Sur les semi-caracteres des groupes de Lie resolubles connexes

Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by $\text{D}$(G) the space of C^∞ functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra U$\text{G}$_$\text{c}$ of the complexification of the Lie algebra of G for which the linear form $\text{? ψ(π(u 1 ?))}$ on $\text{D}$(G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra U$\text{G}$_$\text{c}$.     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative.     

Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group

We show that the kernel of an irreducible unitary representation π of the group algebra L¹(G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups.     

Analyse harmonique sur les espaces symétriques nilpotents

Let G be a connected simply connected nilpotent Lie group and H the set of fixed points of an involution of G; we give the Plancherel formula of the representation Ind_H^G(1) and infer from it the existence of an H-invariant tempered elementary solution for every nonzero G-invariant differential operator on $\text{G}\text{H}$.     

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

 Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that the algebra of the differential operators on the Schwartz-space of τ which commute with τ is isomorphic to the algebra of H-invariant polynomials on the affine space . We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in . We prove also that if is an ideal of , then the finite multiplicities of τ is equivalent to the fact that the algebra is commutative. (Received 15 November 2000)     

Differential recursion relations for Laguerre functions on symmetric cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R⁺. This family forms an orthogonal basis for the subspace of L-invariant functions in L²(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L²(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R⁺.     

The Taylor map on complex path groups

Let W(G) denote the path group of an arbitrary complex connected Lie group. The existence of a heat kernel measure νt on W(G) has been shown in [M. Cecil, B.K. Driver, Heat kernel measure on loop and path groups, preprint, http://www.math.uconn.edu/~cecil/papers/p2.pdf; Infin. Dimens. Anal. Quantum Probab. Relat. Top., submitted for publication]. The present work establishes an isometric map, the Taylor map, from the space of L²(νt)-holomorphic functions on W(G) to a subspace of the dual of the universal enveloping algebra of Lie(H(G)), where H(G) is the Lie subgroup of finite energy paths. This map is shown to be surjective in the case where G is a simply connected graded Lie group.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Bounds on smooth matrix coefficients on L 2-spaces

Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and ${\mathfrak k}Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and \mathfrak k{\mathfrak k}-smooth matrix coefficients of the regular representation L ²(X) under an assumption about supp(L²(X)) ?[^(G)]_K{{\rm supp}(L^2(X)) \cap \hat G_K}. Furthermore, we show that this bound holds for unitary representations that are weakly contained in L ²(X). Our result generalizes a result of Cowling–Haagerup–Howe (J Reine Angew Math 387:97–110, 1988). As an example, we discuss the matrix coefficients of the O(p, q) representation L²(\mathbbR^p+q){L^2(\mathbb{R}^{p+q})}.     

The center of a rational cohomology algebra for some loop spaces with respect to the Pontryagin product

We solve the problem of calculation of the center of the rational cohomology algebra H*(ΩX;?) of the loop space of simply connected four-dimensional manifolds with the Pontryagin product. The well-known results of Milnor and Moore represent H*(ΩX) as a universal enveloping Lie algebra π*(ΩX) endowed with the Whitehead-Samelson bracket. Neisendorfer obtained a representation of the algebra Π*(ΩX) ? ? in terms of generators and relations for the case when X is a simply connected four-dimensional manifold. In this paper, the center Z(H*(ΩX;?)) is calculated using this representation.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set of K-orbits in the dual of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Δ(K, N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.     

Exponential actions and maximal ?-invariant ideals

Let V be an exponential ?-module, ? being an exponential Lie algebra. Put ? = exp ?. Then every orbit of V under the action of ? admits a closed orbit in its closure. If G= exp ? is a nilpotent Lie group and ? an exponential algebra of derivations of ?, then ? = exp ? acts on G, L ¹(G), (?) and the maximal ?-invariant ideals of L ¹(G), resp. of (?) coincide with the kernels Ker Ω, resp. Ker Ω∩ (?), where Ω is a closed orbit of ?^*. Received: 6 December 1996 / Revised version: 7 December 1997     

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let Z be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g^∗ with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G₂.     

Berezin-Toeplitz quantization on Lie groups

Let K be a connected compact semisimple Lie group and KC its complexification. The generalized Segal-Bargmann space for KC is a space of square-integrable holomorphic functions on KC, with respect to a K-invariant heat kernel measure. This space is connected to the “Schrödinger” Hilbert space L²(K) by a unitary map, the generalized Segal-Bargmann transform. This paper considers certain natural operators on L²(K), namely multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the generalized Segal-Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on KC. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.     

Lie properties of symmetric elements in group rings

Let ¹ be an involution of a group G extended linearly to the group algebra KG. We prove that if G contains no 2-elements and K is a field of characteristic $p\ne 2$, then the 1-symmetric elements of KG are Lie nilpotent (Lie n-Engel) if and only if KG is Lie nilpotent (Lie n-Engel).     

Deformation rigidity of the rational homogeneous space associated to a long simple root

As a continuation of our previous works we study the conjecture on the rigidity under Kähler deformation of the complex structure of rational homogeneous spaces G/P of Picard number 1, confirming its validity whenever G/P is associated to a long simple root. For these rational homogeneous spaces the minimal G-invariant holomorphic distribution D is spanned by varieties of minimal rational tangents, and, excepting the symmetric and the contact cases, the complex structure of G/P is completely determined by the nilpotent symbol algebra of the weak derived differential system of D. The problem is reduced, in a sense, to the invariance of this nilpotent symbol algebra under Kähler deformation. In our earlier works in relation to the question of the integrability of distributions spanned by varieties of minimal rational tangents we have established identities on Lie brackets using integral surfaces arising from pencils of rational curves. In the case on hand, at a point o∈G/P we prove that the nilpotent symbol algebra at o is nothing other than the universal Lie algebra generated by D_o subject to these identities on Lie brackets, by verifying that they correspond to finiteness condition in the Serre presentation of the simple Lie algebra G.     

Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T ^*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2(G,K) the maximal number of functionally independent functions from A\C. We prove that (G,K) is equal to the codimension (G,K) of maximal dimension orbits of the Borel subgroup BG ^C in the complex algebraic variety G ^C/K ^C. Moreover, if (G,K)=1, then all G-invariant Hamiltonian systems on T ^*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.