The problem of harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on a (noncommutative) group is to decompose the most “natural” unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(∞) is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(∞) consists of.We deal with unitary representations of a reasonable class, which are in 1-1 correspondence with characters (central, positive definite, normalized functions on U(∞)). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.The main result of the present paper consists in explicitly constructing a 4-parameter family of “natural” representations and computing their characters. We view these representations as a substitute of the nonexisting regular representation of U(∞). We state the problem of harmonic analysis on U(∞) as the problem of computing the spectral measures for these “natural” representations. A solution to this problem is given in the next paper (Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, math/0109194, to appear in Ann. Math.), joint with Alexei Borodin.We also prove a few auxiliary general results. In particular, it is proved that the spectral measure of any character of U(∞) can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups U(N). This fact is a starting point for computing spectral measures.     

When is a symplectic quotient an orbifold?

Let K be a compact Lie group of positive dimension. We show that for most unitary K-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When K is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian K  -manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary _SU2${\mathrm{SU}}_2$-modules yield symplectic quotients that are Z⁺${\mathbb{Z}}^{+}$-graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold.     

On the local case in the Aschbacher theorem for linear and unitary groups

We consider the subgroups H in a linear or a unitary group G over a finite field such that O _r (H) ? Z(G) for some odd prime r. We obtain a refinement of the well-known Aschbacher theorem on subgroups of classical groups for this case.     

Lower large deviations for supercritical branching processes in random environment

Branching processes in random environment (Z _n : n ≥ 0) are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of Z, which means the asymptotic behavior of the probability {1 ≤ Z _n ≤ exp(nθ)} as n → ∞. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where ?(Z ₁ = 0 | Z ₀ = 1) > 0 and also much weaker moment assumptions.     

Continuous families of vectors that are orbits of a unitary group

We characterize functions u from the real line into a Hilbert space that are the orbits of a unitary group {U(t)}_t∈R; that is, u(t)=U(t)u(0), for all real t. One of the characterizations is that u be the Fourier transform of a certain type of vector-valued measure Z; we then use our characterizations to construct Z from u.     

On the normalizers of the unitary subgroup in an integral group ring

In this paper, we investigate the properties of the normalizers of the unitary subgroup u_f(ZG) in an integral group rings. One of our main results is Theorem 2.6 which character¬izes the second normalizer of the unitary subgroup. As a conse¬quence of this theorem, we prove that the second normalizer of u_f(ZG) coincides with the first normalizer when G is a periodic group. Among other results, we give necessary and sufficient conditions for which the unitary subgroup is normal in the unit group when G is periodic and also characterize when all bicyclic units are nontrivial and elements of the normalizer of the unitary subgroup.     

Unitary graphs and classification of a family of symmetric graphs with complete quotients

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.     

Revisits for transient random walk

For a random walk on the integers define R_n as the number of (distinct) states visited in the first n steps and Z_n as the number of states visited in the first n steps which are never revisited. Here we deal with transient walks. The increments of Z_n form a stationary process and various central limit results and an iterated logarithm result are obtained for Z_n from known results on stationary processes. Furthermore, the limit behaviour of R_n is closely related to that of Z_n; this relationship is elucidated and corresponding limit results for R_n are then read off from those for Z_n.     

Decay properties of Markovian bulk-arrival and bulk-service queues with state-independent control

We consider decay properties regarding decay parameter and invariant measures of Markovian bulk-arrival and bulk-service queues with state-independent control. The exact value of the decay parameter, denoted by λz, is firstly revealed. A criterion regarding ）λz-recurrence and λz-positive is obtained. The corresponding λz-subinvariant/invariant measures and λz-subinvariant/invariant vectors are then presented.     

Unitary groups of nondegenerate Hermitian forms and the homotopy groups of pseudospheres

The Witt Extension Theorem states that the unitary group of a finite-dimensional vector space V equipped with a nondegenerate hermitian form acts transitively on the pseudosphere induced by the form. We provide a new, constructive proof of this result for finite-dimensional vector spaces V over R, C, or H. This constructive proof is then used to prove a similar result for the unitary group of a finitely generated free right module over an abelian AW^∗-algebra. The topology of these unitary groups is examined and as an application we determine the homotopy groups π₁ and π₂ of the induced real, complex, and quaternionic pseudospheres.     

The Decay Parameter and Invariant Measures for Markovian Bulk-Arrival Queues with Control at Idle Time

We consider properties regarding decay parameter and invariant measures for unstable Markovian bulk-arrival Queues with control at idle time. The exact value of the decay parameter, denoted by λ _Z , is firstly revealed. An elegant criterion regarding λ _Z -recurrence and λ _Z -positive is obtained. The corresponding subinvariant and invariant measures are considered and the structure of all the subinvariant and invariant measures are presented.     

On the prediction of vector-valued random fields and the spectral distribution of their evanescent component

Let (X_m,n)_(m,n)∈Z² be a C^p-valued wide sense stationary process. We study the prediction theory of such processes according to different total orders on Z². In the case of a “rational order”, we give the spectral distribution of the resulting evanescent component and prove that for two different rational orders, the resulting evanescent components are mutually orthogonal.     

Dynamic Markov bridges motivated by models of insider trading

Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X₁=Z₁. We call X a dynamic bridge, because its terminal value Z₁ is not known in advance. We compute its semimartingale decomposition explicitly under both its own filtration F^X and the filtration F^X,Z jointly generated by X and Z. Our construction is heavily based on parabolic partial differential equations and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading that can be viewed as a non-Gaussian generalization of the model of Back and Pedersen (1998) [3], where the insider’s additional information evolves over time.     

Dilating covariant representations of the non-commutative disc algebras

Let φ be an isometric automorphism of the non-commutative disc algebra An for n?2. We show that every contractive covariant representation of (An,φ) dilates to a unitary covariant representation of (On,φ). Hence the C^∗-envelope of the semicrossed product Anφ_×Z⁺ is Onφ_×Z.     

Normalisation d'un processus de branchement critique dans un environnement aléatoire

Let (Z_n) be a critical branching process in an independent and identically distributed (i.i.d.) random environment. For each fixed environment ω, let C_n=E^ω[Z_n∣Z_n>0] be the conditional expectation of Z_n given Z_n>0. We prove an analogue of Yaglom's law: as n→∞, the conditional law of Z_n/C_n, conditional on Z_n>0, converges to a non-degenerate law on [0,∞). We give also an analogue of Kolmogorov's law, as well as a local limit theorem for the semi-group of probability generating functions. To cite this article: Y. Guivarc'h et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

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.     

Convexity properties of gradient maps

We consider the action of a real reductive group G on a Kähler manifold Z which is the restriction of a holomorphic action of a complex reductive group H. We assume that the action of a maximal compact subgroup U of H is Hamiltonian and that G is compatible with a Cartan decomposition of H. We have an associated gradient map μp:Z→p where g=k⊕p is the Cartan decomposition of g. For a G-stable subset Y of Z we consider convexity properties of the intersection of μp(Y) with a closed Weyl chamber in a maximal abelian subspace a of p. Our main result is a Convexity Theorem for real semi-algebraic subsets Y of Z=P(V) where V is a unitary representation of U.     

On recurrence and transience of self-interacting random walks

Let µ₁,...,µ _k be d-dimensional probabilitymeasures in ? ^d with mean 0. At each time we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.     

Codes over Zm

In this paper we study cyclic codes inZ _m. i.e., ideals inZ _mG, G a finite abelian group, and we give a classification of such codes.We also study the minimum Hamming distance and the generalized Hamming weight of BCH codes overZ _m.