# User:Maximilian Janisch/latexlist/Algebraic Groups/Dixmier mapping

Dixmier map

A mapping first defined for nilpotent Lie algebras by J. Dixmier in 1963 [a6], based on the orbit method of A.A. Kirillov [a12]. In 1966, Dixmier extended his definition to solvable Lie algebras [a7] (here and below, all Lie algebras are of finite dimension over an algebraically closed field of characteristic zero, cf. also Lie algebra; Lie algebra, solvable).

The Dixmier mapping is an equivariant mapping with respect to the adjoint algebraic group \$k\$ from the dual space \$a ^ { x }\$ of a solvable Lie algebra \$8\$ into the space of primitive ideals of the enveloping algebra \$U ( \mathfrak { g } )\$ of \$8\$ (cf. also Universal enveloping algebra; the adjoint algebraic group \$k\$ of \$8\$ is the smallest algebraic subgroup of the group of automorphisms of the Lie algebra \$8\$ whose Lie algebra in the algebra of endomorphisms of \$8\$ contains the adjoint Lie algebra of \$8\$). All ideals of \$U ( \mathfrak { g } )\$ are stable under the action of \$k\$.

The properties of the Dixmier mapping have been studied in detail. In particular, the Dixmier mapping allows one to describe (the) primitive ideals of the enveloping algebra \$U ( \mathfrak { g } )\$ and to describe the centre of \$U ( \mathfrak { g } )\$.

The dual \$a ^ { x }\$ of \$8\$ is equipped with the Zariski topology and the space of primitive ideals of \$U ( \mathfrak { g } )\$ with the Jacobson topology.

The Dixmier mapping (for \$8\$ solvable) is surjective [a9], injective (modulo the action of the adjoint algebraic group \$k\$) [a17], continuous [a5], and even open [a13].

Hence the Dixmier mapping induces a homeomorphism between \$\mathfrak { g } ^ { * } / G\$ and the space \$( U ( g ) )\$ of primitive ideals of \$U ( \mathfrak { g } )\$ and allows a complete classification of the primitive ideals of \$U ( \mathfrak { g } )\$. The openness was an open question for quite a long while.

The Dixmier construction goes as follows: If \$f\$ is a linear form on the Lie algebra \$8\$, one chooses a subalgebra \$h\$ of \$8\$ which is a polarization of \$f\$. This means that the subalgebra \$h\$ is an isotropic subspace of maximal dimension for the skew-symmetric bilinear form \$f ( [ . ] )\$ (on \$8\$); hence the dimension of \$h\$ is one half of \$\operatorname { im } \mathfrak { g } - \operatorname { dim } \mathfrak { g } ( f )\$, where \$\mathfrak { g } ( f )\$ is the stabilizer of \$f\$ in \$8\$ with respect to the co-adjoint action of \$8\$ in \$a ^ { x }\$.

For solvable Lie algebras such polarizations always exist, whereas for arbitrary Lie algebras this is, in general, not the case. Let \$t\$ denote the linear form on \$h\$ defined as the trace of the adjoint action of \$h\$ in \$9 + 5\$. The linear form \$f + 1 / 2 tr\$ on \$h\$ defines a one-dimensional representation of the enveloping algebra \$U ( h )\$. Let \$j\$ denote its kernel and \$I ( f , h )\$ the largest two-sided ideal in \$U ( \mathfrak { g } )\$ contained in \$U ( g ) J\$. This is nothing else but the kernel of the so-called twisted induction from \$U ( h )\$ to \$U ( \mathfrak { g } )\$ of the one-dimensional representation of \$U ( h )\$ given by \$f\$. In the case of a nilpotent Lie algebra, the twist \$1 / 2 tr\$ is zero. The twisted induction on the level of enveloping algebras corresponds to the unitary induction on the level of Lie groups.

The ideal \$I ( f , h )\$ obtained (in the solvable case) in this way is independent of the choice of the polarization [a7], hence this ideal may be denoted by \$I ( f )\$. The ideal \$I ( f )\$ is a (left) primitive ideal, i.e. the annihilator of an irreducible representation (left module) of \$U ( \mathfrak { g } )\$. It is known that for enveloping algebras of Lie algebras, left and right primitive ideals coincide (see [a7] in the solvable case and [a14] in the general case). It should be noted that for solvable Lie algebras \$8\$ all prime ideals (hence especially all primitive ideals) of \$U ( \mathfrak { g } )\$ are completely prime [a7].

For solvable Lie algebras \$8\$, the Dixmier mapping associates to a linear form \$f\$ of \$8\$ this primitive ideal \$I ( f )\$. The \$k\$-equivariance follows immediately from the fact that this construction commutes with automorphisms of \$8\$. For a general description and references, see [a3] and [a8].

The definition has been extended in several directions:

1) To the Dixmier–Duflo mapping [a10], defined for all Lie algebras \$8\$ but only on the set of elements of \$a ^ { x }\$ having a solvable polarization. In particular, this set contains the open set of linear forms whose orbits under \$k\$ have maximal dimension. For solvable Lie algebras one gets the usual definition.

2) To the \$2\$-parameter Duflo mapping [a11]. This mapping is defined for algebraic Lie algebras \$8\$ (cf. also Lie algebra, algebraic). The first parameter is a so-called linear form on \$8\$ of unipotent type, the second parameter is a primitive ideal in the reductive part of the stabilizer in \$8\$ of the first parameter. The mapping goes into the space of primitive ideals of \$U ( \mathfrak { g } )\$. This mapping coincides with the Dixmier mapping if \$8\$ is nilpotent and it can be related to the Dixmier mapping if \$8\$ is algebraic and solvable. For \$8\$ semi-simple, the mapping reduces to the identity. This \$2\$-parameter Duflo mapping is surjective [a11] and it is injective modulo the operation of \$k\$ [a16].

3) The Dixmier mapping for \$s ( n )\$. This was done by W. Borho, using the above Dixmier procedure [a1]. The problem is its being well-defined. Because of the twist in the induction, this Dixmier mapping for \$s ( n )\$ (\$= \mathfrak { g }\$) is continuous only on sheets of \$a ^ { x }\$ but not as mapping in the whole. The sheets of \$a ^ { x }\$ are the maximal irreducible subsets of the space of linear forms whose \$k\$-orbits have a fixed dimension. The Dixmier mapping for \$s ( n )\$ is surjective on the space of primitive completely prime ideals of \$U ( \operatorname { si } ( n ) )\$ [a15] and it is injective modulo \$k\$ [a4].

4) The Dixmier mapping on polarizable sheets (sheets in which every element has a polarization) in the semi-simple case. This was done by W. Borho. This map is well-defined [a2] and continuous, and it is conjectured to be injective modulo \$k\$ (the conjecture is still open, October 1999).

#### References

 [a1] W. Borho, "Definition einer Dixmier–Abbildung für \$sl ( n , C )\$" Invent. Math. , 40 (1977) pp. 143–169 MR442046 Zbl 0346.17014 [a2] W. Borho, "Extended central characters and Dixmier's map" J. Algebra , 213 (1999) pp. 155–166 MR1674672 [a3] W. Borho, P. Gabriel, R. Rentschler, "Primideale in Einhüllenden auflösbarer Lie–Algebren" , Lecture Notes Math. , 357 , Springer (1973) Zbl 0293.17005 [a4] W. Borho, J.C. Jantzen, "Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra" Invent. Math. , 39 (1977) pp. 1–53 MR0453826 Zbl 0327.17002 [a5] N. Conze, M. Duflo, "Sur l'algèbre enveloppante d'une algèbre de Lie résoluble" Bull. Sci. Math. , 94 (1970) pp. 201–208 MR0283037 [a6] J. Dixmier, "Représentations irréductibles des algèbres de Lie nilpotents" An. Acad. Brasil. Ci. , 35 (1963) pp. 491–519 [a7] J. Dixmier, "Representations irreductibles des algebres de Lie résolubles" J. Math. Pures Appl. , 45 (1966) pp. 1–66 MR0200393 [a8] J. Dixmier, "Enveloping algebras" , Amer. Math. Soc. (1996) (Translated from French) MR1451138 MR1393197 Zbl 0867.17001 [a9] M. Duflo, "Sur les extensions des representations irreductibles des algèbres de Lie contenant un ideal nilpotent" C.R. Acad. Sci. Paris Ser. A , 270 (1970) pp. 504–506 MR257160 [a10] M. Duflo, "Construction of primitive ideals in enveloping algebras" I.M. Gelfand (ed.) , Lie Groups and their representations: Summer School of the Bolyai Janos Math. Soc. (1971) , Akad. Kiado (1975) MR399194 [a11] M. Duflo, "Théorie de Mackey pour les groupes de Lie algébriques" Acta Math. , 149 (1982) pp. 153–213 MR0688348 Zbl 0529.22011 [a12] A.A. Kirillov, "Unitary representations of nilpotent Lie groups" Uspekhi Mat. Nauk , 17 (1962) pp. 57–110 (In Russian) MR0142001 Zbl 0106.25001 [a13] O. Mathieu, "Bicontinuity of the Dixmier map" J. Amer. Math. Soc. , 4 (1991) pp. 837–863 MR1126380 MR1115787 Zbl 0743.17013 Zbl 0762.17008 [a14] C. Moeglin, "Ideaux primitifs des algèbres enveloppantes" J. Math. Pures Appl. , 59 (1980) pp. 265–336 MR0604473 Zbl 0454.17006 [a15] C. Moeglin, "Ideaux primitifs completement premiers de l'algèbre enveloppante de \$gl ( n , C )\$" J. Algebra , 106 (1987) pp. 287–366 [a16] C. Moeglin, R. Rentschler, "Sur la classification des ideaux primitifs des algèbres enveloppantes" Bull. Soc. Math. France , 112 (1984) pp. 3–40 MR0771917 Zbl 0549.17007 [a17] R. Rentschler, "L'injectivite de l'application de Dixmier pour les algèebres de Lie résolubles" Invent. Math. , 23 (1974) pp. 49–71
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