Distinction of some induced representations

Editorial

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.

We study the continuity of archimedean zeta integrals, as well as the lower bound of their operator norms, with respect to a natural family of Sobolev norms arising from the functional calculus of harmonic oscillator. The relevant notion and results will be generalized to archimedean Rankin–Selberg integrals of the tempered principal series representations of general linear groups.

Let $\mathrm{B}_n$ denote the set of complex square matrices of order $n$ whose Euclidean operator norms are less than one. Its Shilov boundary is the set $\operatorname{U}(n)$ of all unitary matrices. A holomorphic map $\mathrm{B}_m\to\mathrm{B}_n$ is inner if it sends $\operatorname{U}(m)$ to $\operatorname{U}(n)$. On the other hand we consider the group $\operatorname{U}(n+mj)$ and its subgroup $\operatorname{U}(j)$ that is embedded in $\operatorname{U}(n+mj)$ in the block-diagonal way ($m$ blocks $\operatorname{U}(j)$ and a unit block of size $n$). To any conjugacy class of $\operatorname{U}(n+mj)$ with respect to $\operatorname{U}(j)$ we assign a ‘characteristic function’, which is a rational inner map $\mathrm{B}_m\to\mathrm{B}_n$. We show that the class of inner functions that can be obtained as ‘characteristic functions’ is closed with respect to such natural operations as pointwise direct sums, pointwise products, compositions, substitutions into finite-dimensional representations of general linear groups and so on. We also describe explicitly the corresponding operations on conjugacy classes. Bibliography: 24 titles.

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.

We assume that K is an infinite field, so that there is no difference between formal polynomials and the functions that they give rise to.

Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur–Weyl duality is the key for an equivalence between both categories.

Probabilistic Galois theory over p -adic fields

A theorem of Cassidy states that any Zariski closed differential algebraic subgroup of a simple linear algebraic group, defined over a differential field, is either the whole group or is conjugate to the subgroup of constant matrices. An arithmetic analogue of this theorem is proved in which usual derivations on fields are replaced by certain nonlinear operators on p -adic rings, called " p -derivations."

We consider the special linear group G = SL 2 over a p -adic field, and its diagonal torus M = GL 1 .Parabolic induction of representations from M to G induces a map in equivariant homology, from the Bruhat-Tits building of M to that of G .We compute this map at the level of chain complexes, and show that it is given by parahoric induction (as defined by J.-F.Dat).

We consider stacks of filtered$\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize$p$-adic Galois representations of the absolute Galois group of a$p$-adic field with varying coefficients over an open substack containing all classical points. Further, we study a period morphism (defined by Pappas and Rapoport) from a stack parameterizing integral data, and determine the image of this morphism.

In this chapter, we will discuss the representation theory of GL (2) over a p -adic field, a cornerstone of the modern theory of automorphic forms, emphasizing techniques that are applicable to GL ( n ). We begin by looking at the representation theory of GL (2) over a finite field, where we encounter an essential tool, Mackey theory, which is the calculus of intertwining operators between induced representations. We will see that the irreducible representations of G = GL (2, F ) when F is finite are roughly parametrized by the characters of maximal tori in G . The representations parametrized by maximal split tori are induced representations, those parametrized by nonsplit tori must be constructed by some other method. A convenient method of accomplishing this is afforded by the Weil representation, which we study in detail. In the rest of the chapter, we will adapt the results of Section 4.1 to representations of GL (2, F ) where F is local. In Section 4.2, we introduce the categories of smooth and admissible representations and establish their basic properties. In Section 4.3, we introduce some tools, sheaves, and distributions, which are needed to extend Mackey theory to locally compact groups, as in Bruhat's thesis (1956, 1961). We follow Bernstein and Zelevinsky (1976) in emphasizing these tools. In Section 4.4, we prove the uniqueness of Whittaker models, a fundamental result that was applied in Section 3.5 to the multiplicity one theorem for automorphic forms and to the construction of L-functions. We also introduce the Jacquet module, a functor adjoint to parabolic induction, and establish its basic properties.

Let $G$ be a general linear group over a $p$-adic field. It is well known that Bernstein components of the category of smooth representations of $G$ are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of $G$ by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.

Assuming that the differential field [Formula: see text] is differentially large, in the sense of [León Sánchez and Tressl, Differentially large fields, preprint (2020); arXiv:2005.00888], and “bounded” as a field, we prove that for any linear differential algebraic group [Formula: see text] over [Formula: see text], the differential Galois (or constrained) cohomology set [Formula: see text] is finite. This applies, among other things, to closed ordered differential fields in the sense of [Singer, The model theory of ordered differential fields, J. Symb. Logic 43(1) (1978) 82–91], and to closed[Formula: see text]-adic differential fields in the sense of [Tressl, The uniform companion for large differential fields of characteristic [Formula: see text], Trans. Amer. Math. Soc. 357(10) (2005) 3933–3951]. As an application, we prove a general existence result for parameterized Picard–Vessiot (PPV) extensions within certain families of fields; if [Formula: see text] is a field with two commuting derivations, and [Formula: see text] is a parameterized linear differential equation over [Formula: see text], and [Formula: see text] is “differentially large” and [Formula: see text] is bounded, and [Formula: see text] is existentially closed in [Formula: see text], then there is a PPV extension [Formula: see text] of [Formula: see text] for the equation such that [Formula: see text] is existentially closed in [Formula: see text]. For instance, it follows that if the [Formula: see text]-constants of a formally real differential field [Formula: see text] is a closed ordered[Formula: see text]-field, then for any homogeneous linear [Formula: see text]-equation over [Formula: see text] there exists a PPV extension that is formally real. Similar observations apply to [Formula: see text]-adic fields.

Let E/F be a quadratic extension of p–adic fields and let d, m be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations π and σ of GLd(E) and GLm(E) respectively. We assume that π and σ are conjugate–dual. That is to say and where c is the nontrivial F–automorphism of E. This implies that we can extend π to an unitary representation π of a nonconnected group GLd(E) . Define the same way. We state and prove an integral formula for involving the characters of and . ˜˜This formula is related to the local Gan–Gross–Prasad conjecture for unitary groups.