Bernstein Decomposition Research Articles

On restriction of unitarizable representations of general linear groups and the non-generic local Gan–Gross–Prasad conjecture

We prove the first direction of a recently posed conjecture by Gan, Gross and Prasad, which predicts branching laws that govern restriction from p -adic GL_n to GL_{n-1} of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur–Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid–Mínguez criterion from the p -adic setting.

Hecke algebras for $${{\rm GL}_n}$$ GL n over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).

On semisimple ℓ -modular Bernstein-blocks of a p -adic general linear group

Let G n = GL n ( F ) , where F is a non-archimedean local field with residue characteristic p . Our starting point is the Bernstein decomposition of the representation category of G n over an algebraically closed field of positive characteristic ℓ ≠ p into blocks. In level zero, we associate to each block a replacement for the Iwahori–Hecke algebra which provides a Morita equivalence as in the complex case. Additionally, we explain how this gives rise to a description of an arbitrary G n -block in terms of simple G m -blocks (for m ⩽ n ), parallelling the approach of Bushnell and Kutzko in the complex setting.

Covers for Self-dual Supercuspidal Representations of the Siegel Levi Subgroup of Classical p-adic Groups

We study components of the Bernstein decomposition for a p-adic classical group (with p odd) with inertial support a self-dual positive level supercuspidal representation of a Siegel Levi subgroup. More precisely, we use the method of covers to construct a Bushnell-Kutzko type for such a component. A detailed knowledge of the Hecke algebra of the type should have number-theoretic implications.

Hecke algebras and ${\rm SL}_n$-types

Let $F$ be a non-Archimedean local field. In an earlier paper, we constructed certain types (in the sense of Bushnell and Kutzko) in $SL_n (F)$, in fact enough to describe the non-supercuspidal components of the Bernstein decomposition of this group. We now give an almost fully explicit presentation of the Hecke algebras of these types.

Explicit Plancherel formula for the p-adic group GL( n)

We provide an explicit Plancherel formula for the p-adic group GL( n). We determine explicitly the Bernstein decomposition of Plancherel measure, including all numerical constants. We also prove a transfer-of-measure formula for GL( n). To cite this article: A.-M. Aubert, R. Plymen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Reduced C*-Algebra of the p-Adic Group GL( n) II

The reduced C *-algebra of the p-adic group GL( n) admits a Bernstein decomposition. We give a minimal refinement of this decomposition, and provide structure theorems for the reduced Iwahori–Hecke C *-algebra and the reduced spherical C *-algebra. This leads to a very explicit description of the tempered dual of GL( n) in terms of Bernstein parameters and extended quotients. We also prove that Plancherel measure (on the tempered dual of a reductive p-adic group) is rotation-invariant.

TYPES IN SLn

Let $G = SL_n(F)$ where $F$ is a non-Archimedean local field. This paper concerns the smooth (complex) representation theory of $G$, specifically, the construction of types in $G$, in the sense of Bushnell and Kutzko. The main result describes a type for each non-supercuspidal component of the Bernstein decomposition of $G$. If this is combined with earlier work of Bushnell and Kutzko, it follows that $G$ admits a complete set of types. 2000 Mathematical Subject Classification: 22E50.

Parabolic Induction and the Bernstein Decomposition

We use some fundamental work of Bernstein to study parabolic induction in reductive p-adic groups. In particular, we determine when parabolic induction from a component of the Bernstein decomposition of a Levi subgroup to the corresponding component of the full group is an equivalence of categories.