Representation Rho Research Articles

Rodier type theorem for generalized principal series

Given a regular supercuspidal representation rho of the Levi subgroup M of a standard parabolic subgroup P=MN in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set JH(Ind^G_P(rho )) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation Ind^G_P(rho ), vastly generalizing Rodier structure theorem for P=B=TU Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group W_M=N_G(M)/M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group W_T=N_G(T)/T is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split (G)Sp_{2n} and SO_{2n+1}, and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.

Integrality of volumes of representations

Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation rho :pi _1(M)rightarrow mathrm {Isom}^+({{mathbb {H}}}^n), properly normalized, takes integer values if n is even and ge 4. If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.

Mapping class group dynamics and the holonomy of branched affine structures

We classify, up to few exceptions, the orbit closures of the {mathrm {Mod}}(Sigma )-action on the affine character variety chi ( mathrm {Aff}({mathbb {C}})). We obtain from this classification that the only obstruction for a non-abelian representation rho : pi _1 Sigma longrightarrow mathrm {Aff}({mathbb {C}}) to be the holonomy of a branched affine structure on Sigma is to be Euclidean and not to have positive volume, where Sigma is a closed oriented surface of genus g ge 2.

Arithmetic Properties of the Frobenius Traces Defined by a Rational Abelian Variety (with two appendices by J-P. Serre)

Let A be an abelian variety over Q of dimension g such that the image of its associated absolute Galois representation rho(A) is open in GSp(2g)((Z) over cap). We investigate the arithmetic of the traces a(1,p) of the Frobenius at p in Gal((Q) over bar /Q) under rho(A). In particular, we obtain upper bounds for the counting function #{p <= x : a(1,p) = t} and we prove an Erdos-Kac-type theorem for the number of prime factors of a(1,p). We also formulate a conjecture about the asymptotic behaviour of #{p <= x : a(1,p) = t}, which generalizes a well-known conjecture of Lang and Trotter from 1976 about elliptic curves.

Ordinary representations of G ( Q p ) and fundamental algebraic representations

Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence for G(Q_p) we associate to an n-dimensional ordinary (i.e. Borel valued) representation rho : Gal(Q_p-bar/Q_p) to G-hat(E) a unitary Banach space representation Pi(rho)^ord of G(Q_p) over E that is built out of principal series representations. (Here, E is a finite extension of Q_p.) Our construction is inspired by the "ordinary part" of the tensor product of all fundamental algebraic representations of G. There is an analogous construction over a finite extension of F_p. In the latter case, when G=GL_n we show under suitable hypotheses that Pi(rho)^ord occurs in the rho-part of the cohomology of a compact unitary group. We also prove a weaker version of this result in the p-adic case.

Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j, in the sense that the hyperbolic translation lengths for j are uniformly larger than for rho; conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation rho, whose Euler class can be prescribed. This has applications to compact anti-de Sitter 3-manifolds.

Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei

We investigate regularity properties of molecular one-electron densities rho near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x)) with an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that mu is even C^{2,\alpha}(R^3) for all alpha in (0,1). Placing one nucleus at the origin we study rho in polar coordinates x=r*omega and investigate rho'(r,omega) and rho''(r,omega) for fixed omega as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.

ℓ-adic Representations Associated to Modular Forms over Imaginary Quadratic Fields

Let p be a regular algebraic cuspidal automorphic representation of GL(2) over an imaginary quadratic number field K, and let l be a prime number. Assuming the central character of p is invariant under the nontrivial automorphism of K, it is shown that there is a continuous irreducible l-adic representation rho of Gal((K) over bar /K) such that L(s, rho(v)) = L(s, pi(v)) whenever v is a prime of K outside an explicit finite set.

Open Sets of Maximal Dimension in Complex Hyperbolic Quasi-Fuchsian Space

Let pi(1), be the fundamental group of a closed surface Sigma of genus g > 1. One of the fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations of pi(1) into SU(2, 1), (the triple cover of) the group of holomorphic isometries of H-C(2). In particular, given a discrete, faithful, geometrically finite and purely loxodromic representation rho(0) of pi(1), can we find an open neighbourhood of rho(0) comprising representations with these properties. We show that this is indeed the case when rho(0) preserves a totally real Lagrangian plane.

QuantumSU(2) faithfully detects mapping class groups modulo center

The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted V_A(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter A a primitive 4r-th root of unity (r=k+2). In each of these representations (as well as the general G case), Dehn twists act as transformations of finite order, so none represents the mapping class group M(Y) faithfully. However, taken together, the quantum SU(2) representations are faithful on non-central elements of M(Y). (Note that M(Y) has non-trivial center only if Y is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus = 2.) Specifically, for a non-central h in M(Y) there is an r_0(h) such that if r>= r_0(h) and A is a primitive 4r-th root of unity then h acts projectively nontrivially on V_A(Y). Jones' [J] original representation rho_n of the braid groups B_n, sometimes called the generic q-analog-SU(2)-representation, is not known to be faithful. However, we show that any braid h not= id in B_n admits a cabling c = c_1,...,c_n so that rho_N (c(h)) not= id, N=c_1 + ... + c_n.

Algebraic equations determining quantum dimensions

We show that a quantum dimension Dq( Lambda ) for a representation rho of Uq(G), a quantized universal enveloping algebra of a compact and simple Lie group G, is computed from the algebraic equations which we found recently in studying 2+1-dimensional Chern-Simons theory. We solve the equations explicitly for the typical examples of all compact and simple Lie groups. This method can be applied to super Lie groups such as SU(m,n) and OSp(m,n).

Highest-weight representations of the sl(n+1,C) algebras: Maximal representations

Representations of the sl(n+1, C) Lie algebras are constructed with the help of canonical (boson) realisations of these algebras. For every weight Lambda on the standard Cartan subalgebra of sl(n+1, C) the authors obtain a representation rho Lambda (n+1) (called the maximal representation) which contains an irreducible subrepresentation with Lambda as the highest weight. It is shown that for a major part of the weights Lambda the representations rho Lambda (n+1) themselves are irreducible. The standard construction of the highest-weight representations of semi-simple Lie algebras is based on the so-called elementary representations; comparing with them, the authors maximal representations are given in the explicit form.