1. Introduction
Total Page:16
File Type:pdf, Size:1020Kb
RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS PETER E. TRAPA Abstract. For classical real Lie groups, we compute the annihilators and asso ciated vari- eties of the derived functor mo dules cohomologically induced from the trivial representation. (Generalizing the standard terminology for complex groups, the nilp otent orbits that arise as such asso ciated varieties are called Richardson orbits.) We show that every complex sp ecial orbit has a real form which is Richardson. As a consequence of the annihilator calculations, we give many new in nite families of simple highest weight mo dules with ir- reducible asso ciated varieties. Finally we sketch the analogous computations for singular derived functor mo dules in the weakly fair range and, as an application, outline a metho d to detect nonnormality of complex nilp otent orbit closures. 1. Introduction Fix a complex reductive Lie group, and consider its adjoint action on its Lie algebra g. If q = l u is a parab olic subalgebra, then the G saturation of u admits a unique dense orbit, and the nilp otent orbits which arise in this way are called Richardson orbits (following their initial study in [R]). They are the simplest kind of induced orbits, and they play an imp ortant role in the representation theory of G. It is natural to extend this construction to the case of a linear real reductive Lie group G . Let g denote the Lie algebra of G , write g for its complexi cation, and G for the R R R complexi cation of G . Let denote the Cartan involution of G , write g = k p for R R the complexi ed Cartan decomp osition, and let K denote the corresp onding subgroup of G. Instead of considering nilp otent orbits of G on g , we work on the other side of the Kostant- R R Sekiguchi bijection and consider nilp otent K orbits on p. (As a matter of terminology, we say that such an orbit O is a K -form of its G saturation.) Fix a -stable parab olic subalgebra K q = l u of g. Then the K saturation of u \ p admits a unique dense orbit, and we call the orbits that arise in this way Richardson. It is easy to check that this de nition reduces to the one given ab ove if G is itself complex. R It is convenient to give a slightly more geometric formulation of this de nition. A -stable parab olic q de nes a closed orbit K q of K on G=Q (where Q is the corresp onding parab olic 1 subgroup of G). Let denote the pro jection from G=B to G=Q. Then (K q) has a dense K orbit (say O ) and we may consider its conormal bundle in the cotangent bundle q to G=B . A little retracing of the de nitions shows the image of the conormal bundle to O q under the moment map for T (G=B ) is indeed the K saturation of u \ p, and we thus obtain a second characterization of Richardson orbits: they arise as dense K orbits in the moment map image of conormal bundles to orbits of the form O . q The ab ove geometric interpretation is esp ecially relevant in the context of the represen- tation theory of G . Consider the irreducible Harish-Chandra mo dule (say A ) of trivial R q in nitesimal character attached to the trivial lo cal system on O by the Beilinson-Bernstein q This pap er was written while the author was an NSF Postdo ctoral Fellow at Harvard University. 1 2 PETER E. TRAPA equivalence. Then A is a derived functor mo dule induced from a trivial character and is of q the form considered, for example, in [VZ]; see Section 2.2 for more details. From the preced- ing discussion (esp ecially the fact that K q is closed), it is easy to see that the D -mo dule characteristic variety of A is the closure of the conormal bundle to O . (The de nition of q q the characteristic variety is recalled in Section 2.3 b elow.) Since the moment map image of the characteristic variety of a Harish-Chandra mo dule is its asso ciated variety, we arrive at a third characterization of Richardson orbits: they are the nilp otent orbits of K on p that arise as dense K orbits in the asso ciated varieties of mo dules of the form A . Note that q since the G saturation of the asso ciated variety of a Harish-Chandra mo dule with integral in nitesimal character is sp ecial ([BV2 ]), this interpretation implies that the G saturation of a Richardson orbit is a sp ecial nilp otent orbit for g. The rst result of the present pap er is an explicit computation of Richardson orbits in classical real Lie algebras. In typ e A, this is well-known (see [T3] for instance); we give the answer for other typ es in Sections 3{7. This is not particularly dicult and amounts only to some elementary linear algebra, but the answer do es have the a p osteriori consequence that every complex sp ecial orbit has a Richardson K -form. Theorem 1.1. Fix a special nilpotent orbit O for a complex classical group G. Then there exists a real form G such that some irreducible component of O \ p is a Richardson orbit R of K on the nilpotent cone of p. This result has the avor of a corresp onding result for admissible orbits. Mo dulo some conjectures of Arthur and based on [V4], Vogan gave a simple conceptual pro of that for the split real form of G, every complex sp ecial orbit has a K -form which is admissible. (Without relying on the Arthur conjectures, the result has b een established in a case by case manner for the classical groups by Schwarz [Sc ] and for the exceptional groups by Noel [No] and Nevins [N].) It would b e worthwhile to check Theorem 1.1 for the exceptional groups. A conceptual argument might b e very enlightening. Our second main result concerns the annihilators of the mo dules A (). Using the ex- q plicit form of the computation of Richardson orbits in the classical case, one may adapt an argument from [T2 ] to establish the following result. Theorem 1.2. For the classical groups, the annihilator of any module of the form A is q explicitly computable. The computation, which is carried out in Section 8.6 and is relatively clean, is made in terms of the tableau classi cation of the primitive sp ectrum of U(g) due to Barbasch-Vogan ([BV1 ]) and Gar nkle ([G1 ]{[G4 ]). Using these calculations, one can immediately apply the main techniques of [T2 ] to compute the annihilators and vanishing of many (and p ossibly all) weakly fair A () mo dules of the classical groups. It is imp ortant to recall that these q highly singular mo dules can b e reducible, and implicit in the previous sentence is a metho d to detect cases of such reducibility. In turn, a theorem of Vogan's (see [V3]) asserts that the reducibility of a weakly fair A () is sucient to deduce the nonnormality of the complex q orbit closure that arises as the asso ciated variety of Ann (A ()). Together with the q U(g) reducibility computations, this allows one in principle to deduce the nonnormality of certain orbit closures. It may b e interesting to pursue these ideas in the still op en case of the very 1 even orbits in typ e D . 1 In this context, it is imp ortant to note that this metho d can b e used to work directly with SO (n; C ) (and not O (n; C )) orbits; see Remark 7.3 b elow. RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS 3 The computations of annihilators of weakly fair A () mo dules may still seem rather tech- q nical to those unfamiliar with real groups. Yet they are imp ortant, even for applications to simple highest weight mo dules. We prove the following result, which is logically indep endent from the rest of the pap er, in Section 8.2; the notation is as in Section 2.1. Theorem 1.3. Fix G complex semisimple (not necessarily classical). Suppose I is the 1 annihilator of an A module for some real form G of G. If Ann (L(w )) = I , then q R U(g) AV(L(w )) is irreducible, i.e. is the closure of a unique orbital variety for g. If G is classical, this orbital variety is e ectively computable. This result gives new examples of simple highest weight mo dules with irreducible asso- ciated varieties; using more re ned ideas (which will pursued elsewhere), it leads to many more examples. The pro of of Theorem 1.3 is based on an interesting (but indirect) interac- tion b etween the highest weight category and the category of Harish Chandra mo dules for a real reductive group. It would b e very useful to understand this interaction in a more direct manner. 2. Background and Notation Throughout we retain the notation established in the intro duction for a real reductive Lie group G . R g 2.1. Highest weight mo dules. Let h b e a subalgebra of g. In general, we write ind for h the change of rings functor U(g) . U(h) Fix a Borel b = h n in g, and write for the corresp onding half-sum of p ositive ro ots.