Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

Free group representations from vector-valued multiplicative functions,I

Let Γ denote a noncommutative free group and let Ω stand for its boundary. We construct a large class of unitary representations of Γ. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of Γ ⋉_λ C(Ω). We prove here that each of them is irreducible as a representation of Γ ⋉_λ C(Ω). Actually, as will be shown in further work, each of them is irreducible as a representation of Γ, or is the direct sum of exactly two irreducible, inequivalent Γ-representations. This research was supported by the Italian CNR.     

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000     

Low-dimensional representations of Aut (F 2)

LetF ₂ be the free group of rank two, and Φ₂ its automorphism group. We consider the problem of describing the representations of Φ₂ of degreen for small values ofn. Our main result is the classification (up to equivalence) of all indecomposable representations ρ of Φ₂ of degreen≤4 such that ρ(F ₂) ≠ 1. There are only finitely many such representations, and in all them ρ(F ₂) is solvable. This is no longer true in higher dimensions. Already forn=6 there exists a 1-parameter family of irreducible nonequivalent representations of Φ₂ such that ρ(F ₂) contains a free non-abelian subgroup. We also obtain some new 4-dimensional representations of the braid groupB ₄ which are indecomposable and reducible at the same time. It would be interesting to find some applications of these representations. Supported in part by the NSERC Grant A-5285 Supported in part by an NSERC grant     

Polynomials with a common composite

We develop a conditional entropy theory for infinite measure preserving actions of countable discrete amenable groups with respect to a σ-finite factor. This includes ‘infinite’ analogues of relative Kolmogorov-Sinai, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjointness, Furstenberg decomposition and disjointness theorems, etc. In case of ℤ-action, our concept of relative entropy matches well the ‘absolute’ entropy h _Kr introduced by Krengel. Answering in part his question and a question of Silva and Thieullen, we show that for any non-distal transformation S there exists an infinite measure preserving transformation T with h _Kr(T × S) = ∞ but h _Kr(T) = 0. This project was supported in part by a CRDF grant UM1-2546-KH-03.     

Langlands-shahidi method and poles of automorphicL-functions II

We use Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations, to study the Rankin-SelbergL-functions of GL _k × classical groups. Especially we prove thatL(s, σ ×τ) is holomorphic, except possibly ats=0, 1/2, 1, whereσ is a cuspidal representation of GL _k which satisfies weak Ramanujan property in the sense of Cogdell and Piatetski-Shapiro andτ is any generic cuspidal representation of SO_2l+1. Also we study the twisted symmetric cubeL-functions, twisted by cuspidal representations of GL₂. Partially supported by NSF grant DMS9610387.     

On the stable derivation algebra associated with some braid groups

We shall prove some stability property of the graded Lie algebraD _n of certain derivations associated with pure sphere braid group onn strings; in fact, thatD _n forn≥6. These Lie algebrasD _n are connected with some bigl-adic Galois representations, and the stability property is related to some conjecture of Grothendieck.     

Commensurations of Out(F<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

Let Out(F _n ) denote the outer automorphism group of the free group F _n with n>3. We prove that for any finite index subgroup Γ<Out(F _n ), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(F _n ). We prove that Γ is co-Hopfian: every injective homomorphism Γ→Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(F _n )) is isomorphic to Out(F _n ).     

Equivalence of commutation relations

We consider C*-algebras of commutation relations over the fields ℚ _p, p = 2, 3, 5, …, ∞. We describe all the irreducible separable representations of these algebras. We prove that the algebras are not isomorphic at different p. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 406–412, December, 2008.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

Weak amenability of CAT(0)-cubical groups

We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations π _z for which the quantities z ^{ℓ (g)} are matrix coefficients. Here ℓ is a length function on G obtained from the combinatorial distance function on the complex X.     

On isomorphity of measure-preserving ℤ<Superscript>2</Superscript>-actions that have isomorphic Cartesian powers

Assume that Δ and Π are representations of the group ℤ² by operators on the space L ₂(X, μ) that are induced by measure-preserving automorphisms, and for some d, the representations Δ^⨂d and Π^⨂d are conjugate to each other, Δ(ℤ² \(0, 0)) consists of weakly mixing operators, and there is a weak limit (over some subsequence in ℤ² of operators from Δ(ℤ²)) which is equal to a nontrivial, convex linear combination of elements of Δ(ℤ²) and of the projection onto constant functions. We prove that in this case, Δ and Π are also conjugate to each other. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 193–212, 2007.     

The centralizer algebra of the diagonal action of the group <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(ℂ) in a mixed tensor space

We consider the walled Brauer algebra Br _{k, l}(n) introduced by V. Turaev and K. Koike. We prove that it is a subalgebra of the Brauer algebra and that it is isomorphic, for sufficiently large n ∈ ℕ, to the centralizer algebra of the diagonal action of the group GL_n(ℂ) in a mixed tensor space. We also give the presentation of the algebra Br _{k, l}(n) by generators and relations. For a generic value of the parameter, the algebra is semisimple, and in this case we describe the Bratteli diagram for this family of algebras and give realizations for the irreducible representations. We also give a new, more natural proof of the formulas for the characters of the walled Brauer algebras. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 170–198.     

Simple tensor products

Let ℱ be the category of finite-dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S ₁ ⊗⋅⋅⋅⊗ S _N of simple objects of ℱ is simple if and only S _i ⊗ S _j is simple for any i<j.     

A congruence for the Fourier coefficients of a modular form and its application to quadratic forms

Let F(z)=∑ _n=1^∞ A(n)q ⁿ denote the unique weight 6 normalized cuspidal eigenform on Γ₀(4). We prove that A(p)≡0,2,−1(mod 11) when p≠11 is a prime. We then use this congruence to give an application to the number of representations of an integer by quadratic form of level 4.      

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

Residual properties and almost equicontinuity

A propertyP of a compact dynamical system (X,f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all propertyP systems is a residual property providedP is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from allP systems iff the onlyP systems upon which Λ acts are trivial. We use this to prove that every monothetic group has an action which is weak mixing and topologically ergodic.     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

Sous-groupes discrets de <Emphasis Type="Italic">PU</Emphasis>(2,1) engendrés par <Emphasis Type="Italic">n</Emphasis> réflexions complexes et Déformation

Let PU(2,1) be the group of holomorphic isometries in the hyperbolic complex plane and let G _n be a sub-group of PU(2,1) which is generated by n complex reflections with respect to complex lines in . Under certain conditions, we prove that G _n is discrete. We construct representations ρ of the fundamental group Γ _g of the compact surface Σ _g of genus g, into PU(2,1), we prove they are discrete, faithful and we compute the dimension their deformation space.