The irreducible orthogonal and symplectic Galois representations of a p-adic field (the tame case)

Let F be a p-adic field. If n is a natural number relatively prime to p, then all the irreducible n-dimensional Galois representations are parametrized by admissible characters. This parametrization is used to determine which of these characters are real-valued, and among the real-valued representations to distinguish the orthogonal representations from the symplectic representations.     

ℒ-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field. We study a special kind of p-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez’s formula.     

Homology representations arising from the half cube, II

In a previous work, we defined a family of subcomplexes of the n-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least k, and we proved that the reduced homology of such a subcomplex is concentrated in degree k−1. This homology module supports a natural action of the Coxeter group W(Dn) of type D. In this paper, we explicitly determine the characters (over C) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group Sn by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of Sn agree (over C) with the representations of Sn on the (k−2)-nd homology of the complement of the k-equal real hyperplane arrangement.     

On congruences of m-groups

We indicate a way for constructing m-congruences of an arbitrary m-transitive representation, introduce the notions of m-2-transitive and m-primitive representations, and describe the m-transitive primitive representations in terms of stabilizers. Also we give necessary and sufficient conditions for m-2-transitivity and study some properties of these representations.     

Local representations of braid groups

We study the local linear representations of the braid group B ₃ and the homogeneous local representations of B _n for n ≥ 2. We investigate the connection of these representations with the Burau representation. The linear representations of B _n are constructed from the Wada representation of B _n in the automorphism group of a free group.     

Analysis of a class of nonlinear and non-separable multiscale representations

In this paper, we introduce a particular class of nonlinear and non-separable multiscale representations which embeds most of these representations. After motivating the introduction of such a class on one-dimensional examples, we investigate the multi-dimensional and non-separable case where the scaling factor is given by a non-diagonal dilation matrix M. We also propose new convergence and stability results in L ^p and Besov spaces for that class of nonlinear and non-separable multiscale representations. We end the paper with an application of the proposed study to the convergence and the stability of some nonlinear multiscale representations.     

Explicit Hilbert spaces for certain unipotent representations III

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L²-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L²-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of these representations and describe their decomposition.     

Plancherel Formula for Berezin Deformation of L on Riemannian Symmetric Space

Consider natural representations of the pseudounitary group U(p, q) in the space of holomorphic functions on the Cartan domain (Hermitian symmetric space) U(p, q)/(U(p)×U(q)). Berezin representations of O(p, q) are the restrictions of such representations to the subgroup O(p, q). We obtain the explicit Plancherel formula for the Berezin representations. The support of the Plancherel measure is a union of many series of representations. The density of the Plancherel measure on each piece of the support is an explicit product of Γ-functions. We also show that the Berezin representations give an interpolation between L² on noncompact symmetric space O(p, q)/O(p)×O(q) and L² on compact symmetric space O(p+q)/O(p)×O(q).     

On calibrated representations and the Plancherel Theorem for affine Hecke algebras

This paper has two main purposes. Firstly, we generalise Ram’s combinatorial construction of calibrated representations of the affine Hecke algebra to the multi-parameter case (including the non-reduced BC _n case). We then derive the Plancherel formulae for all rank 1 and rank 2 affine Hecke algebras, using our calibrated representations to construct all representations involved.     

Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.     

On representations of real Jacobi groups

We consider a category of continuous Hilbert space representations and a category of smooth Fr’echet representations,of a real Jacobi group G.By Mackey’s theory,they are respectively equivalent to certain categories of representations of a real reductive group L.Within these categories,we show that the two functors that take smooth vectors for G and for L are consistent with each other.By using Casselman-Wallach’s theory of smooth representations of real reductive groups,we define matrix coefficients for distributional vectors of certain representations of G.We also formulate Gelfand-Kazhdan criteria for real Jacobi groups which could be used to prove multiplicity one theorems for Fourier-Jacobi models.     

Opening the closed text: the poetics of representations of teaching

I provide an analysis of the uses of representations of teaching by embedding such representations within Umberto Eco??s theory of the open work, in which literary works are understood not as carriers of a string of meanings, controlled by the author (??closed works??), but rather as fields of meaning. I contend that the well-established tradition of using representations of teaching for pedagogical purposes corresponds to the use of closed works. In contrast, their use for research purposes corresponds to the use of open works. I develop these considerations through an analysis of several representations of teaching, and show how features of these representations work variously to either open or close the work. I also provide anecdotal evidence that viewers of such ??open works?? may fall victim to a version of the so-called ??intentional fallacy.?? I discuss the implications of this for the integrity of the research enterprise.     

On local newforms for unramified U(2, 1)

Let G be the unramified unitary group in three variables defined over a p-adic field F with p ≠ 2. In this paper, we investigate local newforms for irreducible admissible representations of G. We introduce a family of open compact subgroups {K _n } _n≥0 of G to define the local newforms for representations of G as the K _n -fixed vectors. We prove the existence of local newforms for generic representations and the multiplicity one property of the local newforms for admissible representations.     

Representations of metabelian groups

A knowledge of the simple representation theory of finite abelian groups is useful for understanding the representations of solvable groups, since these provide the one-dimensional representations. The representation theory of metabelian groups (those G with abelian commutator subgroup G′) would seem to be a natural next level.In this paper we shall show that these representations, too, may be simply described in several ways: they are induced from linear representations of some explicity defined subgroups; their degrees may be calculated from a knowledge of the subgroups of G; these degrees depend only on the kernel of the representation (in fact, only on the intersection of this kernel with G′). As an application of these results, we can calculate for metabelian groups a certain measure of group-commutativity studied in an earlier paper [4].     

Hardy-Sobolev Spaces Decomposition in?Signal Analysis

Some fundamental formulas and relations in signal analysis are based on the amplitude-phase representations s(t)=A(t)e ^{i ??(t)} and $\hat{s}(\omega)=B(\omega)e^{i\psi(\omega)}$ , where the amplitude functions A(t) and B(??) and the phase functions ??(t) and ??(??) are assumed to be differentiable. They include the amplitude-phase representations of the first and second order means of the Fourier frequency and time, and the equivalence between two forms of the covariance. A proof of the uncertainty principle is also based on the amplitude-phase representations. In general, however, signals of finite energy do not necessarily have differentiable amplitude-phase representations. The study presented in this paper extends the classical formulas and relations to general signals of finite energy. Under the formulation of the phase and amplitude derivatives based on the Hardy-Sobolev spaces decomposition the extended formulas reveal new features, and contribute to the foundations of time-frequency analysis. The established theory is based on the equivalent classes of the L ² space but not on particular representations of the classes. We also give a proof of the uncertainty principle by using the amplitude-phase representations defined through the Hardy-Sobolev spaces decomposition.     

Monodromy of Fuchsian systems on complex linear spaces

On complex linear spaces, Fuchs-type Pfaffian systems are studied that are defined by configurations of vectors in these spaces. These systems are referred to as R-systems in this paper. For the vector configurations that are systems of roots of complex reflection groups, the monodromy representations of R-systems are described. These representations are deformations of the standard representations of reflection groups. Such deformations define representations of generalized braid groups corresponding to complex reflection groups and are similar to the Burau representations of the Artin braid groups.     

The tame dimension vectors for stars

We consider representations of stars over an algebraically closed field K. We classify those dimension vectors of stars admitting a one parameter family of indecomposable representations and for which, in addition, all families of (not necessarily indecomposable) representations depend on a single parameter. Furthermore, we show how it is possible to construct the corresponding one parameter families of indecomposable representations.     

Elementary transformations of pfaffian representations of plane curves

Let C be a smooth curve in P² given by an equation F=0 of degree d. In this paper we consider elementary transformations of linear pfaffian representations of C. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on C with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of C can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.     

Degenerate group of type A: Representations and flag varieties

We consider the degeneration of a simple Lie group which is a semidirect product of its Borel subgroup and a normal Abelian unipotent subgroup. We introduce a class of highest weight representations of the degenerate group of type A, generalizing the construction of PBW-graded representations of the classical group (PBW is an abbreviation for “Poincaré-Birkhoff-Witt”). Following the classical construction of flag varieties, we consider the closures of orbits of the Abelian unipotent subgroup in projectivizations of the representations. We show that the degenerate flag varieties F _n ^a and their desingularizations R _n can be obtained via this construction. We prove that the coordinate ring of R _n is isomorphic as a vector space to the direct sum of the duals of the highest weight representations of the degenerate group. At the end we state several conjectures on the structure of the highest weight representations of the degenerate group of type A.