Banach Representations Research Articles

Topological group actions by group automorphisms and Banach representations

Abstract We study Banach representability for actions of topological groups on groups by automorphisms (in particular, an action of a group on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL 2 ⁡ ( ℝ ) {\operatorname{SL}_{2}(\mathbb{R})} . The conjugation action of SL n ⁡ ( ℝ ) {\operatorname{SL}_{n}(\mathbb{R})} is not Asplund representable for every n ≥ 4 {n\geq 4} . The linear action of GL n ⁡ ( ℝ ) {\operatorname{GL}_{n}(\mathbb{R})} on ℝ n {{\mathbb{R}}^{n}} , for every n ≥ 2 {n\geq 2} , is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l 1 {l_{1}} ). The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). As a byproduct, we obtain some counterexamples about Banach representations of homogeneous G-actions G / H {G/H} .

On the vanishing of reduced 1-cohomology for Banach representations

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Pr\"ufer rank. Moreover all our results apply to isometric representations in a large class of Banach spaces, including reflexive Banach spaces. As applications, we obtain an ergodic theorem in for integrable cocycles, as well as a new proof of Bourgain's Theorem that the 3-regular tree does not embed quasi-isometrically into any superreflexive Banach space.

Schneider-Teitelbaum duality for locally profinite groups

We define monoidal structures on several categories of linear topological modules over the valuation ring of a local field, and study module theory with respect to the monoidal structures. We extend the notion of the Iwasawa algebra to a locally profinite group as a monoid with respect to one of the monoidal structure, which does not necessarily form a topological algebra. This is one of the main reasons why we need monoidal structures. We extend Schneider--Teitelbaum duality to duality applicable to a locally profinite group through the module theory over the generalised Iwasawa algebra, and give a criterion of the irreducibility of a unitary Banach representation.

Duality theory of $p$-adic Hopf algebras

We show the monoidal functoriality of Schikhof duality, and cultivate new duality theory of $p$-adic Hopf algebras. Through the duality, we introduce two sorts of $p$-adic Pontryagin dualities. One is a duality between discrete Abelian groups and affine formal group schemes of specific type, and the other one is a duality between profinite Abelian groups and analytic groups of specific type. We extend Amice transform to a $p$-adic Fourier transform compatible with the second $p$-adic Pontryagin duality. As applications, we give explicit presentations of a universal family of irreducible $p$-adic unitary Banach representations of the open unit disc of the general linear group and its $q$-deformation in the case of dimension $2$.

On Sobolev norms for Lie group representations

We define Sobolev norms of arbitrary real order for a Banach representation (π,E) of a Lie group, with regard to a single differential operator D=dπ(R2+Δ). Here, Δ is a Laplace element in the universal enveloping algebra, and R>0 depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for D on the space of smooth vectors of E. The main tool is a novel factorization of the delta distribution on a Lie group.

The module multiplier embedding problem

We precise properties, embedding and isomorphism theorems of Banach module multipliers between a Banach algebra and modules on it. This task for Banach dual valued multipliers is a problem of interest because of its connection with the theory of induced Banach representations of Banach algebras and Frobenius reciprocity theorems.

DOMAINS OF HOLOMORPHY FOR IRREDUCIBLE ADMISSIBLE UNIFORMLY BOUNDED BANACH REPRESENTATIONS OF SIMPLE LIE GROUPS

In this note, we address a question raised by B. Krotz on the classification of G-invariant domains of holomorphy for irreducible admissible Banach representations of connected non-compact simple real linear Lie groups G. When G is not of Hermitian type, we give a complete description of such G-invariant domains for irreducible admissible uniformly bounded representations on reflexive Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine such G-invariant domains only when the representations considered are highest or lowest weight representations.

First covering of the Drinfel’d upper half-plane and Banach representations of GL2(ℚp)

We construct some admissible Banach representations of GL_2(Q_p) that conjecturally should correspond to some 2-dimensional tamely ramified, potentially Barsotti-Tate representations of G_{Q_p} via the p-adic local Langlands correspondence. To achieve this, we generalize Breuil's work in the semi-stable case and work on the first covering of Drinfel'd upper half plane. Our main tool is an explicit semi-stable model of the first covering.

Vanishing at infinity on homogeneous spaces of reductive type

Let $G$ be a real reductive group and $Z=G/H$ a unimodular homogeneous $G$ space. The space $Z$ is said to satisfy VAI (vanishing at infinity) if all smooth vectors in the Banach representations $L^{p}(Z)$ vanish at infinity, $1\leqslant p<\infty$. For $H$ connected we show that $Z$ satisfies VAI if and only if it is of reductive type.

Quasi-Regular Representations and Rapid Decay

We study property RD in terms of rapid decay of matrix coefficients. We give new formulations of property RD in terms of a L1-integrability condition of a Banach representation. Combining this new definition with the existence of cyclic subgroups of exponential growth in non-uniform lattices in semisimple Lie groups, we deduce that there exist matrix coefficients associated to several kinds of quasi-regular representations which satisfy a “non-RD condition” for non-uniform lattices. We obtain also that such coefficients can not satisfy the weak inequality of Harish-Chandra.

Borel–Weil Theory for Root Graded Banach–Lie Groups

In this paper we introduce (weakly) root graded Banach‐Lie algebras and corresponding Lie groups as natural generalizations of group like GLn(A) for a Banach algebra A or groups like C(X,K) of continuous maps of a compact space X into a complex semisimple Lie group K. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups P to representations of G on holomorphic sections of homogeneous vector bundles over G/P. One of our main results is an algebraic characterization of the space of sections which is used to show that this space actually carries a natural Banach structure, a result generalizing the finite dimensionality of spaces of sections of holomorphic bundles over compact complex manifolds. We also give a geometric realization of any irreducible holomorphic representation of a (weakly) root graded Banach‐Lie group G and show that all holomorphic functions on the spaces G/P are constant.

The spectral theorem of Banach representation of compact space generated by generalized shift operators

The paper is devoted to description a version of the classical Peter-Weyl theory in the case of generalized shift operators generated by a compact topological space.

Mellin Transforms of Whittaker Functions

AbstractIn this note we show that for an arbitrary reductive Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles and show that they always lie along translates of walls of Weyl chambers.

On the extensions of infinite‐dimensional representations of Lie semigroups

The necessary and sufficient conditions have been obtained for extendability of a Banach representation of a generating Lie semigroup S to a local representation of the Lie group G generated by S when the tangent wedge of S is a Lie semialgebra. The most convenient conditions we obtain correspond to the case of unitary representations. In this case, we give a criterion of global extendability if G is exponential and solvable.

On the relationship between the strong and scalar almost periodicity of Banach representations of the group of reals

In general, the above notions make sense for an arbitrary topological group Go (almost periodic representations of the group Go ; see [1]), but in this note we restrict ourselves to the case Go = R. One of the reasons for this restriction is the absence of a theorem stating that the spectrum of a scalar a.p. function on a group distinct from R is at most countable (see [4]). The following assertion is due to Yu. I. Lyubich: if X is weakly sequentially complete, then each scalar a.p. group acting on X is a.p. However, if X = c (c is the Banach space of all convergent number sequences), then there exists a scalar a.p. group that is not a.p. The following natural question in the theory of functions with ranges in Banach spaces arises: what is the maximal class of Banach spaces for which the affirmative part of the Lyubich assertion holds? This note gives a partial answer to this question.

Unbounded representations of symmetry groups in gauge quantum field theory. II. Integration

Within the gauge quantum field theory of the Wightman–Gårding type, the integration of representations of Lie algebras is investigated. By means of the covariance condition (substitution rules) for the basic fields, it is shown that a form skew-symmetric representation of a Lie algebra can be integrated to a form isometric and in general unbounded representation of the universal covering group of a corresponding Lie group provided the conditions (Nelson, Sternheimer, etc.), which are well known for the case of Hilbert or Banach representations, hold. If a form isometric representation leaves the subspace from which the physical Hilbert space is obtained via factorization and completion invariant, then the same is proved to be true for its differential. Conversely, a necessary and sufficient condition is derived for the transmission of the invariance of this subspace under a form skew-symmetric representation of a Lie algebra to its integral.

Local coefficients as Artin factors for real groups

Introduction. The purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi-split real reductive algebraic groups in [27] with their corresponding Artin factors attached by local class field theory [21]. As a consequence, we establish an identity satisfied by certain normalized intertwining operators. It seems to be useful in applications of the trace formula [1, 29]. More precisely, let G be the group of real points of a quasi-split reductive algebraic group over R. Let A be the set of simple roots defined by a fixed minimal parabolic subgroup Po = M0A0U of G. Fix 6 A, and let P = P9 be the corresponding standard parabolic subgroup of G and write P = MAN for its Langlands decomposition. Fix a nondegenerate character x °f U i ^ (a, H{o)) be an irreducible admissible x-gic Banach (in particular x~generic unitary) representation of M (cf. Section 1). Given v G a£, the complex dual of the Lie algebra of A, let I(v,o,0) be the continuously (quasi-unitarily, if o is unitary) induced representation IndP^Ga® e v 9 and let V(v,o,0) be its space (Section 0). Then F ^ a , ^ = Vfoo^O). Now, let W be the Weyl group of Ao in G. Choose w G W such that w(0) A. Let N~ be the unipotent group opposite to N. Define N$ = U C wN~w~ where w is a representative of w in G. For/ G F(^,a,^)00, define

A note on Banach representations of Moore Groups

A note on Banach representations of Moore Groups

Integral operators in the theory of induced Banach representation II. The bundle approach

Let G be a locally compact group, H a closed subgroup and L a Banach representation of H. Suppose U is a Banach representation of G which is induced by L. Here, we continue our program of showing that certain operators of the integrated form of U can be written as integral operators with continuous kernels. Specifically, we show that: (1) the representation space of a Banach bundle; (2) the above operators become integral operators on this space with kernels which are continuous cross‐sections of an associated kernel bundle.

Irreducible Banach representations of locally compact groups of a certain type

Irreducible Banach representations of locally compact groups of a certain type