Nice Parabolic Subalgebras of Reductive Lie Algebras

This paper gives a classification of parabolic subalgebras of simple Lie algebras over C that are complexifications of parabolic subalgebras of real forms for which Lynch’s vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three dimensional simple Lie algebra). Introduction If g is a semi-simple Lie algebra over C and if p is a parabolic subalgebra of g then there is a Z-grade of g as a Lie algebra, g = ∑ j gj such that p = ∑ j≥0 gj and if n = ∑ j>0 gj then n is the nilradical of p and g1 projects bijectively onto the abelianization (n/[n, n]) of n. The purpose of this article is to give a classification of those parabolic subalgebras such that there is a Richardson element x ∈ n such that x ∈ g1. That is there exists x ∈ g1 such that [p, x] = n. A parabolic subalgebra will be called nice if it satisfies this condition. These parabolic subalgebras are exactly the complexifications of the real parabolic subalgebras whose nilradicals support admissible Lie algebra homomorphisms to iR in the sense of Lynch’s thesis [L]. In that thesis Lynch proved a generalization of Kostant’s vanishing theorem for Whittaker modules [K] (valid for generic Lie algebra homomorphisms to iR of nilradicals of Borel subalgebras of quasisplit real forms). Of course, Lynch’s theorem is vacuous if the nilradical of the parabolic subalgebra admits no such homomorphisms. He introduced the term admissible for the parabolic sublagebras whose nilradicals admit admissible homomorphisms. Thus an admissible parabolic in the sense of Lynch is a real form of a nice parabolic in our sense. It is clear that a parabolic subalgebra is nice if and only if its intersection with each normal subalgebra is nice. Thus it is enough to do the classification for simple Lie algebras over C. The complete classification is given in section 1 and the rest of the paper is devoted to the proof of the correctness of the list (sections 3 and 4), to a description of the corresponding Richardson elements (admissible elements in the sense of Lynch, the proofs of the assertions in section 5 will appear in [B] and to Received by the editors August 27, 2004. First named author supported by the Swiss National Science Foundation, Second named author partially suppoted by an NSF summer grant. c ©1995 American Mathematical Society 1 2 KARIN BAUR, NOLAN WALLACH describe an important special case that is equivalent to classifying even nilpotent elements. In [W] the second named author used his extension of the Lynch results to prove a holomorphic continuation of generalized Jacquet integrals for degenerate principal series under the real analogue of the condition of niceness. These results contain all known cases of continuation, holomorphy and (essentially) uniqueness for Jaquet integrals and Whittaker models. Thus the results of this paper explain both the range of applicability of those results and their limitations. In [L] Lynch studies the classification of his admissible parabolic subalgebras. His results are correct for type An, F4 and G2. The paper [EK] studies the notion of a good grade of a semi-simple algebra. This is a Z-grade of g = ∑ j gj as a Lie algebra such that there is an element x ∈ g2 such that ad(x) is injective on ∑ j<0 gj . Theorem 2.1 implies that a good grade with all odd components equal to 0 defines a nice parabolic ∑ j≥0 gj . Thus a classification of good grades yields as a special case a classification of nice parabolic subalgebras. That said we have decided that the special case was of sufficient importance to have an independent exposition even if the paper [EK] were without any errors (see the remark before Theorem 1.4). Also, our description of the answer for the classical groups is (we hope) simpler than that given in [EK]. In addition, we have endeavored to give enough detail that a serious reader could with little additional effort check that the results are correct. The authors would like to thank Elashvili for informing them of his work with Kac. For obvious reasons there is substantial overlap between our papers. Many of the general results in section 2 can be found in the second named author’s manuscript [W] that has been freely available on his web site (in various forms) for at least two years and in [EK]. 1. Statement of the results If not specified otherwise, g will denote a simple Lie algebra over the complex numbers. Fix a Borel subalgebra b in g, let h ⊂ b be a Cartan subalgebra of g. We will denote the set of simple roots relative to this choice by ∆ = {α1, . . . , αn}. We always use the Bourbaki-numbering of simple roots. Let p ⊂ g be a parabolic subalgebra, p = m ⊕ u (where m is a Levi factor and u the corresponding nilpotent radical of p). After conjugation we can assume that p contains the chosen Borel subalgebra and m ⊃ h. If b has been fixed then we will say that p is standard if p ⊃ b from now on. In particular, if p is standard then it is given by a subset of ∆, namely the simple roots such that both roots spaces g±α belong to the Levi factor of p. Thus such a parabolic subalgebra is described by a n-tuple, (u1, ..., un) in {0, 1}: ones correspond to simple roots with root spaces not in m. Equivalently, a parabolic subalgebra is given by a coloring of the Dynkin diagram of the Lie algebra: a black (colored) node corresponds to a simple root whose root space belongs to m. Here, one has to be very careful since there exist different notations. Our choice was motivated by the coloring for Satake diagrams. Let (u1, ..., un) define the parabolic subalgebra p and and H ∈ h be defined by αi(H) = ui. If we set gi = {x ∈ g|[H,x] = ix} then p = ∑ i≥0 gi. NICE PARABOLIC SUBALGEBRAS OF REDUCTIVE LIE ALGEBRAS 3 1.1. Results in the classical cases. As is usual, we will refer to the simple Lie algebras of type An,Bn,Cn,Dn as the classical Lie algebras and the remaining five simple Lie algebras will be called exceptional. We realize the classical Lie algebras as subalgebras of glN for N = n + 1, 2n + 1, 2n, 2n respectively. With An the trace zero matrices, Bn,Dn the orthogonal Lie algebra of the symmetric form with matrix with all entries 0 except for those on the skew diagonal which are 1 and Cn the symplectic Lie algebra for the symplectic form with matrix whose only nonzero entries are skew diagonal and the first n are 1 and the last n are −1. With this realization we take as our choice of Borel subalgebra the intersection of the corresponding Lie algebra with the upper triangular matrices in glN . We will call a parabolic subalgebra that contains this Borel subalgebra standard. If p is a standard parabolic subalgebra then we refer to the Levi factor that contains the diagonal Cartan subalgebra, h, by m and call it the standard Levi factor. Thus for all classical Lie algebras the standard Levi factor is then in diagonal block form given by a sequence of square matrices on the diagonal. For the orthogonal and symplectic Lie algebras, these sequences are palindromic. If p is a parabolic subalgebra for one of these Lie algebras then if m is the standard Levi factor of the parabolic subalgebra to which it is conjugate then we will say that m is the standard Levi factor. Theorem 1.1. For type An a parabolic subalgebra is nice if and only if the sequence of the block lengths of its standard Levi factor is unimodal. Theorem 1.2. For type Bn a parabolic subalgebra p is nice if and only if its standard Levi factor either has unimodal block lengths or the block lengths are of the form a1 ≤ a2 ≤ ... ≤ ar > b1 = b2 = ... = bs < ar ≥ ... ≥ a1 with b1 = ar − 1 and if r > 1 then ar−1 < ar. Theorem 1.3. For type Cn a parabolic subalgebra is nice if and only if the sequence of block lengths of its standard Levi factor is unimodal and if the number of blocks is odd then each odd block length occurs exactly twice. Before we state our result for Dn a word should be said about the ambiguity in describing parabolic subalgebras in the case of Dn. We note that the intersection of the standard parabolic subalgebra with block sizes [a1, a2, ..., ar, 1, 1, ar, ..., a2, a1] for GL(d) (d = 2(a1+...+ar+1)) with so(d) is the same as the intersection with the standard parabolic with block lengths [a1, a2, ..., ar, 2, ar, ..., a2, a1]. In the following theorem we we will only look at the second version of the parabolic subgroup. The first two types in Theorem 6.4 (i),(ii) in [EK] do not appear in our result directly because of this choice. We should also point out that there is a difference between our statement and that of [EK] involving the last part of this result and Theorem 6.4 (i) and (ii) in [EK]. Their list is missing all cases with s even and ar > 3. In addition to having a (perhaps too) detailed proof of the following result in this paper the authors have done extensive computer computations which agree with our formulation for n ≤ 20. Theorem 1.4. For type Dn a parabolic subalgebra is nice if and only if its standard Levi factor (taking into account the choice made above) has one of the following forms: 1) It is unimodal with an odd number of blocks. 4 KARIN BAUR, NOLAN WALLACH 2) It is unimodal with an even number of block lengths and the odd block lengths occur exactly twice. 3) The block lengths are of the form a1 ≤ a2 ≤ ... ≤ ar > b1 = b2 = ... = bs < ar ≥ ... ≥ a1 with b1 = ar − 1, ar is odd and and if s is even then the odd block lengths occur exactly twice. 1.2. Results in the exceptional cases. In this subsection we will classify the nice parabolic subalgebras for the exceptional simple Lie algebras. The parabolic subalgebra will be given by an n-tuple where n is the rank and the entries are αi(H) where H is the element that gives the grade corresponding to the parabolic subalgebra and the αi are the simple roots in the Bourbaki order. The only nice parabolic subalgebras in type G2, F4 are those that are given by an even sl2-triple (see the next section for the definition). They are listed below. G2 F4 (1, 1) (1, 1, 1, 1) (1, 0) (1, 1, 0, 1) (0, 0) (1, 1, 0, 0) (1, 0, 0, 1) (0, 1, 0, 1) (0, 1, 0, 0) (0, 0, 0, 1) (0, 0, 0, 0) NICE PARABOLIC SUBALGEBRAS OF REDUCTIVE LIE ALGEBRAS 5 The nice parabolic subalgebras of type E are given in the following table: E6 E7 E8 1 (1, 1, 1, 1, 1, 1) (1, 1, 1, 1, 1, 1, 1) (1, 1, 1, 1, 1, 1, 1, 1) 2 (1, 1, 1, 0, 1, 1) (1, 1, 1, 0, 1, 1, 1) (1, 1, 1, 0, 1, 1, 1, 1) 3 (1, 1, 1, 0, 1, 0) (1, 1, 1, 0, 1, 0, 1) (1, 1, 1, 0, 1, 0, 1, 1) 4 (1, 1, 0, 1, 0, 1) (1, 1, 0, 0, 1, 0, 1) (1, 0, 1, 0, 0, 0, 0, 1) 5 (1, 1, 0, 0, 1, 0) (1, 1, 0, 0, 0, 0, 1) (1, 0, 0, 1, 0, 1, 1, 1) 6 (1, 1, 0, 0, 0, 1) (1, 0, 1, 1, 0, 1, 0) (1, 0, 0, 1, 0, 1, 0, 1) 7 (1, 1, 0, 0, 0, 0) (1, 0, 1, 0, 0, 1, 0) (1, 0, 0, 1, 0, 0, 1, 1) 8 (1, 0, 1, 1, 0, 1) (1, 0, 1, 0, 0, 0, 0) (1, 0, 0, 1, 0, 0, 1, 0) 9 (1, 0, 1, 0, 0, 1) (1, 0, 0, 1, 0, 1, 1) (1, 0, 0, 0, 1, 0, 0, 1) 10 (1, 0, 1, 0, 0, 0) (1, 0, 0, 1, 0, 1, 0) (1, 0, 0, 0, 0, 1, 1, 1) 11 (1, 0, 0, 1, 1, 1) (1, 0, 0, 1, 0, 0, 1) (1, 0, 0, 0, 0, 1, 0, 1) 12 (1, 0, 0, 1, 0, 1) (1, 0, 0, 0, 1, 0, 0) (1, 0, 0, 0, 0, 1, 0, 0) 13 (1, 0, 0, 1, 0, 0) (1, 0, 0, 0, 0, 1, 1) (1, 0, 0, 0, 0, 0, 1, 1) 14 (1, 0, 0, 0, 1, 1) (1, 0, 0, 0, 0, 1, 0) (1, 0, 0, 0, 0, 0, 1, 0) 15 (1, 0, 0, 0, 1, 0) (1, 0, 0, 0, 0, 0, 1) (1, 0, 0, 0, 0, 0, 0, 1) 16 (1, 0, 0, 0, 0, 1) (1, 0, 0, 0, 0, 0, 0) (1, 0, 0, 0, 0, 0, 0, 0) 17 (1, 0, 0, 0, 0, 0) (0, 1, 1, 0, 0, 1, 1) (0, 1, 0, 0, 0, 0, 0, 1) 18 (0, 1, 1, 0, 1, 1) (0, 1, 0, 0, 0, 0, 0) (0, 1, 0, 0, 0, 0, 0, 0) 19 (0, 1, 1, 0, 0, 1) (0, 0, 1, 0, 0, 1, 0) (0, 0, 1, 0, 0, 0, 1, 0) 20 (0, 1, 0, 1, 0, 0) (0, 0, 1, 0, 0, 0, 1) (0, 0, 0, 1, 0, 0, 1, 1) 21 (0, 1, 0, 0, 0, 1) (0, 0, 1, 0, 0, 0, 0) (0, 0, 0, 1, 0, 0, 1, 0) 22 (0, 1, 0, 0, 0, 0) (0, 0, 0, 1, 0, 1, 0) (0, 0, 0, 1, 0, 0, 0, 1) 23 (0, 0, 1, 0, 0, 1) (0, 0, 0, 1, 0, 0, 1) (0, 0, 0, 0, 1, 0, 0, 1) 24 (0, 0, 1, 0, 0, 0) (0, 0, 0, 1, 0, 0, 0) (0, 0, 0, 0, 1, 0, 0, 0) 25 (0, 0, 0, 1, 0, 1) (0, 0, 0, 0, 1, 0, 1) (0, 0, 0, 0, 0, 1, 0, 0) 26 (0, 0, 0, 1, 0, 0) (0, 0, 0, 0, 1, 0, 0) (0, 0, 0, 0, 0, 0, 1, 1) 27 (0, 0, 0, 0, 1, 1) (0, 0, 0, 0, 0, 1, 0) (0, 0, 0, 0, 0, 0, 1, 0) 28 (0, 0, 0, 0, 1, 0) (0, 0, 0, 0, 0, 0, 1) (0, 0, 0, 0, 0, 0, 0, 1) 29 (0, 0, 0, 0, 0, 1) (0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0) 30 (0, 0, 0, 0, 0, 0) 2. Characterizations of niceness In this section we study the notion of niceness and prove properties of nice parabolic subalgebras. Recall that a parabolic subalgebra induces a Z-grading of g via the map adH where H ∈ h is defined as in the beginning of section 1. We will also use the notation B for the Killing form of g. Theorem 2.1. Let p ⊂ g with associated grading g = ⊕j∈Zgj. The following are equivalent: (1) p is nice (2) There exists X ∈ g1 such that ad(X) : gj → gj+1 is surjective for all j ≥ 0. (3) There exists X ∈ g1 such that ad(X) : gj → gj+1 is surjective for all j > 0. (4) There exists X ∈ g1 such that ad(X) : gj−1 → gj is injective for all j ≤ 0. Proof. Let X ∈ g1 be such that [p, X ] = n. Then since p = g0 ⊕ g1 ⊕ ...⊕ gr 6 KARIN BAUR, NOLAN WALLACH we must have ad(X)gi = gi+1 for i ≥ 0. This proves the necessity of condition 2). The sufficiency is equally clear. To prove 3) we will show that 3) implies 2). Let Ω be the set of all X ∈ g1 satisfying the condition in 3). Then Ω is Zariski open and dense in g1. We also note that if Λ is the set of all X ∈ g1 such that [X, p] = g1 then it is also Zariski open and dense. Hence Ω ∩ Λ 6= ∅. We have proved that 3) implies 2). We now prove 4). For this we observe that if n = ∑