Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits B. Binegar Department of Mathematics Oklahoma State University Stillwater, OK 74078, USA Nankai Summer School in Representation Theory and Harmonic Analysis June, 2008 B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 1 / 31 Setup g : a semisimple Lie algebra over C h : a Cartan subalgebra of g ∆ = ∆(h; g) : the roots of h in g Π = Π(h; g) : a set of simple roots of ∆ G : the adjoint group of g (a complex algebraic group) N = Ng : the cone of nilpotent elements in g a N = O O2GnN Our problem: parameterizing GnN , the (finite) set of nilpotent orbits B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 2 / 31 Parameterizations of GnN x 2 O 2 GnN x extends to a Jacobson-Morozov standard triple fx; h; yg [x; y] = h ; [h; x] = 2x ; [h; y] = −2y h can be conjugated to element in dominant Weyl chamber Theorem (Kostant, 1959) Suppose Π = fα1; : : : ; αng. A nilpotent orbit O is completely determined by the values [α1 (h) ; α2 (h) ; : : : ; αn (h)]. Moreover, the only possible values of αi (h) are 0, 1, and 2. B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 3 / 31 Weighted Dynkin diagrams Upshot Each nilpotent orbits corresponds to a certain labeling of the nodes of Dynkin diagram of g by one of f0; 1; 2g. Such a labeled Dynkin diagram is called a weighted Dynkin diagram (or WDD). Example WDDs for sl(4) 2 2 2 0 2 0 • • • • • • 0 0 0 • • • 2 0 2 1 0 1 • • • • • • B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 4 / 31 Remarks Relatively few of the 3rank(g) possible diagrams are actually realized as WDDs. No algorithm for predicting which diagrams occur. More accurate to say WDDs provide a unique labeling (rather than a parameterization) of nilpotent orbits. Nevertheless, some general characteristics of an orbit can be read off its WDD. B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 5 / 31 Partition-type classifications When g is of classical type (g = sln, so2n+1, sp2n, or so2n) the nilpotent orbits of g can be parameterized by certain families of partitions. Let N denote the dimension of the standard representation of g. GnN 3 O 3 x −! X 2 M(N; N) The Jordan normal form of X completely determines O. The partition pO corresponding to O is the list of sizes of the irreducible Jordan blocks that occur in the Jordan normal form of X. 0 0 1 0 0 0 0 0 0 1 B 0 0 1 0 0 0 0 0 C B C B 0 0 0 0 0 0 0 0 C B 0 0 0 0 1 0 0 0 C [3; 2; 2; 1] ! B C 2 sl B 0 0 0 0 0 0 0 0 C 8 B C B 0 0 0 0 0 0 1 0 C B C @ 0 0 0 0 0 0 0 0 A 0 0 0 0 0 0 0 0 B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 6 / 31 partition parameterizations Unlike WDD labeling, the partitions pO that do occur can be determined by simple rules. The nilpotent orbits of sln are in a one-to-one correspondence with the set PA = P (n) of partitions of n. The nilpotent orbits of so2n+1 are in a one-to-one correspondence with the set PB (2n + 1) consisting of partitions 2n + 1 such that even parts only occur with even multiplicity. The nilpotent orbits of sp2n are in a one-to-one correspondence with the set PC (2n) consisting of partitions 2n such that odd parts only occur with even multiplicity. The nilpotent orbits of so2n are in a nearly one-to-one correspondence with the set PD (2n) consisting of partitions 2n such that even parts only occur with even multiplicity. Partitions in PD (2n) which consist only of even parts (necessarily each with even multiplicity) are called very even partitions. To each very even partition there corresponds two distinct nilpotent orbits. B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 7 / 31 Remarks Given a partition p 2 PG it is easy to write down a representative matrix X using recipes found, e.g., in Collingwood and McGovern. The closure relations amongst the nilpotent orbits can be inferred directly from the dominance partial ordering of partitions i i 0 X X 0 Op ⊆ Op0 () p ≤dom p ≡ pj ≤ pj 8 i j=1 j=1 There exist nice algorithms for computing dimensions of orbits, induced orbits, Spaltenstein duals, etc. using the partition parameterizations. But partition classification schemes apply only to Lie algebras of classical type. B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 8 / 31 Digression: Standard Levis and Standard Parabolics Definition Let Γ ≡ a subset of the simple roots Π of g, ∆Γ ≡ the subset of ∆ generated by the simple roots α 2 Γ, + ∆(uΓ) ≡ ∆ − ∆Γ. The standard Levi subalgebra corresponding to Γ is the reductive subalgebra lΓ of g given by X lΓ = h + gα α2∆Γ The standard parabolic subalgebra corresponding to Γ is the parabolic subalgebra pΓ of g given by X pΓ = lΓ + gα α2∆(uΓ) B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 9 / 31 Inclusion and Induction There are two fundamental ways of lifting a nilpotent orbit Ol of a Levi subalgebra l to a nilpotent orbit in g. Bala-Carter inclusion: g incl (Ol) := G ·Ol = G-saturation of Ol in g induction: Let p = l + u be any extension of l to a parabolic subalgebra of g. Then g indl (Ol) := unique dense orbit in G · (Ol + u) g Theorem (Lusztig-Spaltenstein) indl (Ol) exists and is independent of the choice of the nilradical u. B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 10 / 31 Distiguished orbits Definition An element x 2 N is distinguished in g if it is not contained in any proper Levi subalgebra of g. If x is distinguished, then so is every element of G · x. It thus makes sense to speak of distinguished orbits. The principal orbit (i.e. the maximal nilpotent orbit) is always distinguished. For sln the principal orbit is the only distinguished orbit. E8, on the other hand, has 11 different distinguished orbits B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 11 / 31 Bala-Carter classification: version I Theorem (Bala-Carter, 1976) Every nilpotent orbit O in g is the Bala-Carter inclusion of a distinguished orbit Ol in a Levi subalgebra l of g. In fact, the correspondence GnN ! fG-conjugacy classes of distinguished orbits of Levi subalgebrasg is one-to-one. There are a number of facts that go into the proof of the above theorem: fminimal Levi subalgebras containing xg ! fmaximal toral subalgebras of gx g =) minimal Levis containing x are all conjugate x is distinguished if and only if gx has no semisimple elements B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 12 / 31 Let x −! fx; h; yg : J-M triple M g = gk ;[h; z] = kz if z 2 gk k2Z If x is distinguished, then dim g0 = dim g2 If x is distinguished, then Ox is even. (g2j+1 = 0) B-C idea: inclusion of distinguished orbits provides an inductive construction of nilpotent orbits. To get this inductive scheme going one still needs a manageable way of classifying the distinguished orbits of a reductive Lie group. B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 13 / 31 Distinguished parabolics and a construction of distinguished orbits Definition A parabolic subalgebra p = l + u of g is called distinguished if dim l = dim (u= [u; u]) : Theorem Every distinguished orbit of l is obtained by parabolic induction of the trivial orbit of a distinguished parabolic algebra p = l0 + u of l. g O distinguishe )O = indl (0) for some distinguished parabolic p = l + u. B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 14 / 31 Bala-Carter classification, version II Theorem There is a natural one-to-one correspondence between nilpotent orbits of g and 0 G-conjugacy classes of pairs (l; pl) where l is a Levi subalgebra of g and pl = l + u is a distinguished parabolic subalgebra of l. The correspondence is given by g l (l; pl) ! incl indl0 (0) Definition A nilpotent orbit obtained by parabolic induction from the trivial orbit of a Levi subalgebra is called a Richardson orbit: g O is Richardson () O = indl (0) for some Levi subalgebra l B. Binegar (Oklahoma State University) Lecture 3: A Combinatorial Parameterization of Nilpotent Orbits Nankai 2008 15 / 31 Classifying distinguished parabolics Theorem The conjugacy classes of parabolic subalgebras of a semisimple Lie algebra are in a one-to-one correspondence with the subsets Γ 2 2Π. The correspondence is given by Γ −! G-conj class of pΓ where pΓ is the standard parabolic corresponding to Γ ⊂ Π Recall Definition A parabolic subalgebra p = l + u of g is called distinguished if dim l = dim (u= [u; u]) : B.