Nilpotent Lie Groups: Fourier Inversion and Prime Ideals

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups G=exp(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G= \hbox {exp}({\mathfrak {g}})$$\end{document} by showing that operator fields defined on suitable sub-manifolds of g∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {g}}^*$$\end{document} are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of L1(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1(G)$$\end{document} as kernels of sets of irreducible representations of G.


Introduction
For a connected, simply connected, nilpotent Lie group G, the description of its spectrum and the Fourier inversion theorem are due to Kirillov [3], who showed that the dual space G of G is in one-to-one correspondence with the space g * /G of co-adjoint orbits of G. R. Howe proved in [2] that for every irreducible unitary representation (π, H π ) of G and every smooth linear operator a on H π there exists a Schwartz function f a on G such that π( f a ) = a. He also showed that the mapping a → f a is linear and continuous with respect to the Fréchet topology of the space B ∞ (H π ) of smooth linear operators on H π and the Fréchet topology on the space S(G) of Schwartz functions on G. In [12], N. Pedersen gave a precise construction on the map a → f a , for a fixed representation, using the trace function.
In this paper, we study a general version of the Fourier inversion theorem for nilpotent Lie groups. More precisely, we generalise the result of Howe's mentioned above by constructing a continuous retract from the space of adapted smooth kernel functions defined on a smooth G-invariant sub-manifold M of g * and supported in a subset G · M of M, where M is a relatively compact open subset of M, into the space S(G). Note that since we work with various representations at the same time, we cannot apply the construction of Pedersen's, unless the manifold is an open subset of g * . The main difficulty with this retract construction is that the spectrum of nilpotent Lie groups is not Hausdorff, thus given a smooth operator field defined on a small (Hausdorff) region in the spectrum, one cannot take an extension as in the usual locally compact Hausdorff space case. We will apply the variable group techniques developed in [11] to prove this result, which we call the Retract Theorem, by induction on the length |I | of the largest index set I for which (B × g * ) I ∩ M = ∅ (defined in Sect. 2.4).
Once we have the Retract Theorem, we can apply it to study the G-prime ideals of the Banach algebra L 1 (G). Here G denotes a Lie subgroup of the automorphism group of G with the property that the G-orbits in g * are all locally closed. The Retract Theorem implies that the Schwartz functions contained in the kernel of a G-orbit in G are dense in the L 1 (G)-kernel of . Using the methods in [7], it follows that every G-prime ideal in L 1 (G) is the kernel of such a G-orbit . This result can be used, for instance, in the study of bounded irreducible representations (π, X ) of a Lie group G on a Banach space X . Restricting the representation π to the nilradical G, one obtains the kernel ker(π |G ) of π |G in the algebra L 1 (G). The ideal ker(π |G ) is then G-prime. If ker(π |G ) is given as the kernel in L 1 (G) of a G-orbit G · π 0 ⊂ G for some π 0 ∈ G, then one can use π 0 to make an analysis of π as Mackey did in the case of unitary representations.
Prime ideals in L 1 (G) have been studied by various authors in different settings. For connected, simply connected, nilpotent Lie groups, J. Ludwig showed in [6] that the closed prime ideals of L 1 (G) coincide with the kernels of the irreducible unitary representations. In 1984, D. Poguntke studied the action of an abelian compact group K on a nilpotent Lie group [13] and characterised the K -prime ideals as kernels of K -orbits. In [4], R. Lahiani and C. Molitor-Braun identified the K -prime ideals with hull contained in the generic part of the dual space of G for a general compact Lie subgroup K of the automorphism group of G. In [7] and [8], it was shown that for an exponential Lie group G, the G-prime ideals are also kernels of G-orbits. In this way the bounded irreducible Banach space representations of an exponential Lie group could be determined.
The paper is organised in the following way: in Sect. 2 we recall the definition of induced representations and of kernel functions, we explain the notion of variable nilpotent Lie groups and their Lie algebras, of index sets for co-adjoint orbits and of adapted kernel functions on a G-invariant sub-manifold of g * . In Sect. 3, we state our main theorem of the paper, the Retract Theorem, and in Sect. 4 we present the proof of the theorem, dividing it into several steps. As an application of the Retract Theorem, in the last section (Sect. 5) we show that every G-prime ideal in L 1 (G) is the kernel of a G-orbit.

Representations and Kernel Functions
Let G = exp(g) be a connected, simply connected, nilpotent Lie group and g be its Lie algebra. All the irreducible unitary representations of G (and hence of L 1 (G)) are obtained (up to equivalence) in the following way: Let l ∈ g * and p = p(l) be an arbitrary polarisation of l in g (a maximal isotropic subalgebra of g for the bilinear form (X, Y ) → l, [X, Y ] ). Let P(l) = exp(p(l)). The induced representation denoted by π l := ind G P(l) χ l on the Hilbert space H l , where dġ is the invariant measure on G/P(l), is unitary and irreducible. Here χ l is the character defined on P(l) by χ l (g) = e −i l,log g for all g ∈ P(l). Two different polarisations for the same l give equivalent representations. The same is true for the case of two linear forms l and l belonging to the same co-adjoint orbit. One particular way to obtain a polarisation is the following: Let {Z 1 , . . . , Z n } denote a Jordan-Hölder basis of g, for 1 ≤ k ≤ n, let g k := span{Z k , . . . , Z n } be the linear span of Z k , . . . , Z n and l k = l| g k for all l ∈ g * . The polarisation , is called the Vergne polarisation at l with respect to the basis Z 1 , . . . , Z n . We refer to [1] for more details on the theory of irreducible representations of nilpotent Lie groups.
Let π l = ind G P(l) χ l . The corresponding representation of L 1 (G), also denoted by π l , is obtained via the formula π l ( f )ξ := G f (x) π l (x)ξ dx, for all ξ ∈ H l . If f ∈ L 1 (G), then π l ( f ) is a kernel operator, i.e. it is of the form where F is the operator kernel given by If f is a Schwartz function, then the kernel function F belongs to C ∞ and satisfies the covariance relation and is a Schwartz function on G/P(l) × G/P(l).

Group Actions
Let G = exp(g) be a connected, simply connected, nilpotent Lie group and A be a Lie subgroup of the automorphism group Aut(G) of G acting smoothly on G. This action will be denoted by The action of A on G induces naturally actions of A on g, g * , G, L 1 (G), and on S(G). These group actions will lead to examples for our retract theory and provide an important application of retracts.

Variable Lie Algebras and Groups
We will prove our main theorem by induction; in our proofs, new parameters and new variations will appear. This may be handled most easily by the concept of variable Lie structures. Such structures were already considered in [5], [11], [10] and [9], among others. Definition 2.3.1 1 Let g be a real vector space of finite dimension n and B be an arbitrary nonempty set. We say that (B, g) is a variable (nilpotent) Lie algebra if (a) For every β ∈ B, there exists a Lie bracket [·, ·] β defined on g such that satisfy the following property: For all β ∈ B and k ≤ max{i, j}, a k i j (β) = 0. This means that {Z 1 , . . . , Z n } is a Jordan-Hölder basis for g β = (g, [·, ·] β ).
2 Assume that B is a smooth manifold. If the structure constants a k i j (β) vary smoothly on B, we say that (B, g) is a smooth variable (nilpotent) Lie algebra. We will denote (B, g) = (g, [·, ·] β ) β∈B for the variable Lie algebra.
For the rest of the paper we will assume that all variable Lie algebras are smooth. If B is reduced to a singleton, we have in fact no dependency on β in B but a fixed Lie algebra. To each variable Lie algebra, we associate a variable Lie group G β . The variable Lie group G := (G β ) β may be identified with the collection of Lie algebras (g, [·, ·] β ) β equipped with the corresponding Campbell-Baker-Hausdorff multiplications. If G = (G β ) β is a (smooth) variable Lie group endowed with a fixed Jordan-Hölder basis, then the corresponding Vergne polarisations, induced representations and operator kernels all depend on β ∈ B and l ∈ g * .
Let (β, l) ∈ B × g * . The Ludwig-Zahir indices I (β, l) defined in [11] can be obtained in the following way: Let g β (l) := {U ∈ g; l, [U, g] β ≡ 0} be the stabiliser of l in g β = (g, [·, ·] β ) and let a β (l) be the maximal ideal contained in g β (l). If a β (l) = g β (l) = g, then χ (β,l) (x) := e −i l,log β x is a character on G β and nothing has to be done. In this case, there are no Ludwig-Zahir indices, i.e. I (β, l) = ∅. Otherwise, let j 1 (β, l) = max{ j ∈ {1, . . . , n}; Z 0 j / ∈ a β (l)}, and We let We then consider which is an ideal of co-dimension one in g β . A Jordan-Hölder basis of (g 1 (β, l), [·, ·] β ) is given by . As previously we may now compute the indices j 2 (β, l), k 2 (β, l) of l 1 := l| g 1 (β,l) with respect to this new basis and construct the corresponding subalgebra g 2 (β, l) with its associated basis {Z 2 i (β, l); i = k 1 (β, l), k 2 (β, l)}. This procedure stops after a finite number r of steps. Let which is called the Ludwig-Zahir index of l in g β with respect to the basis {Z 1 , . . . , Z n }. The construction in [11] shows that the final subalgebra g r (β, l) obtained by this construction coincides with the Vergne polarisation of l in g β with respect to the basis Z 0 (see also [10], [9]). Note that the length |I | = 2r of the index set I = I (β, l) gives us the dimension of the co-adjoint orbit Ad * (G β ) . The vectors Y 1 (β, l), · · · , Y r (β, l) together with the stabiliser g β (l) of l in g β span the polarisation p β (l) = g r (β, l) and Let us introduce the following notations: For any index set I ∈ (N 2 ) r ≡ N 2r with r = 0, · · · , dim(g)/2, we let This last line corresponds to the Pukanszky section associated to the index I . In fact, in [9] it was proved that the indices j s (β, l), k s (β, l) coincide with the Pukanszky indices of the given layer (if one does not make any distinction between the j's and the k's). For many I 's, the subset (B × g * ) I is empty. Hence it is reasonable to define This gives a partition of B × g * into the different layers (B × g * ) I . The set I may be ordered lexicographically: if I = {( j 1 , k 1 ), · · · , ( j r , k r )}, I = {( j 1 , k 1 ), · · · , ( j r , k r )} ∈ I, we say that I < I if either 2r = |I | < |I | = 2r or there exists a ∈ {1, . . . , r } such that which means that either j a < j a or ( j a = j a and k a < k a ).
This allows us to define By induction on the length of the index sets, it is easy to see that for every I ∈ I there exists a smooth function P I on B × g * , which is polynomial in l for fixed β ∈ B such that (B × g * ) I = {(β, l); P I (β, l) = 0 for I > I and P I (β, l) = 0}. (2.3)

Co-adjoint Orbits
For any index set I , we consider the subspace s I of g * which is given by For each β ∈ B, let Then β,I is locally closed in s I , since we have the smooth functions P I , I ∈ I, defined on B × g * as in (2.3).
There exist functions p j : (B × g * ) I × R d → R, j = 1, · · · , n, which are rational in l ∈ g * and polynomial in z ∈ R d for fixed β ∈ B such that for every (β, l) ∈ (B×g * ) I , Furthermore if we write I = {i 1 < · · · < i d }, then p i j (β, l, z) = z j for j = 1, · · · , d, and for i / ∈ I , we have is also contained in M.

Schwartz Functions
Let r ∈ N, we define the space of (generalised) Schwartz functions S(R r , B, G) ≡ S(R r , B, g) ≡ S(R r , B, R n ) to be the set of all functions f from R r × B × G to C such that the functionf defined bỹ is smooth on R r × B × R n and that for any compact subset K of B, any finite collection T 1 , . . . , T s of smooth vector fields defined on the manifold B, and any A 1 , is equipped with the topology defined by the collection of all these seminorms. One may of course also use coordinates of the second kind to define the semi-norms on S(R r , B, G). Note that the space S(R r , B, G) does not depend on the choice of the Jordan-Hölder basis.

Kernel Functions
Let S be a subset of B × g * and L be a smooth manifold. We say that a mapping F : S → L is smooth, if the restriction of F to any smooth manifold N contained in S is smooth. Let B×g * be a smooth variable nilpotent Lie group with Jordan-Hölder basis Z. For any (β, l) ∈ B × g * , denote the Vergne polarisation p β,l at (β, l) associated to Z. We put π(β, l) := ind G P(β,l) χ l , with P(β, l) := exp β p(β, l), for the corresponding family of induced unitary representations. Then the mapping (β, We denote by D c M,r the space of all functions F : R r × M × G × G → C satisfying the following conditions. 1. F satisfies the covariance condition for every (β, l) ∈ M with respect to p(β, l), i.e.
for all α ∈ R r , p, q ∈ P(β, l) and x, y ∈ G.
2. The function F satisfies the following compatibility condition for α ∈ R r , (β, l) ∈ M and x, y, g ∈ G. This compatibility condition reflects the unitary equivalence of the representations π (β,l) and π (β,Ad * The function F has the Schwartz space property, i.e. for any I ∈ I the function F |R r ×M∩(B×s I )×G×G is smooth and that for any smooth differential operator D = D (β,l) on the manifold M, and any The space D c M,r will be equipped with the topology defined by the collection of all these semi-norms. This does of course not depend on the choice of the smooth Malcev basis of g with respect to the smooth family of Vergne polarisations. For an adapted field of kernel functions F on M, denote by op F the field of smooth operators defined through their kernel functions. For (β, l) ∈ M, the operator op F(β,l) acts on the space L 2 (G/P(β, l), χ (β,l) ) in the following way: Remarks 2.7.2.1 a) If we impose the condition that the support of (β, l) be contained in the set G · C 0 for a fixed subset C 0 of M, then we will denote the space of kernel functions by D C 0 M . b) One has a similar definition of the kernel functions if one takes another smooth family of polarisations together with a smooth family of Malcev bases.

The Retract Theorem
In this section, we state the main theorem of the paper which will be proved in the next section.
, ·, ·) as an operator kernel for all (α, (β, l)) ∈ R r × M. Moreover the mapping F → f is continuous with respect to the corresponding function space topologies.
If the variation is trivial, then we get the following theorem.
Proof It suffices to apply the Fourier inversion formula.
where P a (β, l) is the Pfaffian of the polynomial . It follows from [11] that the function f is Schwartz and the Fourier inversion theorem tells us that π (β,l) ( f ) = op F(β,l) for any (β, l) ∈ B × g * .

Proof of the Retract Theorem
The proof of Theorem 3.1 proceeds by induction on the length |I | of the largest index set I for which (B × g * ) I ∩ M = ∅ and it will be done in several steps.

The case I = ∅
Suppose that all the elements (β, l) ∈ M are characters of g β , which means that their index sets are empty. Let us replace the variable group (B, G) by the group (C, G), where C = B as a manifold, and the multiplications coming from C are abelian, i.e. [U, V ] γ = 0 for every U, V ∈ g and γ ∈ C. We identify now the group G with its Lie algebra and then U · γ V = U + V for every U, V ∈ g and γ ∈ C. This also means that χ l is a character on G γ = exp γ g, for all (γ , l) ∈ C × g * . Now take M = M. Let F ∈ S(R r × M) be a kernel function with compact support in the variables (γ , l). As R r × M is a sub-manifold of R r × C × g * , the function F may be extended to a Schwartz function F (in the sense of Sect. 2.6 and 2.7) on R r × C × g * with compact support in the variables (γ , l).

Reducing B
There are two cases where we can reduce the manifold B. Let F be a kernel function defined on R r × M ×G ×G such that its support in (β, l) is contained in G ·M ⊂ M 0 . By assumption, there exists f ∈ S(R r ×B 0 ×G) such that π (β,l) ( f (·, β, ·)) admits F(·, (β, l), ·, ·) as an operator kernel if (β, l) ∈ M 0 . In particular, π (β,l)

Reducing to smoothly varying subspaces depending on B
Let M ⊂ B × g * be a smooth G-invariant sub-manifold of B × g * . Let us fix the largest index where Then, by the definition of the indices ( j 1 , k 1 ), we have It is easy to see that n 1 β is an ideal in g. Let We fix a scalar product ·, · on g such that {Z 1 , . . . , Z n } is an orthonormal basis and we identify c * 1 with c 1 by identifying n r = j 1 +1 a r Z * r ∈ c * 1 with the element n r = j 1 +1 a r Z r of c 1 . Denote by · 2 the Euclidean norm on c 1 (and hence on c * 1 ) with respect to the given scalar product. We also identify with a subspace of c 1 . For all β ∈ B, we write c 1 = n 1 β ⊕ (n 1 β ) ⊥ and define p β to be the orthogonal projection of c 1 onto (n 1 β ) ⊥ . For each β ∈ B, a generating subset of n 1 β is given by Fix 0 ≤ k ≤ s, let I k = {J ⊂ {1, · · · , s}; |J | = k} and for β ∈ B, let It is easy to check that On the other hand, let n 1 := max β∈B M dim(n 1 β ), then we have Note that if we want n 1 β to be of fixed dimension and to have n 1 β , (n 1 β ) ⊥ and p β to vary smoothly with respect to β, we must restrict to B n 1 . But in general B n 1 is not a submanifold of B. Therefore we must find a smooth sub-manifold inside  Furthermore, since the function β → p β (Z β ) 2 2 is smooth on B, we can take β 0 ∈ B and 0 < δ < R < ∞ such that δ < p β 0 (Z β 0 ) 2 < R and by using the reduction argument, we can then assume that the number p β (Z β ) 2 is contained in the interval [δ, R] for any β ∈ B.

On the Manifold M
Let us focus on the manifold M again. Let (β 0 , l 0 ) ∈ M be fixed, but arbitrary. There exist 0 < δ < R < ∞ such that This is due to the fact that M ⊂ (B × g * ) I . According to Remark 4.3.1 we can now assume that Obviously, M red is open in M and thus is a smooth sub-manifold of M. On the other hand, we define

Construction of a New Variable Group
We start this section with an example which will demonstrate the use of a variable group and its variable algebra in the retract construction.

An Example
In this subsection, we will consider the free two-step nilpotent Lie group on four generators.
The co-adjoint orbits and the Pukanszky layers of the orbit spaces of the group G have been determined in the paper [4]. The non-generic index sets are given by the conditions Consider now the layer g * (3,4) , it is given precisely by a i Y * i ∈ g * | a 5 a 10 + a 8 a 7 − a 6 a 9 = 0, a 10 = 0}.
We must find a smooth sub-manifold M inside M red such that for a given smooth compactly supported operator field F(l) on M, we have a Schwartz function f defined on G such that for all u, y 3 ∈ R and l in M (here k F denotes the kernel function of the operator F(l)), and π l ( f ) = 0 for all l in g * (i, j) with (i, j) < (3,4). For simplicity of the notations, let Then we can consider M to be a smooth sub-manifold of (g 0 1 ) * . The construction of the manifold M and of the retract function in the general case will be shown in Sect. 4.5.
In our example, we take a smooth kernel function k(l, u, v), l ∈ M and u, v ∈ R, compactly supported in M and we extend it to a smooth function K (l, u, v) on (g 0 1 ) * which is compactly supported in l ∈ (g 0 1 ) * , vanishes for |l(Y 10 )| ≤ δ and is Schwartz in the variables (u, v). We can then find a function h ∈ S(g 0 1 × R 2 ) such that Finally let Then the function f has the required properties (by the computations in Sect. 4.5).
We have that by the definition of α(β, q) in (4.7). This means that δ(β, q) is an algebra homomorphism of g β = (g, [·, ·] β ) which does not vanish at the vector Z k 1 . Hence the subspace is an ideal of g β of co-dimension one and g = RZ k 1 ⊕ g 1 (β, q). (4.9) Furthermore g 1 (β, q) contains c 1 for any (β, q) ∈ B × c * 1 . In fact, Group (B 1 , G 1 ) In order to construct a new variation in the induction procedure, we put
Obviously the smooth manifold (B × g * ) ≤I,R,δ is diffeomorphic with the manifold R 2 × (B × g * ) 0 ≤I,R,δ . The mapping given by is such a diffeomorphism. Hence every smooth G-invariant sub-manifold M of (B × g * ) ≤I,R,δ can be decomposed into a direct product of R 2 with the smooth manifold For (β, l) ∈ B × g * , one has l 1 ( We remark that for (β, l) and (β, l ) in M with ι 1 (β, l) = ι 1 (β, l ) we have that l and l have the same restriction to g 1 (β, l) = g 1 (β, l ), so they are on the same co-adjoint orbit and l = Ad * (y)l for some y ∈ P(β, l) and hence notations of the multiplication and the exponential map, and we will identify g 1 , g 1 ∈ G 1 = (G 1 ) (β 1 ) β 1 ∈B 1 ≡ g 1 with the corresponding elements in G 1 . In the following computations, the parameters β and (β, v, l| c 1 ) will indicate how to multiply group elements or how to decompose smoothly the group elements, even if it is not marked explicitly. For ι 1 (β, l) = ((β, l, Z k 1 , l |c 1 ), l 1 ) ∈ M 1 , we put for α ∈ R r , u, t ∈ R and g 1 , g 1 ∈ G 1 , where c(β, l) := l, [Z k 1 , Z j 1 ] β = 0 and X = Z k 1 . This function F 1 has its support S 1 := ι 1 (S) contained in G · M 1 , and belongs to D c M 1 . The operator field F 1 is smooth on M 1 , since the mappings F and c are both smooth. By the induction hypothesis, there exists h ∈ S(R r +2 , B 1 , G 1 ) such thatπ ((β 1 ,l| c 1 ),l 1 ) h(α, u, t, β 1 , ·) admits F 1 (α, u, t, (β 1 , l 1 ), ·, ·) as an operator kernel for all (β 1 , l 1 ) ∈ M 0 1 . The construction of the retract function f will now be done in several steps.

Definition of the Retract Function on the Original Group
where Y = Z j 1 and Z = Z β = [X, Y ] β with X = Z k 1 . The integral converges, as h is Schwartz in g 1 (for fixed β 1 ), and it is rapidly decreasing in q ∈ (c 1 ) * , because it is a Fourier transform in Z . For all (β, v, q) ∈ B × R × c * 1 , we then define As f is of rapid decrease in q ∈ c * 1 by construction, we may define f by , (β, 0, q), g)dq, α ∈ R r , β ∈ B, g ∈ G.
One can see that f ∈ S(R r , B, G) (in the sense of Sect. 2.6).
The algorithm used to build the retract function f respects the semi-norms defining the topology of our function spaces. So the retract map F → f is continuous.

G-prime ideals in L 1 (G)
In this section, we will study the structure of the A-prime ideals in L 1 (G) by using the Retract Theorem.

A Retract Defined on Closed Orbits
Let G be a Lie group of automorphisms of a connected, simply connected, nilpotent Lie group G = exp(g) containing the inner automorphisms of G. For instance take any simply connected Lie group G and let G be the nilradical of G.
Let l 0 ∈ g * be fixed, we consider the orbit = l 0 := G · l 0 in g * , let O = O l 0 be the G-orbit of l 0 . We assume that is locally closed in g * . In particular we can write where denotes the closure of in g * and U is a G-invariant open subset of g * . It is then a G-invariant smooth sub-manifold of g * diffeomorphic to the manifold G/G l 0 , where G l 0 denotes the stabiliser G l 0 := {α ∈ G; α ·l 0 = l 0 }. The G-orbit G · (G ·l 0 ) in the orbit space g * /G is then locally closed and homeomorphic to the quotient G/G O , where G O is the stabiliser of the set O in G. In fact, we have that For a Jordan-Hölder basis Z = {Z 1 , · · · , Z n } of g and g ∈ G, let g · Z := {Ad(g)Z 1 , · · · , Ad(g)Z n }, which is again a Jordan-Hölder basis of g. For every index set I , we have the following relation (see [9]): Ad * (g)g * I,g·Z = g * I,Z , g ∈ G. (5.13) For an index set I and a Jordan-Hölder basis Z of g, recall that s I := i∈I RZ * i , I,Z := s I ∩ g * I,Z , and the mapping E I : R d × I,Z → g * I,Z is given by E I (s 1 , t 1 , · · · , s r , t r ; l) := Ad * (exp(s 1 Z j 1 )exp(t 1 Z k 1 ) · · · exp(s r Z j r )exp(t r Z k r ))l.
We have that E I is a bijection and E I (R d × {l}) is the G-orbit of l. Let where p I,Z is the projection of R d × I,Z onto I,Z . For the orbit , we need to construct a finite partition of unity (ψ i ) i∈ consisting of smooth G-invariant functions ψ i : → R + such that for every i ∈ the support of each function ψ i is contained in an open subset of g * I,g i ·Z for some g i ∈ G. In order to do that let ϕ : R → R + be a smooth function with compact support and vanish in a neighbourhood of 0. We define a function ψ : g * ≤I → R + by ψ(l) := ϕ(P I (ϒ(l))) if l ∈ g * I and ψ(l) := 0 if l ∈ g * <I , where P I is a smooth function on B × g * defined in Sect. 2.4. We see that ψ is smooth (since ϕ vanishes in a neighbourhood of 0) and is G-invariant by the construction. Let U I,Z := {l ∈ ; ψ(l) = 0}. Now assume that g * I = g * I,Z be the maximal layer with respect to Z such that ∩ g * I,Z = ∅. We have that ∩ g * I,g·Z = ∅ but ∩ g * I ,g·Z = ∅ for g ∈ G and I > I . Moreover, U I,Z is a non-empty open subset of contained in g * I and ⊂ g∈G Ad * (g)U I,Z .
Let C be a compact subset of g * contained in , then there exists a finite subset ⊂ G such that C ⊂ g∈ Ad * (g)U I,Z .
Hence there is a finite partition of unity (ψ i ) i consisting of smooth G-invariant functions ψ i : → R + such that the support of each function ψ i is contained in Ad * (g i )U I,Z ⊂ g * I,g i ·Z for every g i ∈ . Suppose we have a smooth adapted operator field F on supported on G · C, we can write According to the Retract Theorem, for each i ∈ , there is a (retract) Schwartz function f i on G such that for every l ∈ . Let f := i∈ f i , we have that This is, for every smooth adapted kernel function supported on G ·C, we build a retract function.

G-Prime Ideals
As an application, we show that every G-prime ideal in L 1 (G) is the kernel of a G-orbit. Let us first recall the definition of G-prime ideals. The set h(I) is then closed in Prim * (G) with respect to the Fell topology.
We have the following result for G-prime ideals of L 1 (G) which can be viewed as an application of the Retract Theorem.
Theorem 5.1 Let G be a simply connected, connected nilpotent Lie group and let G be a Lie group of automorphisms of G containing the inner automorphisms, which acts smoothly on the group G, such that every G-orbit in g * is locally closed. If I is a proper G-prime ideal of L 1 (G), then there exists an G-orbit l 0 in g * such that I = ker( l 0 ).

Moreover, the kernel of each G-orbit is a G-prime ideal.
Proof For any G-orbit in g * , the Retract Theorem tells us that the Schwartz functions contained in ker( ) are dense in ker( ) (see [7, proof of Proposition 4.1] and [4]). From the proof of [8,Theorem 1.2.12], it follows that the hull of a prime ideal I is the closure of a G-orbit in Prim * (G) G. On the other hand, the density of S(G) ∩ ker( ) implies that ker( ) N is contained in the minimal ideal J ( ) with hull for some N ∈ N. This tells us that ker( ) N ⊂ J ( ) ⊂ I, since the minimal ideal with hull is contained in every ideal with hull . Since I is G-prime, we have that I = ker( ).
Obviously the ideal ker( ) is G-prime for any G-orbit in g * . To see this, let I 1 and I 2 be two G-invariant ideals of L 1 (G) such that I 1 * I 2 ⊂ ker( ). This means that I 1 * I 2 ⊂ ker( ) ⊂ ker(π l ) for some l ∈ . We have then either I 1 or I 2 is contained in ker(π l ), since π l is irreducible. But if I 1 is contained in ker(π l ), it is also contained in ker(π k·l ) since I 1 is G-invariant. Hence I 1 ⊂ ker( ) and the proof is thus complete.