induction

“…We generalize the definitions of [Eme10a, § 2.1] (which only considers Noetherian A). Our definitions coincide with those of [Sch13] (in terms of pseudocompact objects) since the residue field k is finite.…”

“…Thus π is a smooth A[G]-module, free over A, endowed with an isomorphism π ⊗ A k ∼ −→ π. Assuming End k [G] (π) = k, the functor Def π is known to be pro-representable, as recently shown by one of us ([Sch13]). To allow more flexibility one actually deforms the Pontrjagin dual π∨ := Hom k (π, k) which lives in a category of profinite augmented representations.…”

“…We say that V is unitary if its topology can be defined by a G-invariant norm; we write Ban G (E) unit for the full subcategory consisting of unitary representations. Following [Sch13,§ 6.1] we denote by Ban G (E) ≤1 the category whose objects are E-Banach representations (V, • ) of G for which V ⊂ |E| and gv = v for all g ∈ G and v ∈ V ; the morphisms are the E[G]-linear norm-decreasing maps. Passing to the isogeny category gives an equivalence (cf.…”

“…) We define a covariant functor Def N : Pro(O) → Cat by letting Def N (A) be the essentially small category of lifts of N over A for any A ∈ Pro(O), and Def N (ϕ) : Def N (A) → Def N (A ) be the base change functor for any morphism ϕ : A → A in Pro(O). (3) We define the deformation functor Def N : Pro(O) → Set as the composite i • Def N .The following is one of the main results from[Sch13].Theorem 4.10 (T.S.). Suppose End cont k[G] (N ) = k. Then: (1) Def N is representable: There exists a universal deformation ring R N ∈ Pro(O) along with bijections Def N…”

Functorial properties of generalised Steinberg representations

“…We let χ : G → O × denote the continuous character obtained by composing χ with the canonical lifting k × → O × . We define a continuous character χ univ : G → Λ × by setting χ univ (g) := χ(g)λ(g) for all g ∈ G. Proceeding as in the proof of[Sch13, Proposition 3.11] (here G ab need not be topologically finitely generated because H is required to be open in the definition of Λ), we see that Λ is the universal deformation ring of χ and χ univ is the universal deformation of χ.Corollary 15. The universal deformation ring of St G Q( χ) is Λ.…”