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Bull. Sci. math. 128 (2004) 16 www.elsevier.com/locate/bulsci

Regions in the dominant chamber and nilpotent orbits
Dmitri I. Panyushev
1

Independent University of Moscow, Bol'shoi Vlasevskii per. 11, 121002 Moscow, Russia Received 10 July 2003; accepted 19 July 2003

Abstract Let G be a complex semisimple algebraic group with Lie algebra g. The goal of this note is to show that combining some ideas of Gunnells and Sommers [Math. Res. Lett. 10 (23) (2003) 363373] and Vinberg and Popov [Invariant Theory, in: Algebraic Geometry IV, in: Encyclopaedia Math. Sci., Vol. 55, Springer, Berlin, 1994, pp.123284] yields a geometric description of the characteristic of a nilpotent G-orbit in an arbitrary (finite-dimensional) rational G-module. 2003 Elsevier SAS. All rights reserved.
MSC: 14L30; 17B10; 17B35 Keywords: Semisimple Lie algebra; Periodic grading; Characteristic of a nilpotent orbit; Weyl chamber

Let G be a complex semisimple algebraic group with Lie algebra g. The goal of this note is to show that combining some ideas of [5] and [1] quickly yields a geometric description of the characteristic of a nilpotent G-orbit in an arbitrary (finite-dimensional) rational G-module. Fix a Borel subalgebra b g and a Cartan subalgebra t in it. For each t-weight µ of a G-module V, consider the affine hyperplane Hµ,2 = {x | (x , µ) = 2} tR . These hyperplanes cut the dominant chamber in finitely many regions, and to any region R one may attach a b-stable subspace of V by the following rule: VR =
µIR

Vµ ,

E-mail address: panyush@mccme.ru (D.I. Panyushev).
1 This research was supported in part by R.F.B.I. Grants no. 01-01-00756 and 02-01-01041.

0007-4497/$ see front matter 2003 Elsevier SAS. All rights reserved. doi:10.1016/j.bulsci.2003.07.001

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D.I. Panyushev / Bull. Sci. math. 128 (2004) 16

where IR is the set of weights of V such that (x , µ) > 2 for some (equivalently, any) x R . Given a nilpotent G-orbit O V, consider the closure of the union of all regions R such that O VR =. Let's call this set CO . Our first observation is that CO contains a unique element of minimal length, and this element is just the dominant characteristic of O in the sense of [5, 5.5]. Next, we show that if the representation G GL(V) is associated with either a periodic or a Z-grading of a reductive algebraic Lie algebra, then the condition "O VR = " can be replaced with "O VR is dense in VR ". This new condition determines a smaller set CO CO , but these two sets still have the same element of minimal length. This provides another proof and also a generalization of the main result of [1]. It is worth noting that the representations associated with Zm -gradings are visible, i.e., contain finitely many nilpotent orbits, and in this case different orbits have different characteristics. We also give an example showing that, for an arbitrary visible G-module V, it may happen that different orbits have the same characteristic and that for some orbits O there are no subspaces of the form VR such that O VR is dense in VR . Main notation. is the root system of (g, t) and W is the Weyl group of (t,). + is the set of positive roots and ={1 , ... , p } is the set of simple roots in + . We define tR to be set of all elements of t having real eigenvalues in any G-module (a Cartan subalgebra of a split real form of g). Denote by (, ) a W -invariant inner product p on tR .Using (, ), we identify tR and t . So that, one may think that tR = i =1 Ri . R C ={x V | (x , ) > 0 } is the (open) fundamental Weyl chamber.

1. The characteristic of a nilpotent orbit In this section we recall some results published in the survey article [5, §5]. Unfortunately, that simple approach to questions of stability, optimal one-parameter subgroups, and a stratification of the null-cone remained largely unnoticed by the experts. Let V be a G-module. Write Vµ for the µ-weight space of V. Here µ is regarded as element of tR . Hence µ(x ) = (µ, x ) for any x tR . Suppose h tR , i.e., h is a rational semisimple element. For a G-module V and c Q, we set Vh c ={v V | h·v = cv }, V
h

c=
kc

Vh k ,

and Vh >c =
k>c

Vh k .

For instance, gh 0 is the centralizer of h in g (a Levi subalgebra of g), gh 0 is a parabolic subalgebra of g,and gh >0 is the nilpotent radical of gh 0 . Clearly, Vh c =
µ: µ(h)=c

V

µ

and gh a ·Vh c Vh a + c .

Recall that an element v V, or the orbit G·v , is called nilpotent,if G·v 0. It is easy to verify that, for any h tR , the subspace Vh >0 consists of nilpotent elements. Conversely, the HilbertMumford criterion asserts that any nilpotent G-orbit in V meets a subspace of this form for a suitable h.

D.I. Panyushev / Bull. Sci. math. 128 (2004) 16

3

Definition 1.1. The characteristic of a nilpotent orbit O is a shortest element h tR such that O Vh 2 =. Remark. In principle, one may choose an arbitrary normalization "( c)" in the definition. The choice c = 2 is explained by the fact that for V = g this leads to the usual (Dynkin) characteristic of a nilpotent element. It was shown in [5, 5.5] that each nilpotent orbit has a characteristic. Moreover, if h1 ,h2 tR are two characteristics of O , then they are W -conjugate. Thus, to any nilpotent orbit O V one may attach uniquely the dominant characteristic, which is denoted by hO . If we are given an h tR and u Vh 2 , then it is helpful to have a criterion to decide whether h is a characteristic of G·u. The following result, attributed in [5, Theorem 5.4] to F. Kirwan and L. Ness, solves the problem. Let ZG (h) denote the centralizer of h in G and ZG (h) ZG (h) the reduced centralizer. That is, the Lie algebra of ZG (h) is the orthogonal complement to h in gh 0 = Lie ZG (h). Clearly, Vh c is a ZG (h)-module for any c. Theorem 1.2. Under the previous notation, h is a characteristic of G·u if and only if the projection of u to Vh 2 is not a nilpotent element with respect to the action of ZG (h).

2. Regions and characteristics Let V be to t. For any n u m b er "2 " the hyperpla true. a G-module. Write P (V) for the set of nonzero weights of V with respect µ P (V), consider the affine hyperplane Hµ,2 ={x tR | (x , µ) = 2}.The is determined by the normalization in Definition 1.1. We will be interested in nes meeting the dominant Weyl chamber. It is easily seen that the following is

Lemma 2.1. We have Hµ,2 C = µ has a positive coefficient in the expression
p

µ=
i =1

ai i

(ai Q).

The set of all such hyperplanes cuts C in regions. That is, a region (associated with V) is a connected component of C \ µ Hµ,2 . The set of all regions is denoted by R = R(V). Clearly, the closure of each region is a convex polytope. Given a region R , consider all hyperplanes separating R from the origin, and the corresponding weights in P (V). This set of weights is denoted by IR . More precisely, if x R , then IR = µ P (V) | (x , µ) > 2 . Lemma 2.2. Let R R. (i) if µ IR , + , and µ + + , then µ + IR . (ii) The subspace VR := µIR Vµ V is b-stable.

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D.I. Panyushev / Bull. Sci. math. 128 (2004) 16

(iii) Each G-orbit meeting VR is nilpotent. Proof. (i) obvious; (ii) follows from (i); (iii) we have lim x R and u VR .
t -

exp(t x )·u = 0 for any

Suppose O V is a nilpotent G-orbit. One may attach to O a collection of regions, as follows. Set MO ={R R | O G·VR }={R R | O VR =}, and CO =
R MO

(2.1)

R C.

Thus, CO is a closed subset of C determined by O . Let h CO be an element of minimal length. Proposition 2.3. h is a unique element of minimal length in CO , and h = hO . 2 = and it is a Proof. By the very construction, h has the property that O Vh shortest dominant element with this property. It then follows from results described in Section 1 that h = hO and CO contains a unique element of minimal length. The above construction is inspired by [1], where the case V = g is considered. However, condition (2.1) was slightly different there. Namely, the set of regions attached to O was determined by the condition that O gR be dense in gR . But this stronger condition cannot lead in general to satisfactory results, unless V is a visible G-module. For, the number of subspaces of the form VR is finite and therefore the set of such regions would be empty for infinitely many nilpotent orbits, if V is not visible. Moreover, even if V is visible, it may happen that, for a given nilpotent orbit, there is no subspace VR (R R) such that O VR is dense in VR (see example below). However, one may formally set MO ={R R | O is dense in G·VR } ={R R | O VR is dense in VR }, and CO =
R MO

(2.2)

R C.

~ Clearly, MO MO and CO CO . We also define hO to be an element of CO of minimal length (if CO =!). Example 2.4. Here we give an example of a visible module such that (i) MO = for some O , and (ii) hO1 = hO2 for different nilpotent orbits.

D.I. Panyushev / Bull. Sci. math. 128 (2004) 16

5

Let G = SL(V1 ) в SL(V2 ), dim V1 = dim V2 = 2, and V = (V1 V2 ) V1 . Identifying V with the space of 2 by 3 matrices, we write v= m p n q x y

for a generic element in V.Here V1 V2 = m p n q 0 0 .

There are five nilpotent orbits in V, and representatives of the non-trivial orbits are: O2 : v2 = O4 : v4 = 00 00 10 00 1 , 0 1 , 0 O3 : v3 = O5 : v5 = 1 0 1 0 00 , 00 00 . 01

One has dim Oi = i for 2 i 5. Since rk G = 2, we have tR is R2 and it is not hard to depict all the regions and determine the characteristics. The dominant chamber is the positive quadrant. There are four lines of the form Hµ,2 meeting the positive quadrant, which correspond to the roots 1 ,1 + 2 ,1 - 2 , -1 + 2 . Here i is the simple root of SL(Vi ). Hence one gets six regions marked by Roman numbers. We have MO2 = , MO3 ={II, III}, MO4 ={IV,V}, MO5 ={VI}. Since O4 O2 and O3 O2 , we conclude that h3 = (1, 1), h2 = h4 = (2, 0),and h5 = (2, 4). The elements hi are circled in Fig. 1. ~ Therefore one should not expect that hO is always defined and that different orbits have different characteristics. At the rest of the section, we give a sufficient condition for this to happen. Let l be a reductive algebraic Lie algebra. Consider a Zm -grading of l, where m N or m = . That is, we have l = i Zm li if m is finite, and l = i Z li is a Z-grading in the second case. Here l0 is reductive and each li is an l0 -module. Let G be a connected (reductive) group with Lie algebra l0 , and set V = l1 . Then we shall say that the representation G GL(V) is associated with a Zm -grading (of l). By a famous result of Vinberg [3, § 2], V is a visible G-module in this situation.

Fig. 1.

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D.I. Panyushev / Bull. Sci. math. 128 (2004) 16

Theorem 2.5. Suppose the representation G GL(V) is associated with a Zm -grading. Then different nilpotent G-orbits in V have different characteristics and for any nilpotent ~ G-orbit O V we have hO = hO . Proof. (1) Let e V = l1 be a nilpotent element, and O = G·e. By a generalization of the MorozovJacobson theorem [4, Theorem 1(1)], there is an sl2 -triple (e,h,f ) such that h l0 = g and f l-1 . The rational semisimple element h determines a Z-grading of l, and we have e Vh 2 lh 2 . It is well known that, in the Lie algebra l, we have ZL (h)·e is closed in lh 2 . Hence, by the RichardsonVinberg lemma [3, § 2], ZG (h)·e is closed in Vh 2 . Without loss of generality, one may assume that h is a dominant element in tR . Then, in view of Theorem 1.2, h = hO is the dominant characteristic of O . Next, ~ L·e lh 2 is dense in lh 2 and hence O Vh 2 is dense in Vh 2 . Thus, h = hO . (2) That different nilpotent orbits have different characteristics stems from [4, Theorem 1(4)]. This result applies, in particular, to the adjoint representations (m = 1), where one obtains another proof for the main result in [1]. Another interesting case is that of the little adjoint G-module, if g is a simple Lie algebra having roots of different lengths. Let s be the short dominant root. Then the simple G-module with highest weight s is called the little adjoint. It is denoted by gla .The set P (gla ) is s , the set of short roots. This module is always associated with a Zm -grading (m = 2 for Bp , Cp , F4 ; m = 3 for G2 ). Therefore the set of regions R(gla ) allows us to determine the characteristics of the nilpotent G-orbits in gla . The arrangement of hyperplanes Hµ,2 (µ + ) inside of C was studied in [2], s where it was shown that there is a bijection between the set of regions R(gla ) and the set of all b-stable subspaces of gla without semisimple elements. We also give in [2] an explicit formula for the number #R(gla ).

References
[1] P. Gunnells, E. Sommers, A characterization of Dynkin elements, Math. Res. Lett. 10 (23) (2003) 363373. [2] D. Panyushev, Short antichains in root systems, semi-Catalan arrangements, and B -stable subspaces, Eur. J. Combin. (in press). [3] .B. Vinberg, Gruppa Veil graduirovannoi algebry Li, Izv. Akad. Nauk SSSR Ser. Matem. 40 (1976) 488526 (Russian). English translation: E.B. Vinberg, The Weyl group of a graded Lie algebra, Math. USSR-Izv. 10 (1976) 463495. [4] .B. Vinberg, Klassifikaci odnorodnyh nilpotentnyh lementov poluprostoi gra duirovannoi algebry Li, in: Trudy Seminara po Vekt. i Tenz. Analizu, T. 19, MGU, Moskva, 1979, pp. 155177 (Russian). English translation: E.B. Vinberg, Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Selecta Math. Sovietica 6 (1987) 1535. [5] .B. Vinberg, V.L. Popov, Teori invariantov, in: Sovremennye Problemy Matematiki. Fundamentalnye Napravleni , T. 55, VINITI, Moskva, 1989, pp. 137309 (Russian). English translation: V.L. Popov, E.B. Vinberg, Invariant theory, in: Algebraic Geometry IV, in: Encyclopaedia Math. Sci., vol. 55, Springer, Berlin, 1994, pp. 123284.