Analysis of restrictions of unitary representations of a nilpotent Lie group

Let G be a connected simply connected nilpotent Lie group, K an analytic subgroup of G and π an irreducible unitary representation of G. Let DπK(G) be the algebra of differential operators keeping invariant the space of C^∞ vectors of π and commuting with the action of K on that space. In this paper, we assume that the restriction of π to K has finite multiplicities and we show that DπK(G) is isomorphic to a subalgebra of the field of rational K-invariant functions on the co-adjoint orbit Ω(π) associated to π, and for some particular cases, that DπK(G) is even isomorphic to the algebra of polynomial K-invariant functions on Ω(π). We prove also the Frobenius reciprocity for some restricted classes of nilpotent Lie groups, especially in the cases where K is normal or abelian.     

Dual topology of the Heisenberg motion groups

Let ? _n be the (2n + 1)-dimensional Heisenberg group, and let T ⁿ be the n-dimensional torus acting on ? _n by automorphisms. In this paper, we describe the space of admissible coadjoint orbits of the Heisenberg motion group G _n = T ⁿ ? ? _n and we determine the topology of this space. We show that the bijection between the unitary dual ? _n of G _n and its admissible coadjoint orbit space is a homeomorphism.     

The Dirac Equation in Geometric Quantization

The coadjoint orbit of the restricted Poincaré group corresponding to a mass m and spin 1/2 is described. The orbit is quantized using the geometric quantization. To include the discrete symmetries, one has to induce the irreducible representation of the restricted Poincaré group obtained by the quantization procedure to the full Poincaré group. The new representation is reducible and the reduction to an irreducible representation corresponds to the Dirac equation. Communicated by Rafael D. Benguria submitted 02/01/02, accepted: 14/02/02     

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.     

On the motivic class of the stack of bundles

Let G be a split connected semisimple group over a field. We give a conjectural formula for the motivic class of the stack of G-bundles over a curve C, in terms of special values of the motivic zeta function of C. The formula is true if C=P¹ or G=SLn. If k=C, upon applying the Poincaré or called the Serre characteristic by some authors the formula reduces to results of Teleman and Atiyah-Bott on the gauge group. If k=Fq, upon applying the counting measure, it reduces to the fact that the Tamagawa number of G over the function field of C is |π₁(G)|.     

Equivariant cohomology of real flag manifolds

Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I₀(P) and K=Gp is the stabilizer of p∈P. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K₀⊂K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K₀ is connected and the action of K₀ on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H^∗(X). In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.     

On the characters of exponential groups

Let G be an exponential solvable group, O an orbit of the coadjoint representation and T the corresponding irreducible unitary representation of G. A polynomial function P, such that P ¦ O is positive and semi-invariant, determines a positive, self-adjoint operator A on the space of T. Using the resulution of singularities by H. Hironaka, one shows, under suitable conditions on O, that the function t → Tr(A^tT()A^t)( ε C_c^∞(G), fix) admits a meromorphic analytic continuation, with poles on the real axis.     

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π₊ of G how does π₊ decompose on restriction to H. Here G is GL⁺(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL⁺(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K^∗), π₊ is an irreducible, admissible supercuspidal representation of GL⁺(2,F) and H=K^∗ under an embedding of K^∗ into GL(2,F).     

The group marriage problem

Let G be a permutation group acting on [n]={1,…,n} and V={Vi:i=1,…,n} be a system of n subsets of [n]. When is there an element g∈G so that g(i)∈Vi for each i∈[n]? If such a g exists, we say that G has a G-marriage subject to V. An obvious necessary condition is the orbit condition: for any nonempty subset Y of [n], there is an element g∈G such that the image of Y under g is contained in _?y∈YVy. Keevash observed that the orbit condition is sufficient when G is the symmetric group Sn; this is in fact equivalent to the celebrated Hall's Marriage Theorem. We prove that the orbit condition is sufficient if and only if G is a direct product of symmetric groups. We extend the notion of orbit condition to that of k-orbit condition and prove that if G is the cyclic group Cn where n?4 or G acts 2-transitively on [n], then G satisfies the (n−1)-orbit condition subject to V if and only if G has a G-marriage subject to V.     

On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt

Let K be a complete discrete valued field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G such that the induced extension of residue fields kL/kK is separable. Hesselholt (2004) [2] conjectured that the pro-abelian group {H¹(G,Wn(OL))}_n∈N is zero, where OL is the ring of integers of L and W(OL) is the ring of Witt vectors in OL w.r.t. the prime p. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt?s conjecture for all Galois extensions.     

On differentiable vectors for representations of infinite dimensional Lie groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations π:G→GL(V) of an infinite dimensional Lie group G on a locally convex space V. The first class of results concerns the space V^∞ of smooth vectors. If G is a Banach-Lie group, we define a topology on the space V^∞ of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V^∞ is a Fréchet space. This applies in particular to C^∗-dynamical systems (A,G,α), where G is a Banach-Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function 〈π(g)v,v〉 is smooth. The second class of results concerns criteria for Ck-vectors in terms of operators of the derived representation for a Banach-Lie group G acting on a Banach space V. In particular, we provide for each k∈N examples of continuous unitary representations for which the space of Ck+1-vectors is trivial and the space of Ck-vectors is dense.     

Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep2MatK(G) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π₀(G), π₁(G) and [α]∈H³(π₀(G),π₁(G)) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z]∈H²(π₀(G),K^∗)) of the first homotopy group π₀(G) as well as its category of representations on finite sets both live in Rep2MatK(G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep2MatK(G).     

Embedding and splitting ordinary differential equations in normal form

We introduce an embedding of real or complex n-dimensional space Kⁿ as an algebraic variety V which is determined by the action of a linear one-parameter group. Every analytic vector field on Kⁿ corresponds to some embedded vector field on V. For a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non-autonomous linear systems. Examples and applications are mostly concerned with Poincaré-Dulac normal forms. Embeddings provide a natural setting for perturbations of symmetric systems, in particular of systems in normal form up to some degree.     

Companionship and knot group representations

A companionship argument is used to give a constructive geometric proof of a key result concerning the knot homomorph problem: Given elements μ and λ in a group G, is there a knot K in S³ and a surjective representation ρ:π₁(S³−K)→G, such that ρ(m)=μ and ρ(l)=λ, where m and l are the meridian and longitude of K. The result presented here is that if for some μ that normally generates G, the pair (μ,μⁿ) is realizable, where n is the order of H₁(G;Z, then the pair (v,vⁿ) is realizable for any normal generator v.     

Linear preservers of orthogonal decomposable Tensors

Let G be a subgroup of the symmetric group S_m and V be an n-dimensional unitary space where nm. Let V(G) be the symmetry class of tensors over V associated with G and the identity character. Let D(G) be the set of all decomposable elements of V(G) and O(G) be its subset consisting of all nonzero decomposable tensors x ₁ ^?…^? x_m such that {x ₁,…,x_m } is an orthogonal set. In this paper we study the structure of linear mappings on V(G) that preserve one of the following subsets: (i)O(G), (ii) D(G)\(O(G)?{0}).     

Asymptotics of functions satisfying the γ-weak inequality with applications to uniformly bounded representations

Let G be a real reductive group of class $\text{H}$, and π a uniformly bounded representation of G on a Hilbert space having infinitesimal character. We then show that the K-finite matrix elements of π decay “exponentially” on G provided that the infinitesimal character of π is in general position. Further we show that π is infinitesimally equivalent to a subquotient of a cuspidal principal series representation π_Q,ω,ν where ν belongs to a tube domain defined by ?Q. These facts follow from the asymptotics of functions satisfying the γ-weak inequality.     

Symmetric powers of modular representations,hilbert series and degree bounds

Let G = Z _p be a cyclic group of prime order p with a representation G → GL(V) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert series of the invariant ring K[V]^G. Following Almkvist and Fossum in broad outline, we start by giving a shorter, self-contained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our results generalize to groups of the type Z _p ×H with |H| coprime to p. Moreover, we prove that for any finite group G whose order is divisible by p but not by p ² the invariant ring A,K[V]G is generated by homogeneous invariants of degrees at most dim (V).(|G| – 1).     

Order estimates for irreducible projections in L2 of a nilmanifold

Let π be an irreducible representation occurring in $\text{L}$²(Г?N), where N is a nilpotent Lie group and Γ is a discrete, cocompact subgroup. The projection onto the π-equivariant subspace is given by convolution against a distribution D_π. For certain π, we obtain an estimate on the order of D_π. The condition on π involves an extension of the “canonical objects” associated to elements of the Kirillov orbit of π; there does not appear to be an example in the literature where it is not satisfied.     

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

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})}.