Parabolic Subgroups of Real Direct Limit Lie Groups Elizabeth Dan-Cohen∗, Ivan Penkov† & Joseph A. Wolf‡ 31 December 2008 9 0 0 2 n Abstract a J 2 Let GR be a classical real direct limit Lie group, and gR its Lie algebra. The parabolic subalgebras of the complexification gC were described by the first two authors. In the present ] paper we extend these results to gR. This also gives a description of the parabolic subgroups of T GR. Furthermore,wegiveageometriccriterionforaparabolicsubgroupPCofGCtointersectGR R in aparabolicsubgroup. This criterioninvolvestheGR–orbitstructureofthe flagind–manifold . h GC/PC. t a MSC 2000 : 17B05;17B65. m [ 1 1 Introduction and Basic Definitions v 5 9 We start with the three classical simple locally finite countable–dimensional Lie algebras gC = 2 0 l−i→mgn,C, and their real forms gR. The Lie algebras gC are the classical direct limits, sl(∞,C) = . limsl(n;C), so(∞,C) = limso(2n;C) = limso(2n+1;C), and sp(∞,C) = limsp(n;C), where the 1 −→ −→ −→ −→ 0 direct systems are given by the inclusions of the form A7→ (A 0). See [1] or [2]. We often consider 0 0 9 the locally reductive algebra gl(∞;C) = limgl(n;C) along with sl(∞;C). 0 −→ v: Therealformsoftheseclassicalsimplelocallyfinitecountable–dimensionalcomplexLiealgebras Xi gC have been classified by A. Baranov in [1]. A slight reformulation of [1, Theorem 1.4] says that the following is a complete list of the real forms of gC. r a If gC = sl(∞;C), then gR is one of the following: sl(∞;R) = limsl(n;R), the real special linear Lie algebra, −→ sl(∞;H) = limsl(n;H), the quaternionic special linear Lie algebra, where sl(n;H):= gl(n;H)∩ −→ sl(2n;C), su(p,∞) = limsu(p,n), the complex special unitary Lie algebra of finite real rank p, −→ su(∞,∞) = limsu(p,q), the complex special unitary Lie algebra of infinite real rank. −→ ∗Research partially supported byDFG Grant PE 980/2-1. †Research partially supported byDFG Grant PE 980/2-1. ‡Research partially supported byNSFGrant DMS 06 52840. 1 If gC = so(∞;C), then gR is one of the following: so(p,∞) = limso(p,n), the real orthogonal Lie algebra of finite real rank p, −→ so(∞,∞) = limso(p,q), the real orthogonal Lie algebra of infinite real rank, −→ so∗(2∞) = limso∗(2n), with so∗(2n) = {ξ ∈ sl(n;H) | κ (ξx,y)+κ (x,ξy) = 0 ∀x,y ∈ Hn}, −→ n n where κ (x,y) := xℓiy¯ℓ = txiy¯. Equivalently, so∗(2n) = so(2n;C) ∩ u(n,n) with so(2n;C) n ℓ defined by (u,v) = n(u v +u w ) and u(n,n) by hu,vi = n(u v −u v ). P1 2j−1 2j 2j 2j−1 1 2j−1 2j−1 2j 2j P P If gC = sp(∞;C), then gR is one of the following: sp(∞;R) = limsp(n;R), the real symplectic Lie algebra, −→ sp(p,∞) = limsp(p,n), the quaternionic unitary Lie algebra of finite real rank p, −→ sp(∞,∞) = limsp(p,q), the quaternionic unitary Lie algebra of infinite real rank. −→ If gC = gl(∞;C), then gR is one of the following: gl(∞;R) = limgl(n;R), the real general linear Lie algebra, −→ gl(∞;H) = limgl(n;H), the quaternionic general linear Lie algebra, −→ u(p,∞) = limu(p,n), the complex unitary Lie algebra of finite real rank p, −→ u(∞,∞) = limu(p,q), the complex unitary Lie algebra of infinite real rank. −→ The defining representations of gC are characterized as direct limits of minimal–dimensional nontrivial representations of simple subalgebras. It is well known that that sl(∞;C) and gl(∞;C) have two inequivalent defining representations V and W, whereas each of so(∞;C) and sp(∞;C) hasonlyone(uptoequivalence)V. Inparticulartherestrictions toso(∞;C)orsp(∞;C)ofthetwo definingrepresentationsofsl(∞;C)areequivalent. TherealformsgR listedabovealsohavedefining representations, as detailed below, which are particular restrictions of the defining representations of gC. We denote an element of Z≧0∪{∞} by ∗. Suppose that gR is sl(∞;R) or gl(∞;R). The defining representation spaces of gR are the finitary (i.e. with finitely many nonzero entries) column vectors VR = R∞ and the finitary row vectors WR = R∞. The algebra of gR–endomorphisms of VR or WR is R. The restriction of the pairing of V and W is a nondegenerate gR–invariant R–bilinear pairing of VR and WR. The defining representation space VR of gR = so(∗,∞) consists of the finitary real column vectors. The algebra of gR–endomorphisms of VR (the commuting algebra) is R. The restriction of the symmetric form on V to VR is a nondegenerate gR–invariant symmetric R–bilinear form. The defining representation space VR of gR = sp(∞;R) consists of the finitary real column vectors. The algebra of gR–endomorphisms of VR is R. The restriction of the antisymmetric form on V to VR is a nondegenerate gR–invariant antisymmetric R–bilinear form. In both of these cases the defining representation of gR is a real form of the defining represen- tation of gC, i.e. V = VR⊗C. Suppose that gR is su(∗,∞) or u(∗,∞). Then gR has two defining representations, one on the space VR = C∗,∞ of finitary complex column vectors and the other on the space WR of finitary 2 complex row vectors. Thus the two defining representations of gC remain irreducible as a represen- tations of gR, the respective algebras of gR–endomorphisms of VR and WR are C, and V = VR and W = WR. The pairing of V and W defines a gR–invariant hermitian form of signature (∗,∞) on VR. Suppose that gR is sl(∞;H) or gl(∞;H). The two defining representation spaces of gR consist of the finitary column vectors VR = H∞ and finitary row vectors WR = H∞. The algebra of gR–endomorphisms of VR or WR is H. The defining representations of gC on V and W restrict to irreducible representations of gR, and VR = H∞ = C∞ +C∞j = C2∞ = V. The pairing of V and W is a nondegenerate gR–invariant R–bilinear pairing of VR and WR. The defining representation space VR = H∗,∞ of sp(∗,∞) consists of the finitary quaternionic vectors. The algebra of sp(∗,∞)–endomorphisms of VR is H. The form on VR is a nondegenerate sp(∗,∞)–invariant quaternionic–hermitian form of signature (∗,∞). In this case VR = H∗,∞ = C2∗,2∞ = V. The defining representation space VR = H∞ of so∗(2∞) consists of the finitary quaternionic vectors. The algebra of so∗(2∞)–endomorphisms of VR is H. The form on VR is the nondegenerate so∗(2∞)–invariant quaternionic–skew–hermitian form κ which is the limit of the forms κ . In this n case again VR = H∞ = C2∞ = V. The Lie ind–group (direct limit group) corresponding to gl(∞;C) is the general linear group GL(∞;C), which consists of all invertible linear transformations of V of the form g = g′+Id where g′ ∈ gl(∞;C). The subgroup of GL(∞;C) corresponding to sl(∞;C) is the special linear group SL(∞;C), consisting of elements of determinant 1. The connected ind–subgroups of GL(∞;C) whose Lie algebras are so(∞;C) and sp(∞;C) are denoted by SO (∞;C) and Sp(∞;C). 0 In Section 2 we recall the structure of parabolic subalgebras of complex finitary Lie algebras from [4]. A parabolic subalgebra of a complex Lie algebra is by definition a subalgebrathat contains a maximal locally solvable (that is, Borel) subalgebra. Parabolic subalgebras of complex finitary Lie algebras are classified in [4]. We recall the structural result that every parabolic subalgebra is a subalgebra (technically: defined by infinite trace conditions) of the stabilizer of a taut couple of generalized flags in the defining representations, and we strengthen this result by studying the non–uniqueness of the flags in the case of the orthogonal Lie algebra. As in the finite–dimensional case, we define a parabolic subalgebra of a real locally reductive Lie algebra gR as a subalgebra pR whose complexification pC is parabolic in gC = gR ⊗R C. It is a well–known fact that already in the finite–dimensional case a parabolic subalgebra of gR does not neccesarily contain a subalgebra whose complexification is a Borel subalgebra of gC. In Section 3 we prove our main result. It extends the classification in [4] to the real case. The key difference from the complex case is that one must take into account the additional structure of a defining representation space of gR as a module over its algebra of gR–endomorphisms. In Section 4 we give a geometric criterion for a parabolic subalgebra of gC to be the complex- ification of a parabolic subalgebra of gR. The criterion is based on an observation of one of us from the 1960’s, concerning the structure of closed real group orbits on finite–dimensional complex flag manifolds. We recall that result, appropriately reformulated, and indicate its extension to flag ind–manifolds. 3 2 Complex Parabolic Subalgebras 2A Generalized Flags Let V and W be countable–dimensional right vector spaces over a real division algebra D = R, C or H, together with a nondegenerate bilinear pairing h·,·i : V ×W → D. Then V and W are endowed with the Mackey topology, and the closure of a subspace F ⊂ V is F⊥⊥, where ⊥ refers to the pairing h·,·i. A set of D–subspaces of V (or W) is called a chain in V (or W) if it is totally ordered by inclusion. A D–generalized flag is a chain in V (or W) such that each subspace has an immediate predecessor or an immediate successor in the inclusion ordering, and every nonzero vector of V (or W) is caught between an immediate predecessor–successor pair [5]. Definition 2.1. [4] A D–generalized flag F in V (or W) is said to be semiclosed if for every immediate predecessor–successor pair F′ ⊂ F′′ the closure of F′ is either F′ or F′′. ♦ If C is a chain in V (or W), then we denote by C⊥ the chain in W (or V) consisting of the perpendicular complements of the subspaces of C. We fix an identification of V and W with the defining representations of gl(∞;D) as follows. To identify V and W with the defining representations of gl(∞;D), it suffices to find bases in V and W dual with respect to the pairing h·,·i. If D 6= H, the existence of dual bases in V and W with respect to any nondegenerate D–bilinear pairing is a result of Mackey [9, p. 171]. Now suppose that D = H. Then there exist C–subspaces VC ⊂ V and WC ⊂ W such that V = VC⊕VCj and W = WC ⊕WCj. The restriction of h·,·i to VC ×WC is a nondegenerate C–bilinear pairing. The result of Mackey therefore implies the existence of dual bases in VC and WC, which are also dual bases of V and W over H. In all cases we identify the right multiplication of vectors in V by elements of D with the action of the algebra of gR–endomorphisms of VR. Definition 2.2. [4] Let F and G be D–semiclosed generalized flags in V and W, respectively. We say F and G form a taut couple if F⊥ is stable under the gl(∞;D)–stabilizer of G and G⊥ is stable under the gl(∞;D)–stabilizer of F. If we have a fixed isomorphism f : V → W then we say that F is self–taut if F and f(F) form a taut couple. ♦ If one has a fixed isomorphism between V and W, then there is an induced bilinear form on V. A semiclosed generalized flag F in V is self–taut if and only if F⊥ is stable under the gl(∞;D)– stabilizer of F, where F⊥ is taken with respect to the form on V. Remark 2.3. Fix a nondegenerate bilinear form on V. If V is finite dimensional, a self–taut generalized flag in V consists of a finite number of isotropic subspaces together with their perpen- dicular complements. In this case, the stabilizer of a self–taut generalized flag equals the stabilizer of its isotropic subspaces. If V is infinite dimensional, the non–closed non–isotropic subspaces in a self–taut generalized flag in V influence its stabilizer, but it is still true that every subspace is either isotropic or coisotropic. Indeed, let F be a self–taut generalized flag, and let F ∈ F. By [4, Proposition 3.2], F⊥ is a union of elements of F if it is a nontrivial proper subspace of V. Hence F ∪{F⊥} is a chain that contains both F and F⊥. Thus either F ⊂ F⊥ or F⊥ ⊂ F, so F is either isotropic or coisotropic. ♦ 4 We will need the following lemma when we pass to consideration of real parabolic subalgebras. Lemma 2.4. Suppose that D = H. Fix H–generalized flags F in V and G in W. Then F and G form a taut couple if and only if they are form taut couple as C–generalized flags. Proof. It is immediate from the definition that F and G are semiclosed C–generalized flags if and only if they are semiclosed H–generalized flags. The proof of [4, Proposition 3.2] holds in the quaternionic case as well. Thus if F and G form a taut couple as either C–generalized flags or H– generalizedflags,thenaslongasF⊥ isanontrivialpropersubspaceofW,itisaunionofelementsof G for any F ∈ F. Thus F⊥ is stable under both the gl(∞;C)–stabilizer and the gl(∞;H)–stabilizer of G for any F ∈ F. Similarly, if G ∈ G then G⊥ is stable under both the gl(∞;C)–stabilizer and the gl(∞;H)–stabilizer of F. 2B Trace Conditions Let g be a locally finite Lie algebra over a field of characteristic zero. A subalgebra of g is locally solvable (resp. locally nilpotent) if every finitesubsetof gis contained ina solvable(resp. nilpotent) subalgebra. Thesumofalllocally solvableidealsisagain alocally solvableideal, thelocally solvable radical of g. If r is the locally solvable radical of g then r∩[g,g] is a locally nilpotent ideal in g. Indeed, note that r∩[g,g] = (r∩[g,g])∩g for any exhaustion g = g by finite–dimensional n n n n subalgebras g , and furthermore(r∩[g,g])∩g is nilpotent for all n by standard finite–dimensional n n S S Lie theory. Let g be a splittable subalgebra of gl(∞;D), that is, a subalgebra containing the Jordan com- ponents of its elements), and let r be its locally solvable radical. The linear nilradical m of g is defined to be the set of all nilpotent elements in r. Lemma 2.5. Let g be a splittable subalgebra of gl(∞;D). Then its linear nilradical m is a locally nilpotent ideal. If D = R, then the complexification mC is the linear nilradical of gC. Proof. If ξ,η ∈ m they are both contained in the solvable radical of a finite–dimensional subalgebra of g, soξ+η and [ξ,η] are nilpotent. Thus, by Engel’s Theorem, m is a locally nilpotent subalgebra of g. Although it is only stated for complex Liealgebras, [4, Proposition 2.1] shows that m∩[g,g]= r∩[g,g], so [m,g] ⊂ [r,g] ⊂ r∩[g,g], and thus m is an ideal in g. This proves the first statement. For the second let r be the locally solvable radical of g and note that rC is the locally solvable radical of gC, so the assertion follows from finite–dimensional theory. Definition 2.6. Let g be a splittable subalgebra of gl(∞;F) where F is R or C, and and let m be its linear nilradical. A subalgebra p of g is defined by trace conditions on g if m ⊂ p and [g,g]/m ⊂ p/m ⊂ g/m, in other words if there is a family Tr of Lie algebra homomorphisms f : g → F with joint kernel equal to p. Further, p is defined by infinite trace conditions if every f ∈ Tr annihilates every finite–dimensional simple ideal in [g,g]/m. ♦ 5 We write Trp for the maximal family Tr of Definition 2.6. On the group level we have cor- responding determinant conditions and infinite determinant conditions. Note that infinite trace conditions and infinite determinant conditions do not occur when g and G are finite dimensional. 2C Complex Parabolic Subalgebras Recallthataparabolic subalgebra ofacomplexLiealgebraisbydefinitionasubalgebrathatcontains a Borel subalgebra, i.e. a maximal locally solvable subalgebra. Theorem 2.7. [4] Let gC be gl(∞,C) or sl(∞,C), and let V and W be its defining representation spaces. A subalgebra of gC (resp. subgroup of GC) is parabolic if and only if it is defined by infinite trace conditions (resp. infinite determinant conditions) on the gC–stabilizer (resp. GC–stabilizer) of a (necessarily unique) taut couple of C–generalized flags F in V and G in W. Let gC be so(∞,C) or sp(∞,C). and let V be its defining representation space. A subalgebra of gC (resp. subgroup of GC) is parabolic if and only if it is defined by infinite trace conditions (resp. infinite determinant conditions) on the gC–stabilizer (resp. GC–stabilizer) of a self–taut C–generalized flag F in V. In the sp(∞,C) case the flag F is necessarily unique. In contrast to the finite dimensional case, the normalizer of a parabolic subalgebra can be larger than the parabolic algebra. For example, Theorem 2.7 implies that sl(∞,C) is parabolic in gl(∞;C), since it is the elements of the stabilizer of the trivial generalized flags {0,V} and {0,W} whose usual trace is 0. To understand the origins of this example, one should consider the explicit constructionin[6]ofalocally nilpotentBorelsubalgebraofgl(∞;C). Thenormalizerofaparabolic subalgebra equals the stabilizer of the corresponding generalized flags [4], which is in general larger than the parabolic subalgebra because of the infinite determinant conditions. The self–normalizing parabolics are thus those for which Trp = 0. This is in contrast to the finite–dimensional setting, where there are no infinite trace conditions, and all parabolic subalgebras are self–normalizing. In [4] the uniqueness issue was discussed for gl(∞,C), sl(∞,C), and sp(∞,C), but not for so(∞,C). In the orthogonal setting one can have three different self–taut generalized flags with the same stabilizer (see [3] and [7], where the non–uniqueness is discussed in special cases.) Theorem 2.8. Let p be a parabolic subalgebra given by infinite trace conditions on the so(∞;C)– stabilizer of a self–taut generalized flag F in V. Then there are two possibilities: 1. F is uniquely determined by p; 2. there are exactly three self-taut generalized flags with the same stabilizer as F. The latter case occurs precisely when there exists an isotropic subspace L ∈ F with dimCL⊥/L = 2. The three flags with the same stabilizer are then • {F ∈ F |F ⊂ L or L⊥ ⊂ F} • {F ∈ F |F ⊂ L or L⊥ ⊂ F}∪M 1 • {F ∈ F |F ⊂ L or L⊥ ⊂ F}∪M 2 6 where M and M are the two maximal isotropic subspaces containing L. 1 2 Proof. The main part of the proof is to show that p determines all the subspaces in F, except a maximal isotropic subspace under the assumption that F has a closed isotropic subspaces L with dimCL⊥/L = 2. Let A denote the set of immediate predecessor–successor pairs of F such that both subspaces in the pair are isotropic. Let F′ denote the predecessor and F′′ the successor of each pair α ∈ A. α α Let M denote the union of all the isotropic subspaces in F, i.e. M = F′′. If M 6= M⊥, then α∈A α M has an immediate successor W in F. Note that W is not isotropic, by the definition of M. S Furthermore, one has W⊥ = M since F is a self–taut generalized flag. If M = M⊥, let us take W = 0. Let C denote the set of all γ ∈ A such that F′ is closed. For each γ ∈ C, it is seen in [4] that γ the coisotropic subspace (F′′)⊥ has an immediate successor in F. For each γ ∈ C, let G′′ denote γ γ the immediate successor of (F′′)⊥ in F. It is also shown in [4] that (G′′)⊥ = F′. γ γ γ Since F is a self–taut generalized flag, F is uniquely determined by the set of subspaces {F′′ |α ∈ A}∪{G′′ |γ ∈ C such that G′′ is not closed}∪{W}. α γ γ We use separate arguments for these three kinds of subspaces to show that they are determined by p, except for a maximal isotropic subspace and W under the assumption that F has a closed isotropic subspace L with dimCL⊥/L = 2. We must also show that we can determine from p whether or not F has a closed isotropic subspace L with dimCL⊥/L = 2. Let p denote the normalizer in so(∞;C) of p. We use the classical identification so(∞;C) ∼= Λ2(V) where u ∧ v corresponds to the linear transformation x 7→ hx,viu − hx,uiv. With this identificeation, following [4] one has p= F′′∧(F′)⊥+ F′′∧G′′ +Λ2(W). α α γ γ α∈A\C γ∈C X X e Let α ∈ A, and let x ∈ F′′\F′. Then one may compute α α p·x = F′′∧(F′)⊥ + F′′∧G′′ +Λ2(W) ·x α α γ γ (cid:16)α∈XA\C γX∈C (cid:17) e = F′′⊗(F′)⊥ + F′′⊗G′′ ·x α α γ γ (cid:16)α∈XA\C γX∈C (cid:17) = F′′ ∪ F′′ . α γ (cid:16)x∈/(F[α′)⊥⊥ (cid:17) (cid:16)x∈/([G′γ′)⊥ (cid:17) As a result F′ if α ∈ A\C p·x= α (Fα′′ if α ∈ C. e So far we have shown the following. Ifx∈ p·x, then F′′ = p·x. If x ∈/ p·x, then F′′ = (p·x)⊥⊥. α α Furthermore, if x ∈/ M, then p·x is not isotropic, unless there exists a closed isotropic subspace e e e e e 7 L ∈ F with dimCL⊥/L = 2, and x is an element of M1 or M2. We now consider the union of the subspaces p·x, where the union is taken over x ∈ V for which p·x is isotropic. If there does not exist L as described, then these subspaces will be the nested isotropic subspaces computed above, and indeedetheir union is M. If L exists, then these subspaces weill exhaust L, and furthermore M1 and M will both appear in the union. Hence the union of the isotropic subspaces of the form p·x 2 for x ∈ V when L exists is L⊥. As a result, if the union of all the isotropic subspaces of the form p·x for x ∈ V is itself isotropic, then we conclude that no such L exists and we have construceted the subspace M. If that union is not isotropic, then we conclude that there exists a closed isotropic esubspaceL ∈ F with dimCL⊥/L = 2, and theunion isL⊥. In the latter case, L is recoverable from p, as it equals L⊥⊥. We have now shown that we can determine whether F has a closed isotropic subspace L with dimCL⊥/L = 2, that Fα′′ is determined by p for all α ∈ A in the latter case, and that F′′ is determined by p for all α∈ A such that F′′ ⊂ L in the former case. α α We now turn our attention to a non–closed subspace G′′ for γ ∈ C. Since G′′ is not closed, the γ γ codimension of F′′ in G′′ is infinite. Thus if there exists L ∈ F as above, then F′′ ⊂ L. So we have γ γ γ already shown that F′′, and indeed F′ as well, are recoverable from p whether or not there exists γ γ L ∈ F. Let x∈ (F′)⊥ \(F′′)⊥. Then there exists v ∈ F′′ such that hv,xi =6 0, and one has γ γ γ (v∧G′′)·x= {(v∧y)·x| y ∈ G′′}= {hx,yiv−hx,viy |y ∈ G′′}. γ γ γ Since v ∧ G′′ ⊆ p and v ∈ F′′, we see that G′′ = (v ∧ G′′)· x+F′′ ⊂ p ·x+ F′′ ⊂ G′′. Hence γ γ γ γ γ γ γ G′′ = p·x+F′′, and we conclude that G′′ is recoverable from p. γ γ γ e e Finally, we must show that p determines W under the assumption that no subspace L ∈ F e as above exists. We have already shown that M is recoverable from p under this assumption. If M = M⊥, then W = 0. We claim that W = p·x+M for any x ∈ M⊥ \M when M 6= M⊥. Indeed, let X be any vector space complement of M in W. Since x ∈/ M and W⊥ = M, one has hx,Xi =6 0. Furthermore, the restriction of theesymmetric bilinear form on V to X is symmetric and nondegenerate. Then Λ2(X)·x= X because dimCX ≧ 3. Since Λ2(X) ⊂ p, we conclude that p·x+M = W. Thus W can be recovered from p. IfF isaself–tautgeneralized flagwithoutanyisotropicsubspaceL ∈ F suchethatdimCL⊥/L = e 2, then we have now shown that F is uniquely determined by p. Finally, suppose that there does exist an isotropic subspace L ∈ F such that dimCL⊥/L = 2. Then we have shown that every subspace of F which does not lie strictly between L and L⊥ is determined by p. There are exactly two maximal isotropic subspaces M and M containing L, and both M and M are stable under 1 2 1 2 the so(∞;C)–stabilizer of L. Hence the three self-taut generalized flags listed in the statement are precisely the self–taut generalized flags whose stabilizers equal the stabilizer of F. 3 Real Parabolic Subalgebras Recall that a parabolic subalgebra of a real Lie algebra gR is a subalgebra whose complexification is a parabolic subalgebra of the complexified algebra gC. LetgC beoneofgl(∞,C),sl(∞,C),so(∞,C),andsp(∞,C), andletgR bearealformofgC. Let GR be the corresponding connected real subgroup of GC. When gR has two inequivalent defining 8 representations, we denote them by VR and WR, and when gR has only one defining representation, we denote it by VR. Let D denote the algebra of gR–endomorphisms of VR. Theorem 3.1. Suppose that gR has two inequivalent defining representations. A subalgebra of gR (resp. subgroup of GR) is parabolic if and only if it is defined by infinite trace conditions (resp. infinite determinant conditions) on the gR–stabilizer (resp. GR–stabilizer) of a taut couple of D– generalized flags F in VR and G in WR. Suppose that gR has only one defining representation. A subalgebra of gR (resp. subgroup) of GR is parabolic if and only if it is defined by infinite trace conditions (resp. infinite determinant conditions) on the gR–stabilizer (resp. GR–stabilizer) of a self–taut D–generalized flag F in VR. Proof. We will prove the statements for the Lie algebras in question. The statements on the level of Lie ind–groups follow immediately, since infinite determinant conditions on a Lie ind–group are equivalent to infinite trace conditions on its Lie algebra. Suppose that pR is a parabolic subalgebra of gR. By definition, the complexification pC is a parabolic subalgebra of gC. Theorem 2.7 implies that pC is defined by infinite trace conditions TrpC on the gC–stabilizer of a taut couple of generalized flags in V and W or on a self–taut generalized flaginV. AsTrpC isstableundercomplexconjugationitisthecomplexification oftherealsubspace (TrpC)R := {t ∈ TrpC | τ(t) = t} where τ comes from complex conjugation of gC over gR. We will use this to show case by case that pR is defined by trace conditions on the gR–stabilizer of the appropriate generalized flag(s). The first cases we treat are those where the defining representation space VR is the fixed point set of a complex conjugation τ : V → V. The real forms fitting this description are sl(∞;R), so(∞,∞), so(p,∞), sp(∞;R), and gl(∞;R). Consider the sl(∞;R) case, and note that the proof also holds in the gl(∞;R) case. Let F and G be the taut couple of generalized flags in V and W given in Theorem 2.7, and note that WR is the fixed points of complex conjugation τ : W → W. Evidently τ(pC) = pC, so τ(F) = F and τ(G) = G by the uniqueness claim of Theorem 2.7. Since the generalized flags F and G are τ–stable, every subspace in them is τ–stable. (Explicitly, for any F ∈ F, we have τ(F) ∈ F, so either τ(F) ⊂ F or F ⊂ τ(F). Since τ2 = Id, we have F = τ(F) for anyF ∈F.) Hence every subspaceinF and G has arealform, obtained as theintersection with VR and WR, respectively. The generalized flags FR := {F ∩VR | F ∈ F} and GR := {G∩WR |G ∈ G} form a taut couple as R–generalized flags in VR and WR. Now pR is defined by the infinite trace conditions (TrpC)R on the sl(∞;R)–stabilizer of the taut couple FR and GR of generalized flags in VR and WR. If gR is so(∗,∞) or sp(∞;R), Theorem 2.7 implies that pC is defined by infinitetrace conditions on the gC–stabilizer a self–taut generalized flag F in V. The arguments of the sl(∞;R) case show that F is τ–stable, provided that τ(pC) = pC forces τ(F) = F. That is ensured by the uniqueness claim in Theorem 2.7 for the symplectic case, and by Theorem 2.8 in the orthogonal cases where uniqueness holds. Uniqueness fails precisely when gR = so(∞,∞) and there exists an isotropic subspace L ∈ F with dimC(L⊥/L) = 2. We may assume that F is the first of the three generalized flags listed in the statement of Theorem 2.8. Then τ(F) is one of the three generalized flags listed in the statement of Theorem 2.8, and since F is contained in any of those three, the subspaces of F are all τ–stable. Finally, the generalized flag FR := {F ∩VR | F ∈ F} in VR is self–taut, and pR is defined by the infinite trace conditions (TrpC)R on its gR–stabilizer. 9 Second, suppose that gR = su(∗,∞). Note that the arguments for su(∗,∞) apply without change to u(∗,∞). By Theorem 2.7, pC is given by infinite trace conditions TrpC on the gl(∞;C)– stabilizer of ataut coupleF andG of generalized flags in V and W. Thereexists an isomorphism of gR–modules f : V → W. Both G and f(F)are stabilized bypR, hencealso bypC, so theuniqueness claim of Theorem 2.7 tells us that G = f(F). Thus F is self–taut. We conclude that pR is given by the infinite trace conditions (TrpC)R on the stabilizer of the self–taut generalized flag F. ThethirdcaseweconsideristhatofgR = sl(∞;H). Notethatthegl(∞;H)caseisprovedinthe same manner. Then gC = sl(2∞;C), where we have the identifications V = C2∞ = C∞ +C∞j = H∞ = VR and W = WR. The quaternionic scalar multiplication v 7→ vj is a complex conjugate– linear transformation J of C2∞ of square −Id, and the complex conjugation τ of gC over gR is given by ξ 7→ JξJ−1 = J−1ξJ. Let F and G be the unique taut couple given by Theorem 2.7. Since pC = τ(pC), we have F = J(F) and G = J(G). Since J2 = −Id, every subspace of F and G is preserved by J. In other words F and G consist of H-subspaces of VR and WR. The fact that F and G form a taut couple of C–generalized flags in V and W implies via Lemma 2.4 that they form a taut couple of H–generalized flags in VR and WR. Hence pR is defined by the infinite trace conditions (TrpC)R on the stabilizer of the taut couple F, G. Thefourthcase we consideris thatof sp(∗,∞). ThenVR has aninvariant quaternion–hermitian form of signature (∗,∞) and a complex conjugate–linear transformation J of square −Id as de- scribed above. Let F be the unique self–taut generalized flag in V given by Theorem 2.7. By the uniqueness of F, we have F = J(F), so as before F consists of H–subspaces of VR. Lemma 2.4 implies that F is self–taut when considered as an H–generalized flag in VR. Hence pR is defined by the infinite trace conditions (TrpC)R on the stabilizer of F. The fifth and final case and is that of gR =so∗(2∞). Any subspace of V which is stable under the C-conjugate linear map J which corresponds to x 7→ xj is an H–subspace of VR. Let F be a self–taut generalized flag in V as given by Theorem 2.7. Since gC = so(∞;C), Theorem 2.8 says that either F is unique or there are exactly three possibilities for F. When F is unique, we must have F = J(F), so F is an H–generalized flag. When F is not unique, we may assume that F is the first of the three generalized flags listed in the statement of Theorem 2.8, the one with an immediate predecessor–successor pair L ⊂ L⊥ where L is closed and dimC(L⊥/L) = 2. Then J(F) has the same property so J(F) = F. In all cases Lemma 2.4 implies that F is self–taut when considered as an H–generalized flag. Hence pR is defined by the infinite trace conditions (TrpC)R on the so∗(2∞)–stabilizer of the self–taut H–generalized flag F. Conversely, suppose that pR is defined by infinite trace conditions TrpR on the gR–stabilizer of a taut couple FR, GR or a self–taut generalized flag FR, as appropriate. Either V = VR ⊗ C or V = VR. Suppose first that V = VR ⊗ C. Let F := {F ⊗ C | F ∈ FR}. If gC has only one defining representation V, then F is a self–taut generalized flag in V, and pC is defined by the infinite trace conditions TrpR ⊗C on the gC–stabilizer of F. Now suppose that gC has two inequivalent defining representations. IfgR alsohas two inequivalent definingrepresentations, let G := {G⊗C | G∈ GR}. If gR has only one defining representation, then let G be the image of F under the gR–module isomorphism V → W. Then F, G are a taut couple, and pC is defined by the infinite trace conditions TrpR ⊗C on the gC–stabilizer of F, G. 10