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Branching coefficients of holomorphic representations and Segal-Bargmann transform  [PDF]
Genkai Zhang
Mathematics , 2001,
Abstract: Let $\mathbb D=G/K$ be a complex bounded symmetric domain of tube type in a Jordan algebra $V_{\mathbb C}$, and let $D=H/L =\mathbb D\cap V$ be its real form in a Jordan algebra $V\subset V_{\mathbb C}$. The analytic continuation of the holomorphic discrete series on $\mathbb D$ forms a family of interesting representations of $G$. We consider the restriction on $D$ of the scalar holomorphic representations of $G$, as a representation of $H$. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group $L$ is a spherical subgroup of $K$ and we find a canonical basis of $L$-invariant polynomials in components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal-Bargmann transform of those $L$-invariant polynomials are, under the spherical transform on $D$, multi-variable Wilson type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on $D$, when extended to a neighborhood in $\mathbb D$, in terms of the $L$-spherical holomorphic polynomials on $\mathbb D$, the coefficients being the Wilson polynomials.
Yang-Mills theory over surfaces and the Atiyah-Segal theorem  [PDF]
Daniel A. Ramras
Mathematics , 2007, DOI: 10.2140/agt.2008.8.2209
Abstract: In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(\Gamma) of a compact Lie group $\Gamma$ to the complex K-theory of the classifying space $B\Gamma$. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson's deformation $K$--theory spectrum $\K (\Gamma)$ (the homotopy-theoretical analogue of $R(\Gamma)$). Our main theorem provides an isomorphism in homotopy $\K_*(\pi_1 \Sigma)\isom K^{-*}(\Sigma)$ for all compact, aspherical surfaces $\Sigma$ and all $*>0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.
The Atiyah--Segal completion theorem in twisted K-theory  [PDF]
Anssi Lahtinen
Mathematics , 2008, DOI: 10.2140/agt.2012.12.1925
Abstract: In this note we prove the analogue of the Atiyah-Segal completion theorem for equivariant twisted K-theory in the setting of an arbitrary compact Lie group G and an arbitrary twisting of the usually considered type. The theorem generalizes a result by C. Dwyer, who has proven the theorem for finite G and twistings of a more restricted type. While versions of the general result have been known to experts, to our knowledge no proof appears in the current literature. Our goal is to fill in this gap. The proof we give proceeds in two stages. We first prove the theorem in the case of a twisting arising from a graded central extension of G, following the Adams-Haeberly-Jackowski-May proof of the classical Atiyah-Segal completion theorem. After establishing that the theorem holds for this special class of twistings, we then deduce the general theorem by a Mayer-Vietoris argument.
Canonical Basis and Macdonald Polynomials  [PDF]
Jonathan Beck,Igor Frenkel,Naihuan Jing
Mathematics , 1998,
Abstract: In the basic representation of $U_q(\hat{sl}(2))$ realized via the algebra of symmetric functions we compare the canonical basis with the basis of Macdonald polynomials with $q=t^2$. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the transition matrix are necessarily bar invariant. We also discuss the positivity and integrality of these coefficients. For level $k$, we expect a similar relation between the canonical basis and Macdonald polynomials with $q^2=t^{k}.$
A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory  [PDF]
Yuki Arano,Yosuke Kubota
Mathematics , 2015,
Abstract: We investigate the homological ideal $\mathfrak{J}_G^H$, the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah-Segal completion theorem with the comparison of $\mathfrak{J}_G^H$ with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah-Segal completion theorem for groupoid equivariant $\mathrm{KK}$-theory, McClure's restriction map theorem, permanence property of the Baum-Connes conjecture under extensions of groups and a class of $\mathfrak{J}_G$-injective objects coming from $\mathrm{C}^*$-dynamical systems, continuous Rokhlin property.
On the choice of a basis of invariant polynomials of a Finite Reflection Group. Generating Formulas for $\widehat P$-matrices of groups of the infinite series $S_n$, $A_n$, $B_n$ and $D_n$  [PDF]
Vittorino Talamini
Mathematics , 2015,
Abstract: Let $W$ be a rank $n$ irreducible finite reflection group and let $p_1(x),\ldots,p_n(x)$, $x\in\mathbb{R}^n$, be a basis of algebraically independent $W$-invariant real homogeneous polynomials. The orbit map $\overline p:\mathbb{R}^n\to\mathbb{R}^n:x\to (p_1(x),\ldots,p_n(x))$ induces a diffeomorphism between the orbit space $\mathbb{R}^n/W$ and the set ${\cal S}=\overline p(\mathbb{R}^n)\subset\mathbb{R}^n$. The border of ${\cal S}$ is the $\overline p$ image of the set of reflecting hyperplanes of $W$. With a given basic set of invariant polynomials it is possible to build an $n\times n$ polynomial matrix, $\widehat P(p)$, $p\in\mathbb{R}^n$, sometimes called $\widehat P$-matrix, such that $\widehat P_{ab}(p(x))=\nabla p_a(x)\cdot \nabla p_b(x)$, $\forall\,a,b=1,\ldots,n$. The border of ${\cal S}$ is contained in the algebraic surface $\det(\widehat P(p))=0$, sometimes called discriminant, and the polynomial $\det(\widehat P(p))$ satisfies a system of differential equations that depends on an $n$-dimensional polynomial vector $\lambda(p)$. Possible applications concern phase transitions and singularities. If the rank $n$ is large, the matrix $\widehat P(p)$ is in general difficult to calculate. In this article I suggest a choice of the basic invariant polynomials for all the reflection groups of type $S_n$, $A_n$, $B_n$, $D_n$, $\forall\,n\in \mathbb{N}$, for which I give generating formulas for the corresponding $\widehat P$-matrices and $\lambda$-vectors. These $\widehat P$-matrices can be written, almost completely, as sums of block Hankel matrices. Transformation formulas allow to determine easily both the $\widehat P$-matrix and the $\lambda$-vector in any other basis of invariant polynomials. Examples of transformations into flat bases, $a$-bases, and canonical bases, are considered.
Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials  [PDF]
Mark Davidson,Gestur Olafsson,Genkai Zhang
Mathematics , 2002,
Abstract: Let $\mathcal{D}=G/K$ be a complex bounded symmetric domain of tube type in a complex Jordan algebra $V$ and let $\mathcal{D}_{\mathbb{R}}=H/L\subset \mathcal{D}$ be its real form in a formally real Euclidean Jordan algebra $J\subset V$. We consider representations of $H$ that are gotten by the generalized Segal-Bargmann transform from a unitary $G$-space of holomorphic functions on $\mathcal{D}$ to an $L^2$-space on $\mathcal{D_{\mathbf{R}}}$. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to $\mathcal{D}$ of the spherical functions on $\mathcal{D}_{\mathbb{R}}$ and find the expansion in terms of the $L$-spherical polynomials on $\mathcal{D}$, which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an $L^2$-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on $\mathcal D$.
Quasi-invariant and super-coinvariant polynomials for the generalized symmetric group  [PDF]
Jean-Christophe Aval
Mathematics , 2007,
Abstract: The aim of this work is to extend the study of super-coinvariant polynomials, to the case of the generalized symmetric group $G_{n,m}$, defined as the wreath product $C_m\wr\S_n$ of the symmetric group by the cyclic group. We define a quasi-symmetrizing action of $G_{n,m}$ on $\Q[x_1,...,x_n]$, analogous to those defined by Hivert in the case of $\S_n$. The polynomials invariant under this action are called quasi-invariant, and we define super-coinvariant polynomials as polynomials orthogonal, with respect to a given scalar product, to the quasi-invariant polynomials with no constant term. Our main result is the description of a Gr\"obner basis for the ideal generated by quasi-invariant polynomials, from which we dedece that the dimension of the space of super-coinvariant polynomials is equal to $m^n C_n$ where $C_n$ is the $n$-th Catalan number.
A basis for the diagonally signed-symmetric polynomials  [PDF]
José Manuel Gómez
Mathematics , 2013,
Abstract: Let n>0 be an integer and let B_{n} denote the hyperoctahedral group of rank n. The group B_{n} acts on the polynomial ring Q[x_{1},...,x_{n},y_{1},...,y_{n}] by signed permutations simultaneously on both of the sets of variables x_{1},...,x_{n} and y_{1},...,y_{n}. The invariant ring M^{B_{n}}:=Q[x_{1},...,x_{n},y_{1},...,y_{n}]^{B_{n}} is the ring of diagonally signed-symmetric polynomials. In this article we provide an explicit free basis of M^{B_{n}} as a module over the ring of symmetric polynomials on both of the sets of variables x_{1}^{2},..., x^{2}_{n} and y_{1}^{2},..., y^{2}_{n} using signed descent monomials.
Galois invariant smoothness basis  [PDF]
Jean-Marc Couveignes,Reynald Lercier
Mathematics , 2008,
Abstract: This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field $\bK$, one looks for a smoothness basis for $\bK^*$ that is left invariant by automorphisms of $\bK$. For a broad class of finite fields, we manage to construct models that allow such a smoothness basis. This work aims at accelerating discrete logarithm computations in such fields. We treat the cases of codimension one (the linear sieve) and codimension two (the function field sieve).
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