Global solvability of the laplacians on pseudo-Riemannian symmetric spaces

Let G be a noncompact semisimple Lie group with finite center and H an open subgroup of the fixed point group of an involution of G. $\text{G}\text{H}$ becomes a pseudo-Riemannian manifold. We prove that the Laplacian P on $\text{G}\text{H}$ is globally solvable in the sense that $\text{PC}{}^{\mathrm{\infty }}\text{(}\text{G}\text{H}\text{) = C}{}^{\mathrm{\infty }}\text{(}\text{G}\text{H}\text{)}$. This generalizes the global solvability of the Casimir operators on non-compact semisimple Lie groups with finite center due to J. Rauch and D. Wigner.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

Square-integrable representations and the dual topology

We show that the square-integrable factor representations of a connected locally compact group G are precisely the normal representations whose kernels in $\text{C}{}^1\text{(G)}$ are open points of Prim_r(G) (the support of Plancherel measure). Related results hold for certain other groups. We also settle questions of Dixmier, and Duflo and Moore, by giving examples of square-integrable irreducible representations (of totally disconnected groups) which are not open in the reduced dual.     

Représentations de Gelfand–Graev pour les groupes réductifs non connexesGelfand–Graev representations for non connected reductive groups

Let G be a connected reductive group defined over $\text{F}{}_{\mathrm{q}}$ and let F be the corresponding Frobenius endomorphism. Let σ be a quasi-central rational automorphism of G. We define in this article Gelfand–Graev representations of the group $\text{G}{}^{\mathrm{F}}\text{=G}{}^{\mathrm{F}}\text{.〈σ〉}$ when σ is unipotent and when it is semi-simple. We show that they have similar properties to Gelfand–Graev representations of the group G^F. To cite this article: K Sorlin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 179–184.     

Representations of differential operators on a Lie group

In this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space $\text{B}$. Let $\text{E}$ be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of $\text{E}$ into operators with the Gårding vectors as a common invariant domain. We study elements in $\text{E}$ of the form

$\text{P=}\text{∑}\text{1}\text{r}\text{X}{}^2{}_{\mathrm{j}}\text{|X}{}_0$

with the X_j,'s in the Lie algebra $\text{G}$.If the elements X₀, X₁,…, X_r generate $\text{G}$ as a Lie algebra then we show that the space of C^∞-vectors for U is precisely equal to the C^∞-vectors for the closure $\text{dU(P)}\text{,}\text{of}\text{dU(P)}$. This result is applied to obtain estimates for differential operators.The operator $\text{dU(P)}$ is the infinitesimal generator of a strongly continuous semigroup of operators in $\text{B}$. If X₀ = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators $\text{dU(Σ}{}_1{}^{\mathrm{r}}\text{X}{}_{\mathrm{j}}{}^2\text{+ (?1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{X}{}_0$ with X₀ in the real linear span of the X_j's. An application to quantum field theory is given.Finally, the new characterization of the C^∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C^∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for $\text{G}$ (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators.     

Qualitative behavior of special functions on compact symmetric spaces

For a symmetric space $\text{G}\text{K}$ of compact type, the highest-weight vectors for representations of G occurring in $\text{L}{}^2\text{(G}\text{K)}$ become heavily concentrated near certain submanifolds of $\text{G}\text{K}$ as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators qλ = P_λqP_λ, where $\text{P}{}_{\mathrm{\lambda }}\text{: L}{}^2\text{(}\text{G}\text{K}\text{) → V}{}_{\mathrm{\lambda }}$ is an orthogonal projection onto a G-irreducible summand, and q: G/K → $\text{R}$ is a continuous function acting on $\text{L}{}^2\text{(G}\text{K)}$ by multiplication.     

Unbounded derivations commuting with compact group actions. II

Let ($\text{A}$, G, α) be a $\text{C}{}^1$ dynamical system and let δ be a closed ¹ derivation in $\text{A}$ which commutes with α and satisfies $\text{A}$^G ? ker(δ). If $\text{A}$ is a separable Type I $\text{C}{}^1$ algebra and G is a second countable compact group, then δ generates a strongly continuous one parameter group of ¹ automorphisms of $\text{A}$.     

On nuclear C1-algebras

A C¹-algebra is called nuclear if there is a unique way of forming its tensor product with any other C¹-algebra. Takesaki [17] showed that all C¹-algebras of type I and all inductive limits of such algebras are nuclear, but that the C¹-algebra $\text{C}{}_{\mathrm{r}}{}^1\text{(G)}$ generated by the left regular representation of G on l²(G) is nonnuclear, where G is the free group on two generators. In this paper an extension property for tensor products of C¹-algebras is introduced, and is characterized in terms of the existence of a certain family of weak expectations on the algebra. Nuclearity implies the extension property, and this is used to show that for a discrete group $\text{G, C}{}_{\mathrm{r}}{}^1\text{(G)}$ is nuclear if and only if G is amenable.An approximation property in the dual of a C¹-algebra is introduced, and shown to imply nuclearity. It is not clear whether the converse implication holds, but it is proved that the known nuclear C¹-algebras satisfy the approximation property.     

Infinite Eulerian tessellations

An Eulerian path in a graph G is a path π such that (1) π traverses each edge of G exactly once in each direction, and (2) π does not traverse any edge once in one direction and then immediately after in the other direction. A tessellation T of the plane is Eulerian if its l-skeleton G admits an Eulerian path. It is shown that the three regular tessellations of the Euclidean plane are Eulerian. More generally, if T is a tessellation of the plane such that each face has 2t least p sides and each vertex has degree (number of incident edges) at least q, where $\text{1/p+1/q≤}\text{1}\text{2}$, then, except possibly for the case p = 3 and q = 6, T is Eulerian. Let $\text{T}{}^1$ be the truncation of T. If every vertex of T has degree 3, then $\text{T}{}^1$ is not Eulerian. If every vertex has degree 4, or degree at least 6, then T is Eulerian.     

Levi-flat extensions from a part of the boundary

Let G be a bounded domain in $\text{C}\text{×}\text{R}$ such that $\text{G×}\text{R}\text{?}\text{C}{}^2$ is strictly pseudoconvex and U an open subset of bG. We define an open subset $\text{Ω}{}^{\mathrm{U}}$ of $\text{G}$ with the property $\text{Ω}{}^{\mathrm{U}}\text{∩bG=U}$ such that the following extension theorem holds true: for every ?∈C(U) there exist two functions $\text{Φ}{}^{\mathrm{\pm }}\text{∈C(Ω}{}^{\mathrm{U}}\text{)}$ such that Φ^±|_U=? and the graphs Γ(Φ^±) of Φ^± are Levi-flat over $\text{Ω}{}^{\mathrm{U}}\text{∩G}$. Moreover, for each $\text{Φ∈C(Ω}{}^{\mathrm{U}}\text{)}$ such that Φ|_U=? and Γ(Φ) is Levi-flat over $\text{Ω}{}^{\mathrm{U}}\text{∩G}$ one has Φ^??Φ?Φ⁺ on $\text{Ω}{}^{\mathrm{U}}$. We also show that if G is diffeomorphic to a 3-ball and U is the union of simply-connected domains each of which is contained either in the “upper” or in the “lower” part of bG (with respect to the u-direction), then $\text{Ω}{}^{\mathrm{U}}$ is the maximal domain of Levi-flat extensions for some function ?∈C(U). To cite this article: N. Shcherbina, G. Tomassini, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Extensions of factorial states of C1-algebras

Let A be a $\text{C}{}^1$-algebra, B be a $\text{C}{}^1$-subalgebra of A, and φ be a factorial state of B. Sometimes, φ may be extended to a factorial state of A by a tensor product method of Sakai (“$\text{C}{}^1$-algebras and $\text{W}{}^1$-algebras, Springer-Verlag, Berlin/Heidelberg/ New York 1971”). Sometimes, there is a weak expectation of A into $\text{π}{}_{\mathrm{\phi }}\text{(B)}$, and then factorial extensions may be found by a method of Sakai and Tsui (Yokohama Math. J.29 (1981), 157–160). These two methods are shown to have the same effect, and the factorial extensions produced by them are analysed.     

The Poisson transform and representations of a free group

Let G be a free group with r generators, 1 < r < ∞. All the eigenfunctions of an operator on G which plays the same role of the Laplace Beltrami operator on semisimple Lie groups are characterized. Furthermore, an analytic family of representations π_z of G on functions on the boundary Ω is considered, defined by $\text{π}{}_{\mathrm{z}}\text{(x)?(ω) = p}{}^{\mathrm{z}}\text{(x, ω)?(x}{}^{\mathrm{?1}}\text{ω)}$, where p(x, ω) is the Poisson kernel relative to the action of G on Ω. It is proved that, for 0 < s = Re z < 1, π_z is uniformly bounded on an appropriate Hilbert space $\text{H}{}_{\mathrm{s}}\text{(Ω)}$. Finally the uniform boundedness of other special representations of G, obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of GL(2, Q_p) is proved.     

An inertial decomposition and related range results

With quasicommutative n-square complex matrices A₁,…,A_s and s-square hermitian G=(g_ij), relationships are given between the image $\text{Σ}{}^{\mathrm{s}}{}_{\mathrm{i,j=1}}\text{g}{}_{\mathrm{ij}}\text{A}{}_{\mathrm{i}}\text{HA}{}^1{}_{\mathrm{j}}$ of a linear transformation on $\text{H}$_n being positive definite and the action of H on generalized inertial decompositions of $\text{C}$ⁿ.     

Certain conjugacy classes of units in integral group rings of metacyclic groups

Let G be the metacyclic group of order pq given by

$\text{G = 〈σ, τ: σ}{}^{\mathrm{p}}\text{= 1 = τ}{}^{\mathrm{q}}\text{, τστ}{}^{\mathrm{?}}\text{= σ}{}^{\mathrm{j}}\text{〉}$

where p is an odd prime, q ≥ 2 a divisor of p ? 1, and where j belongs to the exponent q mod p. Let V denote the group of units of augmentation 1 in the integral group ring $\text{Z}$G of G. In this paper it is proved that the number of conjugacy classes of elements of order p in V is

$\text{(p ? 1)}{}^{\mathrm{q?1}}\text{μ}{}_0\text{H}\text{vq}$

where ν, μ₀, and H are suitably defined numbers.     

Some pathologies in the Mackey analysis for a certain nonseparable group

Suppose G is a separable locally compact group and N is a closed normal subgroup. If the dual N? is smooth and the orbit space $\text{N}\text{?}\text{G}$ is smooth for the natural action of G on $\text{N}\text{?}\text{(L}{}^{\mathrm{x}}\text{(n) = L(xnx}{}^{\mathrm{?1}}\text{))}$, the method of G. W. Mackey (Acta Math.99 (1958), 265–311) gives a fairly simple procedure for constructing the dual ?. In this paper we examine an example which shows that the nonseparable case is much more complicated. In the example, N is abelian, $\text{N}\text{?}\text{G}$ is finite and even when the stabilizer is N there are many irreducible representations of G associated with the same orbit.     

Sufficiency and invariance

Suppose that a statistical decision problem is invariant under a group of transformations g?G. T (X) is equivariant if there exists $\text{g}{}^1\text{? G}{}^1$ such that $\text{T(g(X)) = g}{}^1\text{(T((X))}$. We show that the minimal sufficient statistic is equivalent and that if T(X) is an equivariant sufficient statistics and d(X) is invariant under G, then $\text{d}{}^1\text{(T) = Ed(X)∥T}$ is invariant under $\text{G}{}^1$.     

Harmonic analysis on hyperboloids

The regular representation of O(n, N) acting on $\text{L}{}^2\text{(}\text{O(n, N)}\text{O(n, N ? 1)}\text{)}$ is decomposed into a direct integral of irreducible representations. The homogeneous space $\text{O(n, N)}\text{O(n, N ? 1)}$ is realized as the Hyperboloid $\text{H = {(x, t) ? R}{}^{\mathrm{n\; +\; N}}\text{: ¦ t ¦}{}^2\text{? ¦ x ¦}{}^2\text{= 1}}$. The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □_h on H, which is the tangential part of the operator □ = Δ_x ? Δ_t on R^{n + N}. The spectrum of □_h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □u = 0 which vanish in the region $\text{¦ t ¦ < ¦ x ¦}$, and a continuous part $\text{H}$_?. As a representation of O(n, N), $\text{H}$_? ⊕ $\text{H}$_? is unitarily equivalent to the regular representation on L² of the cone $\text{{(x, t) : ¦ x ¦}{}^2\text{= ¦ t ¦}{}^2\text{}}$, and the intertwining operator is obtained by solving the equation □u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL(2, R).     

The KMS condition and spectral passivity in group duality

Let (A, G, α) be a C¹-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in $\text{G}\text{?}$ and a finite constant κ such that $\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}{}^1\text{x}{}_{\mathrm{i}}\text{) ? κ}\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}{}^1\text{)}$ whenever x_i lies in a spectral subspace $\text{R}{}^{\mathrm{\alpha }}\text{(Ω}{}_{\mathrm{i}}\text{)}$, where $\text{Ω}{}_1\text{+ … + Ω}{}_{\mathrm{n}}\text{? Ω}$. This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G.     

Random near-regular graphs and the node packing problem

Nemhauser and Trotter [12] proposed a certain easily-solved linear program as a relaxation of the node packing problem. They showed that any variables receiving integer values in an optimal solution to this linear program also take on the same values in an optimal solution to the (integer) node packing problem. Let π be the property of graphs defined as follows: a graph G has property π if and only if there is a unique optimal solution to the linear-relaxation problem, and this solution is completely fractional. If a graph G has property π then no information about the node packing problem on G is gained by solving the linear relaxation. We calculate the asymptotic probability that a certain type of ‘sparse’ random graph has property π, as the number of its nodes tends to infinity. Let m be a fixed positive integer, and consider the following random graph on the node set {1,2 …, n}). We join each node, j say, to exactly m other nodes chosen randomly with replacement, by edges oriented away from j; we denote by G_n(m) the undirected graph obtained by deleting all orientations and allowing all parallel edges to coalesce. We show that, as n → ∞,

$\text{P(G}{}_{\mathrm{n}}\text{(m)}\text{has property}\text{π)→}\text{0}\text{if}\text{m = 1,}\text{1}\text{if}\text{m ? 3,}$

and we conjecture that $\text{P(G}{}_{\mathrm{n}}\text{(2)}\text{has property}\text{π)→ (1–2e}{}^{\mathrm{?2}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

On a theorem of Osborne and Warner. Multiplicities in the cuspidal spectrum

Osborne and Warner have given a formula for the multiplicity of an integrable discrete series representation in $\text{L}{}^2\text{(}\text{G}\text{Γ}\text{)}$ when G is real rank one semi-simple Lie group and Γ is a discrete subgroup of co-finite volume. We simplify and evaluate this formula to show that for most G (as above) the multiplicity is the formal degree of the representation times the volume of $\text{G}\text{Γ}$. When it is not we give a simple interpretation of the difference.