We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group ${\mathbb R {}}^*_+=\{r:r>0\}$ of automorphisms of $P_0$. This construction is determined by a a faithful unitary representation of $P_0$ (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as $r\to 0$. We apply this construction to the groups of currents of the maximal parabolic subgroups of the groups of motions of the $n$-dimensional real and complex Lobachevsky spaces. The obtained representations of the groups of parabolic currents can be uniquely extended to the groups of currents with values in the semisimple groups O(n,1) and U(n,1). This gives a new description of the representations of the groups of currents of these groups constructed in the 70s and realized in the Fock space. The key role in our construction is played by the so-called special representation of the parabolic subgroup $P$ and the remarkable $σ$-finite measure (Lebesgue measure) $\mathcal L$ in the space of distributions.
We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible unitary representations in terms of this classification, and the fusion rules for the semidirect product.
For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl-Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.
A Lie group G is called a trace class group if for every irreducible unitary representation R of G and every C-infinity function f with compact support the operator R(f) is of trace class. In this note we prove that the semidirect product of R^n and a real semisimple algebraic subgroup G of GL(n;R) is a trace class group only if G is compact. The converse has been shown elsewhere. We also make a descent start with the study of semidirect products with Heisenberg-type groups.
It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincaré groups in arbitrary dimension. Finally we compare our results with those in…
In the 1970's, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey's analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum-Connes conjecture with trivial coefficients for such groups. Here we extend Higson's results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our m…
We discuss some natural maps from a unitary group U(n) to a smaller group U(n-m) (these maps are versions of the Livshic characteristic function). We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces (values of the integrals are products of Gamma-functions). These integrals generalize Hua Loo Keng integrals. We construct inverse limits of unitary groups equipped with analogues of the Haar measure and evaluate some integrals over these inverse limits.
Let $(π, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $γ\: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $π^\#(g,t) = π(g) U_t$ defines a continuous unitary representation of the semidirect product group $G \rtimes_γ\mathbb R$. The first main theorem of the present note provides criteria for the invariance of the space $\mathcal H^\infty$ of smooth vectors of $π$ under the operators $U_f = \int_\mathbb R f(t)U_t\, dt$ for $f \in L^1(\mathbb R)$, resp., $f \in \mathcal S(\mathbb R)$. Using thi…
George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between "most" irreducible (tempered) representations of $G$ and "most" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspond…
We study unitary representations of groups in Krein spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Krein subspaces and their (reproducing) kernels and a variant of Choquet's theorem.
Let $G$ and $T$ be topological groups, $α: T \to \Aut(G)$ a homomorphism defining a continuous action of $T$ on $G$ and $G^\sharp := G \rtimes_αT$ the corresponding semidirect product group. In this paper we address several issues concerning irreducible continuous unitary representations $(π^\sharp, \cH)$ of $G^\sharp$ whose restriction to $G$ remains irreducible. First we prove that, for $T = \R$, this is the case for any irreducible positive energy representation of $G^\sharp$, i.e., for which the one-parameter group $U_t := π^\sharp(\1,t)$ has non-negative spectrum. The passage from irreduc…
We obtain necessary and sufficient conditions for the admissible vectors of a new unitary non irreducible representation $U$. The group $G$ is an arbitrary semidirect product whose normal factor $A$ is abelian and whose homogeneous factor $H$ is a locally compact second countable group acting on a Riemannian manifold $M$. The key ingredient in the construction of $U$ is a $C^1$ intertwining map between the actions of $H$ on the dual group $\hat A$ and on $M$. The representation $U$ generalizes the restriction of the metaplectic representation to triangular subgroups of $Sp(d,\R)$, whence the n…
We show the Alon-Tarsi conjecture on Latin squares is equivalent to a very special case of a conjecture made independently by Hadamard and Howe, and to the non-vanishing of some interesting integrals over SU(n). Our investigations were motivated by geometric complexity theory.
For every finite dimensional Lie supergroup $(G,\mathfrak g)$, we define a $C^*$-algebra $\mathcal A:=\mathcal A(G,\mathfrak g)$, and show that there exists a canonical bijective correspondence between unitary representations of $(G,\mathfrak g)$ and nondegenerate $*$-representations of $\mathcal A$. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations. For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated $C^*$-algebra is CCR. In particular, …
A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators idπ(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g. Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group γ. For t…
In this article, for nilpotent orbits of ramified quasi-split unitary groups with two Jordan blocks, we give closed formulas for their Shalika germs at certain equi-valued elements with half-integral depth previously studied by Hales. These elements are parametrized by hyperelliptic curves defined over the residue field, and the numbers we obtain can be expressed in terms of Frobenius eigenvalues on the $\ell$-adic $H^1$ of the curves, generalizing previous result of Hales on stable subregular Shalika germs. These Shalika germ formulas imply new results on stability and endoscopic transfer of …
The restriction of an irreducible unitary representation $π$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $τ$ of $H$ with multiplicities $m(π,τ)\in\mathbb{N}\cup\{\infty\}$. We show that on the smooth vectors of $π$, the direct integral is pointwise defined. This implies that $m(π,τ)$ is bounded above by the dimension of the space $\operatorname{Hom}_H(π^\infty|_H,τ^\infty)$ of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between t…
We introduce Jack (unitary) characters and prove two kinds of formulas that are suitable for their asymptotics, as the lengths of the signatures that parametrize them go to infinity. The first kind includes several integral representations for Jack characters of one variable. The second identity we prove is the Pieri integral formula for Jack characters which, in a sense, is dual to the well known Pieri rule for Jack polynomials. The Pieri integral formula can also be seen as a functional equation for irreducible spherical functions of virtual Gelfand pairs. As an application of our formulas, …
In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in the class expansion of monomial symmetric functions in Jucys-Murphy elements, from which we recover the first order asymptotics of polynomial integrals over $\U(N)$ as $N \rightarrow \infty$. Our results on class expansion include an analogue of Macdonald's result for the top connection coefficients of the class algebra, a generalization of Stanley and Olsh…
In this paper, we obtained character formulas of irreducible unitary representations of $U(n, n+1)$ by using Howe's correspondence and the Cauchy--Harish-Chandra integral. The representations of $U(n, n+1)$ we are dealing with are obtained from a double lifting of a representation of $U(1)$ via the dual pairs $(U(1), U(1, 1))$ and $(U(1, 1), U(n, n+1))$.
For a local field $F$ we consider tamely ramified principal series representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module $V^{I_0}$ admits an integral structure, then such an integral structure can be chosen in a particularly well organized manner, in particular its modular reduction can be made completely explicit.
For a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions. In this paper we explicitly calculate archimedean zeta integrals of this type for U(n,1), assuming that the corresponding archimedean component of the automorphic representation is a holomorphic discrete series.
Let $N$ be a simply connected, connected nilpotent Lie group which admits a uniform subgroup $Γ.$ Let $α$ be an automorphism of $N$ defined by $α\left( \exp X\right) =\exp AX.$ We assume that the linear action of $A$ is diagonalizable and we do not assume that $N$ is commutative. Let $W$ be a unitary wavelet representation of the semi-direct product group $\left\langle \cup_{j\in\mathbb{Z}}α^{j}\left( Γ\right) \right\rangle \rtimes\left\langle α\right\rangle $ defined by $W\left( γ,1\right) =f\left( γ^{-1}x\right) $ and $W\left( 1,α\right) =\left\vert \det A\right\vert ^{1/2}f\left( αx\right) …
We propose a relative trace formula approach and state the corresponding fundamental lemma toward the global restriction problem involving Bessel or Fourier-Jacobi periods of unitary groups $\mathrm{U}_n\times\mathrm{U}_m$, extending the work of Jacquet-Rallis for $m=n-1$ (which is a Bessel period). In particular, when $m=0$, we recover a relative trace formula proposed by Flicker concerning Kloosterman/Fourier integrals on quasi-split unitary groups. As evidence for our approach, we prove the fundamental lemma for $\mathrm{U}_n\times\mathrm{U}_n$ in positive characteristics.
We consider representations of rational Cherednik algebras which are particular ideals in the ring of polynomials. We investigate convergence of the integrals which express the Gaussian inner product on these representations. We derive that the integrals converge for the minimal submodules in types B and D for the singular values suggested by Cherednik with at most one exception, hence the corresponding modules are unitary. The analogous result on unitarity of the minimal submodules in type A was obtained by Etingof and Stoica, we give a different proof of convergence of the Gaussian product i…
Let $\mathfrak{S}_\infty$ be the infinity permutation group and $Γ$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $Γ^\infty$ by automorphisms, so one can form a semidirect product $Γ^\infty\rtimes \mathfrak{S}_\infty$, known as the {\it wreath} product $Γ\wr\mathfrak{S}_\infty$ of $Γ$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $II_1-$factor-representations of $Γ\wr\mathfrak{S}_\infty$ in terms of finite characters of $Γ$. Our approach is based on extending Okounkov's classification method for admissible representations of $\mathfrak{S}_\inft…
Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as solutions of certain Diophantine equations given by the cluster variables in the cluster algebra. We show that every cluster gives rise to a frieze vector and that the frieze vector determines the cluster. We also study friezes of type Q as homomorphisms from the cluster algebra to an arbitrary integral domain. In particular, we show that every positive inte…
Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of non-integral level using the Hecke algebraic method developed by Allen Moy for GSp(4).
We introduce a new class of symmetric space orbital integrals important for applications in certain relative trace formulas appearing in the theory of automorphic representations. We verify a fundamental lemma for $U_2\times U_2\hookrightarrow U_4$ via an explicit calculation, showing strong evidence that there is a general theory of endoscopy lurking in this situation.
Let F be a number field, A_F its ring of adeles, and let π_n and π_{n+1} be irreducible, cuspidal, automorphic representations of SO_n(A_F) and SO_{n+1}(A_F), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to π_n and π_{n+1} is equivalent to the non-vanishing of L(1/2, π_n x π_{n+1}). More recently, Atsushi Ichino and Tamotsu Ikeda gave a refinement of this conjecture as well as a proof of the first few cases (n = 2,3). Their conjecture gives an explicit relationship between the aforementioned L-value and period int…
In this paper, we establish an explicit isomorphism between the symmetric group algebra and the path algebra of the Young graph. Specifically, we construct a family of matrix units in the group algebra. As a main application of this construction, we obtain new formulas, alternative to Weingarten calculus, for the integral of a polynomial over the unitary group with respect to the Haar measure. In particular, we obtain a closed formula for the law of moments of the first k rows of the unitary group.
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.
Let $E/F$ be a quadratic extension of $p$-adic fields and let $d$, $m$ be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations $π$ and $σ$ of $GL_d(E)$ and $GL_m(E)$ respectively. We assume that $π$ and $σ$ are conjugate-dual. That is to say $π\simeq π^{\vee,c}$ and $σ\simeq σ^{\vee,c}$) where $c$ is the non trivial $F$-automorphism of $E$. This implies, we can extend $π$ to an unitary representation $\tildeπ$ of a nonconnected group $GL_d(E)\rtimes {1,θ}$. Define $\tildeσ$ the same way. We state and prove an integral formula for $ε(1/2,π\times σ,ψ_E)$…
We state and prove an extension of the global Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture to the case of some Eisenstein series on unitary groups $U_n\times U_{n+1}$. Our theorems are based on a comparison of the Jacquet-Rallis trace formulas. A new point is the expression of some interesting spectral contributions in these formulas in terms of integrals of relative characters. As an application of our mains theorems, we prove the global Gan-Gross-Prasad and the Ichino-Ikeda conjecture for Bessel periods of unitary groups.
Locally affine Lie algebras are generalizations of affine Kac--Moody algebras with Cartan subalgebras of infinite rank whose root system is locally affine. In this note we study a class of representations of locally affine algebras generalizing integrable highest weight modules. In particular, we construct such an integrable representation for each integral weight not vanishing on the center and show that, over the complex numbers, we thus obtain unitary representations w.r.t. a unitary real form. We also use Yoshii's recent classification of locally affine root systems to derive a classificat…
Indecomposable characters of the wreath product of a torus by the infinite block-diagonal symmetric group
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine（乌克兰国家科学院B.维尔金低温物理与工程研究所）；Institute of Science and Technology Austria (ISTA)（奥地利科学与技术研究所）
We consider a class of matrix integrals over the unitary group $U(N)$ with an infinite set of couplings characterized by a series $f(q)=\sum_{n \ge 1} a_n q^n$, with $a_n \in \mathbb{Z}$. Such integrals arise in physics as the partition functions of free four-dimensional gauge theories on $S^3$ and, in particular, as the superconformal index of super Yang-Mills theory. We show that any such model can be expressed in terms of a system of free fermions in an ensemble parameterized by the infinite set of couplings. Integrating out the fermions in a given quantum state leads to a convergent expans…
After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C) and (D) of SU(3)-subgroups is studied. We provide a minimal set of generators for one of these groups which has order 162. These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones-Kauffman version of SU(2) Chern-Simons theory at level 4. In light of these new generators, we study the structu…
For a smooth projective unitary representation of a locally convex Lie group G, the projective space of smooth vectors is a locally convex Kaehler manifold. We show that the action of G on this space is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)-extension of G obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, and give some integrality conditions on the image of the momentum map.
This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration meth…
