Representations of solvable Lie algebras with filtrations A.N.Panov ∗ 1 Introduction. Main definitions This paper is devoted to classification of irreducible representations of Lie algebras. Complete classification is known only for Lie algebras of small dimension [1, 2]. This classification is not constructible and cannot be generalized to more complicated Lie algebras. A more reasonable approach is to classify not irreducible representations but their kernels (primitive ideals) in the universal enveloping algebra. Started from [3] there were many works in this direction. An other possibility is to study some category of representations and irredu- cible representation. For instance, the category of representations of highest weight for semisimple Lie algebras. Irreducible representations in this category are irreducible factors of Verma modules [3, 4]. In the case of solvable Lie algebras one well known a family of irreducible representations Ms(f), induced from the Vergne polarization [5]. In this paper we consider the category of representations of Lie algebras with fixed filtrations. We prove that every irreducible representation in this category have the form ∗ Ms(f) for some f ∈ g (theorem 2.4). That is the mapping Ms : f 7→ Ms(f) ∗ from g to Irr(g, s) is surjective. The mapping Ms is extended to bijection of the factor set g∗/R to Irr(g, s) (theorem 3.4 and definition 3.5). Remark that two representations of the form Ms(f) are equivalent in the category arXiv:1202.6479v1 [math.RT] 29 Feb 2012 of representations of Lie algebras with filtrations (i.e. R-equivalent) if and only if they are equivalent in the usual sense (theorem 3.4). The classes of R-equivalence are described in Theorem 3.7. Using classes of R-equivalence on g∗ we find spectra of the representation induced by irreducible representation of subalgebra (theorem 4.3), spectra of restrictions of irreducible representation on subalgebra, (theorem 4.4) and spectra of the tensor product of two irreducible representations (theorem 4.5). ∗ In the last section the connection between the map Ms : g /R → Irr(g, s) and the Dixmier mapping is investigated. The constructed theory is an analog ∗The paper is supported by RFBR grants 08-01-00151-a, 09-01-00058-a and by ADTP grant 3341 1 A.A.Kirillov’s orbit method (see [3, 6, 7]) in the category of Lie algebras with filtration. In 1979 this paper was deposed in All-Union Institute for Science and Technical Information (VINITY), see [8]. Turn to formulation of main definitions of the paper. 1.1. Let field K be a field of zero characteristic and g be a solvable Lie algebra over K. A filtration sg on g is a chain of ideals g = g0 ⊃ g1 ⊃ ... ⊃ gk−1 ⊃ gk, (1) such that gk = {0} and codimension of gi in gi−1 as less or equal to 1. A Lie algebra with filtration is a Lie algebra g with the its fixed filtration sg. To simplify notations we denote filtration by s and a Lie algebra g with filtration s by (g, s). Remark. 1) A filtration on g always exists, since any finite dimensional (in particular, adjoint) representation of g can be performed in triangular form [3, 1.3.12]. 2) In this definition we allow that some ideal may appear several times. This makes it possible to restrict a filtration on a subalgebra (see 1.6) and to consider a restriction of a representations of a Lie algebra with filtration on a subalgebra. We say that two filtrations on a Lie algebra are equal, if they consists of the same ideals and may differ only by multiplicity of their entrance in filtration. Let (g′, s′) and (g′′, s′′) be two Lie algebras with filtrations. A homomorphism φ : (g′, s′) → (g′′, s′′) of Lie algebras with filtrations is a homomorphism ′ ′′ ′ ′′ φ : g → g such that φ(gi) ⊂ gi . 1.2. A chain of ideals g0 ⊃ g1 ⊃ ... ⊃ gk produces the chain of subalgebras U(g0) ⊃ U(g1) ⊃ ... ⊃ U(gk) of the universal enveloping algebra U(g). Here we put U(gk) = K, since gk = {0}. We denote by U(g, s) the universal enveloping algebra U(g) with the fixed filtration. 1.3. Let V be a linear space over a field K ( infinite dimensional, in general). A filtration sV in V is a chain of embedded subspaces V = V0 ⊃ V1 ⊃ ... ⊃ Vk. To simplify notations we denote a filtration by s and a Lie algebra with filtration s by (V, s). Remark. Note here that in these case there is no restrictions on the codimen- sions of Vi in Vi−1 and on dimensions of Vk. Assume that for every i we are given a representation ρi of the Lie algebra gi in linear space Vi, such that the subspace Vi+1 is invariant with respect to 2 the restriction of ρi on gi+1. Then we say that this is a representation (ρ, s) of the Lie algebra with filtration (g, s) or that (V, s) is a U(g, s)-module. We say that two representations (ρ′, s) and (ρ′′, s) of Lie algebras (g, s) in the spaces (V, s) and (W, s) are equivalent, if there exists an isomorphism ′ ′′ C : V → W such that C(Vi)= Wi and Cρ (x)= ρ (x)C for every x ∈ g. 1.4. A representation (ρ, s) is irreducible, if every ρi is an irreducible representation of gi in Vi. We denote the set of equivalence classes of irreducible representations of (g, s) by Irr(g, s). 1.5. If (V, s) is a linear subspace with filtration and W is a subspace in V , then the filtration of V induces a filtration of W : W = W0 ⊃ W1 ⊃ ... ⊃ Wk, where Wi = Vi ∩ W . The factor space V/W is also equipped with a natural filtration (V0 + W )/W ⊃ (V1 + W )/W ⊃ ... ⊃ (Vk + W )/W, that coincides with V0/W0 ⊃ V1/W1 ⊃ ... ⊃ Vk/Wk. If (V, s) is a U(g, s)-module and W is a U(g)-submodule of V , then we say that (W, s) is a U(g, s)-submodule of (V, s). A module Ts over U(g, s) is called a subfactor module of (V, s), if there exists two U(g, s)-submodules (W, s) and (W ′, s) such that W ⊃ W ′ and factor ′ of (W, s) by (W , s) is isomorphic to Ts. The set of all irreducible subfactor modules of (V, s) is called the spectrum of (V, s) and is denoted by Spec(V, s). 1.6. Let h be a Lie subalgebra of g. Then filtration of g produces a filtration g0 ∩ h ⊃ g1 ∩ h ⊃ ... ⊃ gk ∩ h of h. We shall also denote it by s. If (ρ, s) is a representation of (g, s), then the restrictions of ρi on hi = gi ∩h in Vi defines a representation of (h, s) in (V, s). 1.7. Let h be a subalgebra of g and let (W, s) be an arbitrary U(h, s)-module. Recall that the induced module ind(W, g) (see [3, 5.1]) is defined as linear space U(g) ⊗U(h) W with the left action of U(g). We consider a chain of subspaces in ind(W, g) ind(W0, g) ⊃ ind(W1, g1) ⊃ ... ⊃ ind(Wk, gk). Since every ind(Wi, gi) is a U(gi)-module, ind(W, g, s) is a U(g, s)-module. 3 1.8. Let us define the tensor product of representations of Lie algebras with filtrations. Let g be a Lie algebra with filtration s given as g0 ⊃ g1 ⊃ ... ⊃ gk. We denote the direct sum of two copies of g by g × g (instead of the usual notation g ⊕ g). The Lie algebra g × g has a filtration g0 × g0 ⊃ g0 × g1 ⊃ g1 × g1 ⊃ ... ⊃ gk × gk, denoted by s × s. If V , W are U(g)-modules, then we denote by V ×W the linear space V ⊗W considered as a U(g × g)-module. If (V, s) and (W, s) are U(g, s)-modules, then V × W is a U(g × g, s × s)- module with filtration V0 × W0 ⊃ V0 × W1 ⊃ V1 × W1 ⊃ ... ⊃ Vk × Wk. The Lie algebra g embedded into g × g diagonally and the filtration s × s induces the filtration g0 ⊃ g1 ⊃ g1 ⊃ ... ⊃ gk on g which coincides with s. The restriction of representation V × W to (g, s) turns V × W into a U(g, s)- module, which we denote by (V, s) ⊗ (W, s). 1.9. Let g be an arbitrary Lie algebra, g∗ the dual space of g and f ∈ g∗. A subalgebra p is called a polarization of f, if p is a maximal isotropic subspace with respect to a skew symmetric form f([x, y]), that is, f([p, p]) = 0 and 1 f f dim p = 2(dim g +dim g ), where g is the stabilizer of f in g. The restriction of f to its polarization is a one-dimensional representation (character) of the polarization. Let (g, s) be a Lie algebra with filtration. For any element f ∈ g∗ one can construct the polarization, that is called the Vergne polarization, by the following formula k fi pv(f)= gi , Xi=0 ∗ fi where fi is a restriction of f to gi , and gi is the stabilizer of fi in gi (see [5], [3, 1.12.10]). Denote by M(f) the module over U(g), induced by the restriction of f to pv(f). The module M(f) is irreducible [3, 6.1.1]. The intersection pv(f) ∩ gi coincides with the Vergne polarization pvi(fi) of fi in gi [3, 1.12.10].