K-invariants in the universal enveloping algebra of the desitter group

Let G be a non-compact connected semisimple Lie group with finite center and let G^K denote the centralizer of a maximal compact subgroup K of G inG, the universal enveloping algebra over of the Lie algebra of G. In [4] Lepowsky defines an injective anti-homo morphism P:G ^KK ^MA, where M is the centralizer in K of a Cartan subalgebraa of the symmetric pair (G,K),K andA are the universal enveloping algebras over corresponding to K anda, respectively, andK ^M is the centralizer of M inK. The subalgebra P(G ^K) ofK ^MA has considerable significance in the infinite dimensional representation theory of G. In this paper we explicitly compute P(G ^K) when G=S0_o(4,1), and show how this result leads to the determination of all irreducible representations of G and its universal covering group Spin(4,1).Partially supported by CONICET (Argentina) grants.     

Pivoting techniques for symmetric Gaussian elimination

Summary In order to factorize an indefinite symmetric matrixG of the formG=LDL ^T whereL is a trivially invertible matrix andD is a diagonal matrix, we introduce a new kind of pivoting operation. The algorithm suggested maintains the stability and efficiency of the standard Cholesky decomposition whileG need not be positive definite. The problem of factorizingG+uu ^T whereu is a vector, scalar and the factors ofG are known, is also discussed.This research was supported in part by the Israeli National Council for Research and Development     

Arithmetic distance on compact symmetric spaces

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are larger groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple.Using Helgason spheres, S(), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G ^p(K ⁿ ), K=, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L be the geometric transformation group of M. Let L={: MM: is a diffeomorphism and preserves arithmetic distance}. Then L=L     

Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

Let be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andA ² () the standard weighted Bergman space of holomorphic functions on square-integrable with respect to the measureh(z, z) ^–p dz. Extending the recent result of Axler and Zheng for =D, =p=2 (the unweighted Bergman space on the unit disc), we show that ifS is a finite sum of finite products of Toeplitz operators onA ² () and is sufficiently large, thenS is compact if and only if the Berezin transform ofS tends to zero asz approaches . An analogous assertion for the Fock space is also obtained.The author's research was supported by GA AV R grant A1019701 and GA R grant 201/96/0411.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

On theg-convergence of nonlinear elliptic operators related to the dirichlet problem in variable domains

A concept ofG-convergence of operatorsA _s:W _s W _s ^* to an operatorA:W W ^* is introduced and studied under a certain relationship between Banach spacesW _s,s=1,2, ..., and a Banach spaceW. It is shown that conditions establishing this relationship for abstract spaces are satisfied by the Sobolev spacesW ^k,m ( _s) andW ^k,m(), where { _s} is a sequence of perforated domains contained in a bounded region R ⁿ. Hence, the results obtained for abstract operators can be applied to the operators of the Dirichlet problem in the domains _s.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 948–962, July, 1993.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

Perturbation bounds on the polar decomposition

The polar decomposition of ann ×n-matrixA takes the formA=MH whereM is orthogonal andH is symmetric and positive semidefinite. This paper presents strict bounds, (with no order terms), on the perturbationsM,H ofM andH respectively, whenA is perturbed byA. The bounds onM can also be applied to the orthogonal Procrustes problem.     

A symmetric numerical range for matrices

Summary For each normv on ⁿ, we define a numerical rangeZ _v, which is symmetric in the sense thatZ _v=Z_vD, wherev ^D is the dual norm.We prove that, fora ⁿⁿ,Z _v(a) contains the classical field of valuesV(a). In the special case thatv is thel ₁-norm,Z _v(a) is contained in a setG(a) of Gershgorin type defined by C. R. Johnson.Whena is in the complex linear span of both the Hermitians and thev-Hermitians, thenZ _v(a),V(a) and the convex hull of the usualv-numerical rangeV _v(a) all coincide. We prove some results concerning points ofV(a) which are extreme points ofZ _v(a).Part of this research was done while the authors were at the Mathematische Institut, Technische Universität, München, West Germany. The first author presented these results at the Seminar on Matrix Theory (Positivity and Norms) held in Munich in December, 1974. The second author also acknowledges support from the National Science Foundation under grant GP 37978X.     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.     

RUC-decompositions in symmetric operator spaces

We show that ifX is a Banach space of type 2 andG is a compact Abelian group, then any system of eigenvectors {x }_G (with respect to a strongly continuous representation ofG onX) is an RUC-system. As an application, we exhibit new examples of RUC-bases in certain symmetric spaces of measurable operators.Research supported by the Australian Research Council     

Asymptotic cone of semisimple orbits for symmetric pairs

Let G be a reductive algebraic group over C and denote its Lie algebra by g. Let Oh be a closed G-orbit through a semisimple element h∈g. By a result of Borho and Kraft (1979) [4], it is known that the asymptotic cone of the orbit Oh is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer ZG(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair.More precisely, let θ be an involution of G, and K=Gθ a fixed point subgroup of θ. Then we have a Cartan decomposition g=k+s of the Lie algebra g=Lie(G) which is the eigenspace decomposition of θ on g. Let {x,h,y} be a normal sl₂ triple, where x,y∈s are nilpotent, and h∈k semisimple. In addition, we assume , where denotes the complex conjugation which commutes with θ. Then is a semisimple element in s, and we can consider a semisimple orbit Ad(K)a in s, which is closed. Our main result asserts that the asymptotic cone of Ad(K)a in s coincides with , if x is even nilpotent.     

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN= ₁ ⁿ m _iX_i+ _n ^+1s m _iX_i,D= _n ^+1s l _iX_i+ _s ^+1r n _iX_i, and theX _i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.     

Dimensions of irreducible modules over twisted group algebras

LetN be a normal subgroup of a finite groupG, letF be an algebraically closed field, letZ ²(G, F ^*) and letV be an irreducible module over the twisted group algebraF . If charF=p>0 divides (GN), assume thatG/N isp-solvable. It is proved that dim _F V divides (GN)d whered is the dimension of an irreducible constituent ofV _N. The special case where=1 andN is abelian yields a well-known theorem of Dade [3]. Another special case, namely whereN is abelian, charF(GN) and the restriction of ofNxN is a coboundary is a generalization of the main result of Ng [5].     

Craig-Sakamoto's theorem for the Wishart distributions on symmetric cones

A version of Craig-Sakamoto's theorem says essentially that ifX is aN(O,I _n ) Gaussian random variable in ⁿ, and ifA andB are (n, n) symmetric matrices, thenXAX andXBX (or traces ofAXX andBXX) are independent random variables if and only ifAB=0. As observed in 1951, by Ogasawara and Takahashi, this result can be extended to the case whereXX is replaced by a Wishart random variable. Many properties of the ordinary Wishart distributions have recently been extended to the Wishart distributions on the symmetric cone generated by a Euclidean Jordan algebraE. Similarly, we generalize there the version of Craig's theorem given by Ogasawara and Takahashi. We prove that ifa andb are inE and ifW is Wishart distributed, then Tracea.W and Traceb.W are independent if and only ifa.b=0 anda.(b.x)=b.(a.x) for allx inE, where the. indicates Jordan product.Partially supported by NATO grant 92.13.47.     

Automorphism groups of finite distributive lattices with a given sublattice of fixed points

For a latticeL, aL is a fixed point ofL if and only iff(a)=a for every automorphismf ofL. Let Aut (L) andS (L) denote the group of automorphisms ofL and the sublattice of fixed points ofL, respectively. It is shown that ifG is a finite group other thanZ ₂ ² ,Z ₂ ³ ,Z ₂ ⁴ ,Z ₃ ² or the quaternion group of order 8 andL is a finite automorphism free distributive lattice that is not near Boolean then there is a finite distributive latticeL such that Aut (L)G andS(L)L.The support of the National Research Council of Canada is gratefully acknowledged.     

Compactification of symmetric varieties

The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a wonderful compactification ofG/H. We prove the existence of such a compactification for arbitraryk. We also prove cohomology vanishing results for line bundles on the compactification. Dedicated to the memory of C. Chevalley     

The controllability of the equation x = ux

The equation x=uv, wherex R_n andu GM _n (M_n is the ring of all n × n real matrices), is considered. The equation is called weakly controllable if for arbitrary pointsa, b R ⁿ these exist pointsa and b' as near toa and b, respectively, as we like and a control transforming a into b. In this note algebraic criteria are given for the complete and the weak controllability of such equations in the case where the limiting set G is closed with respect to the operation of matrix multiplication and the G-module R_n is semisimple.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 253–259, February, 1978.     

Iterative Pexider equation modulo a subset

Summary LetX, Y, Z be arbitrary nonempty sets,E be a nonempty subset ofZ ^z andK be a groupoid. Assume that {F _t} _{t K} Z ^X, {G _t} _{t K} Y ^X, {H _t} _{t K} Z ^Y are families of functions satisfying the functional equationF _st = k_(s,t) H_s G_t for (s, t) D(K), whereD(K) stands for the domain of the binary operation on the groupoidK andk _(s,t) E for (s, t) D(K). Conditions are established under which the equation can be reduced to the corresponding Cauchy equation. This paper generalizes some results from [4] and [1].