Introduction Permutative representations of $\mathcal O_N$ Monomial representations and integers modulo $N$ Cycles of irreducible $\mathcal O_N$ representations and their atoms of UHF$_N$ representations Relations of finite type and sub-Cuntz states The shift representation The universal permutative multiplicity-free representation Some specific examples of the cycle and atom structure: The mod $N$ case Some specific examples of the cycle and atom structure: The mod $\mathbf N$ case The general mod $\mathbf N$ situation Concluding remarks Bibliography List of symbols.

Iterated Function Systems and Permutation Representations of the Cuntz Algebra

Let L be a left invariant sub-Laplacian on a connected Lie group G of polynomial volume growth, and let {EA, )A*> O} be the spectral resolution of L and m a bounded Borel measurable function on [0, oo) . In this article we give a sufficient condition on m for the operator m(L) = fo' m(A) dEA to extend to an operator bounded on LP(G), 1 0 such that IUn" ? cnC, n E N. Then G becomes unimodular. Furthermore, there is an integer D > 0, such that (cf. Guivarc'h [7]) lUn -nD (n -oo). Notice that every connected nilpotent Lie group has polynomial volume growth. Let X1, . .. , Xn be left invariant vector fields on G that satisfy Hormander's condition, i.e., they generate together with their successive Lie brackets [Xi,, [X)2, [... , XJ ... ], at every point of G, the tangent space of G . To those vector fields is associated, in a canonical way, the control distance d(., *) (cf. [22, 23], which are our basic references for results concerning the Harmonic analysis on G). This distance is left invariant and compatible with the topology on G. We put Ixl = d(e, x) and Br(x) = {y E G: d(x, y) 0. Then, we know that there is d E N, not depending on x (cf. [15, 22]), such that (1) IBr(X)H rd (r--O), IBr(X) rD (r -oo). We consider the sub-Laplacian L=-_(X2 + ..+ 2). L is a left-invariant hypoelliptic second-order differential operator (cf. [9]). It is also positive and selfadjoint (having as domain of definition the set {f E Received by the editors April 24, 1992 and, in revised form, July 6, 1992. 1991 Mathematics Subject Classification. Primary 22E25, 22E30, 43A80.

/pdf/spectral-multipliers-on-lie-groups-of-polynomial-growth-q6kgyi64ld.pdf

Spectral multipliers on Lie groups of polynomial growth

A criterion is given for the LP boundedness of a class of spec- tral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theo- rem of Hormander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction.

/pdf/bounds-for-spectral-multipliers-on-nilpotent-groups-4fum7dfjko.pdf

bounds for spectral multipliers on nilpotent groups

Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds

The paper is a survey of relationships among the following possible properties of a Riemannian homogeneous space X=G/K: Selberg's property of weak symmetry, commutativity of the algebra of K-invariant measures on X, commutativity of the algebra of G-invariant differential operators on X, commutativity of the Poisson algebra of G-invariant functions on the cotangent bundle of the space X, and (if G is a reductive group) the property of the spectrum of the linear representation of the group G on the algebra of polynomial functions on X being multiplicity-free. Diverse results on structure and classification are presented, including the author's classification of irreducible Riemannian homogeneous spaces of Heisenberg type for which the Poisson algebra of invariant functions on the cotangent bundle is commutative.

https://iopscience.iop.org/article/10.1070/RM2001v056n01ABEH000356/pdf

Commutative homogeneous spaces and co-isotropic symplectic actions

Let G be a locally compact group, and let K be a compact subgroup of Aut(G) , the group of automorphisms of G. There is a natural action of K on the convolution algebra L (G), and we denote by LK(G) the subalgebra of those elements in L (G) that are invariant under this action. The pair (K, G) is called a Gelfand pair if LI(G) is commutative. In this paper we consider the case where G is a connected, simply connected solvable Lie group and K C Aut(G) is a compact, connected group. We characterize such Gelfand pairs (K, G), and determine a moduli space for the associated K-spherical functions.

/pdf/on-gelfand-pairs-associated-with-solvable-lie-groups-1eg6k91a1p.pdf

On Gelfand Pairs Associated with Solvable Lie Groups

Bounded K-spherical functions on Heisenberg groups

/pdf/spectra-for-gelfand-pairs-associated-with-the-heisenberg-415bxuh708.pdf

Spectra for Gelfand pairs associated with the Heisenberg group

Let K be a closed subgroup of U(n) acting on the (2n+ 1)dimensional Heisenberg group Hn by automorphisms. One calls (K,Hn) a Gelfand pair when the integrable K-invariant functions on Hn form a commutative algebra under convolution. We prove that this is the case if and only if the coadjoint orbits for G := K n Hn which meet the annihilator k⊥ of the Lie algebra k of K do so in single K-orbits. Equivalently, the representation of K on the polynomial algebra over C is multiplicity free if and only if the moment map from C to k∗ is one-to-one on K-orbits. It is also natural to conjecture that the spectrum of the quasi-regular representation of G on L(G/K) corresponds precisely to the integral coadjoint orbits that meet k⊥. We prove that the representations occurring in the quasi-regular representation are all given by integral coadjoint orbits that meet k⊥. Such orbits can, however, also give rise to representations that do not appear in L(G/K).

/pdf/a-geometric-criterion-for-gelfand-pairs-associated-with-the-1zcrkeb14x.pdf

A geometric criterion for gelfand pairs associated with the heisenberg group

Let /. be a positive. Rockland operator of homogeneous degree y. The minimum eigenvalue of dtr(L) increases as the yth power of the homogeneous distance from the origin of the orbit corresponding to w. Let G be a connected, simply connected nilpotent Lie group with Lie algebra g. Let A be a diagonalizable operator on g with eigenvalues 1 = y, < • • • < y , such that for each r > 0, the operator 8r = rA is an automorphism of g and, hence, determines an automorphism, again denoted by 8r, of G. Let | ■ | denote a homogeneous gauge on G; i.e., | • | is a continuous, nonnegative function on G with \g\ = 0 if, and only if, g = e, and satisfying |ôrg| = r\g\ for r > 0. Let g* be the dual of g, and let 8* denote the adjoint of 8r acting on g*. We define | • | on g by \X\ = |exp X\ and on g * as follows: fix a basis {Xx,... ,Xd) of g consisting of eigenvectors of A, and let {X*,.. .,XJ] be the dual basis of g*. For £ g g* we set |£| = |£(f, X¡)%\. One easily sees that |ôrA'| = r\X\ and |S*£| = r\£\ for X g g, | g g*, and r > 0. Let g -* Ad*g be the coadjoint representation of G on g*, and for £ g g* let 0(1) = Ad*G • I Set |0(£)| = ini{[!'| |«'e O(è)}. By Kirillov theory [1] the equivalence classes of irreducible unitary representations of G, G, can be identified with {0(£)|£ e 0*}Given £ g g*, we denote by it^ the element of G corresponding to 0(£), an<3 by dir( the resulting representation of <%( g), the universal enveloping algebra of g. An element L of ^(g) is said to be homogeneous of degree y if L{f °8r) = ry(Lf)°8r for all smooth functions of/. As in [2], a homogeneous operator L is called a Rockland operator if dn^L) is injective (on the space of smooth vectors) for each | ¥= 0. L is said to be positive if (L/, /) ^ 0 for all / G Q^G). The proof by Nelson and Steinspring [6] that dm(L) is essentially self adjoint for any elliptic operator and any unitary representation it uses, in fact, only the hypoellipticity of L. Thus, by the theorem of Helffer and Nourrigat [3] dir^L) is essentially selfadjoint for any Rockland operator. Therefore, if L is a positive, Rockland operator, both L and drrAL) are infinitesimal generators of contraction semigroups. In [2] Folland Received by the editors October 31. 1983 and. in revised form May 22, 1984 and July 14, 1984. 1980 Mathematics Subject Classification. Primary 22E30, 43A55. 'This research was funded in part by National Science Foundation Grant MCS810078. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

/pdf/minimum-eigenvalues-for-positive-rockland-operators-3i60ilno7n.pdf

Joe Jenkins

Papers

On Gelfand Pairs Associated with Solvable Lie Groups

Bounded K-spherical functions on Heisenberg groups

Spectra for Gelfand pairs associated with the Heisenberg group

A geometric criterion for gelfand pairs associated with the heisenberg group

Minimum eigenvalues for positive, Rockland operators