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Restrictions of Generalized Verma Modules to Symmetric Pairs

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Restrictions of Generalized Verma Modules to Symmetric Pairs Restrictions of generalized Verma modules to symmetric pairs Toshiyuki Kobayashi∗ Abstract We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs (g; g0). In general, Verma modules may not contain any simple mod- ule when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple (g; g0; p) such that the 0 restriction Xjg0 always contains simple g -modules for any g-module X lying in the parabolic BGG category Op attached to a parabolic subal- gebra p of g. Formulas are derived for the Gelfand{Kirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction Xjg0 is generically multiplicity-free for any p and any X 2 Op if and only if (g; g0) is isomorphic to (An; An−1), (Bn; Dn), or (Dn+1; Bn). Explicit branching laws are also presented. Keywords and phrases: branching law, symmetric pair, Verma module, high- est weight module, flag variety, multiplicity-free representation 2010 MSC: Primary 22E47; Secondary 22F30, 53C35. Contents 1 Program 2 2 Branching problem of Verma modules 4 ∗Partially supported by Institut des Hautes Etudes´ Scientifiques, France and Grant-in- Aid for Scientific Research (B) (22340026), Japan Society for the Promotion of Science 1 3 Discretely decomposable branching laws 6 4 Branching problems for symmetric pairs 12 5 Multiplicity-free branching laws 24 1 Program Branching problems in representation theory ask how irreducible modules decompose when restricted to subalgebras. In the context of the Bernstein{ Gelfand{Gelfand category O of a semisimple Lie algebra g, branching prob- lems are seemingly simple, however, it turns out that the restrictions behave 0 0 wildly in general. For instance, the restrictions Xjg1 and Xjg2 of a g-module X lying in O may be completely different even when two reductive subal- 0 0 gebras g1 and g2 are conjugate to each other as the following observations indicate (see Examples 4.14, 4.15 for more details): 0 0 Observation 1.1. The restriction Xjg1 does NOT contain any simple g1- 0 0 module, whereas Xjg2 decomposes into an algebraic direct sum of simple g2- modules. 0 Observation 1.2. The Gelfand{Kirillov dimension of any simple g1-module 0 0 0 occurring in Xjg1 is larger than that of any simple g2-module in Xjg2 . The understanding of such phenomena requires a precise formulation of branching problems. Among others we begin by asking what is a `well- 0 posed' framework of branching problems for the restriction Xjg0 where g is a (generalized) reductive subalgebra of g and X lies in the category O: 0 Problem A. When does the restriction Xjg0 contain a simple g -module? 0 Further, we raise the following problems when Xjg0 contains simple g -modules. 0 Problem B. Find the `size' of simple g -modules occurring in Xjg0 . 0 Problem C. Estimate multiplicities of simple g -modules occurring in Xjg0 . Problem D. Find branching laws, in particular, for multiplicity-free cases. 2 Let us explain briefly our main results. We write B for the full flag variety of g, and G0 for the set of conjugacy classes of g0 under the group G := Int(g) of inner automorphisms. Then the `framework' of the restriction 0 Xjh for X 2 O and h 2 G is described by means of the quotient space Gn(B × G0) under the diagonal action of G. More generally, we formulate a proper framework to discuss Problems A to D in Theorem 2.1 in the parabolic BGG category Op (see Subsection 2.1) for an arbitrary parabolic subalgebra p of g. After discussing basic results in this framework in the generality that g0 is an arbitrary reductive subalgebra in g, we highlight the case where (g; g0) is a symmetric pair to get finer results, keeping differential geometric applications in mind. It includes the `group case' (g1 ⊕ g1; diag(g1)) as a special example, for which the branching laws describe the decomposition of the tensor product of two representations (e.g. fusion rules). For symmetric pairs (g; g0), the cardinality of G-orbits on B × G0 is finite, and we give a complete answer to Problem A in the category Op in terms of the finite set 0 Gn(P × G ). Namely, we prove in Theorem 4.1 that the restriction Xjg0 contains simple g0-modules for any X 2 Op if and only if (p; g0) lies in a closed G-orbit on P × G0. Turning to Problem B, we make use of the associated varieties (see e.g. [5, 14]) as a coarse measure of the `size' of g0-modules. We see that the 0 associated variety Vg0 (Y ) of a simple g -module Y occurring in the restriction Xjg0 is independent of Y if X is a simple g-module. The formulas of Vg0 (Y ) and its dimension (Gelfand{Kirillov dimension) are derived in Theorem 4.12. Concerning Problem C, it is notorious in the category of unitary represen- tations of real reductive groups that the multiplicities in the branching laws may be infinite when restricted to symmetric pairs, see [6]. In contrast, we prove in Theorem 4.16 that multiplicities are always finite in the branching laws with respect to symmetric pairs in the category O. Particularly interesting branching laws are multiplicity-free cases where 0 any simple g -module occurs in the restriction Xjg0 at most once. We give two general multiplicity-free theorems with respect to symmetric pairs (g; g0) in the parabolic category Op: 1) p special, (g; g0) general (Theorem 5.1), 2) p general, (g; g0) special (Theorem 5.4), and then find branching laws corresponding to closed orbits in Gn(P × G0). 3 This is the first article of our project on a systematic construction of equivariant differential operators in parabolic geometry. In subsequent pa- pers, Theorem 4.1 (a solution to Theorem A) plays a foundational role in dealing with • a construction of conformally equivariant differential operators in parabolic geometry, • a generalization of the Rankin{Cohen bracket operators. Actual calculations are carried out by using algebraic branching formulas (Theorem 5.2) together with an analytic machinery that we call the `F - method' in [9]. In Section 5.1 we have studied parabolic subalgebras p with abelian nilpo- tent radical. The case of parabolic subalgebras p with Heisenberg nilpotent radical (e.g. Example 4.14) may be thought of as a generalization of Section 5.1. Using Theorems 3.10 and 4.1, we can give a complete classification of the triples (g; p; gτ ) and the closed orbits in Gτ nG=P (see the framework of The- orem 2.1) with discrete decomposable and multiplicity-free branching laws. The calculation is more involved, and will be reported in another paper. Partial results of this article were presented at the conference in honor of Vinberg's 70th birthday at Bielefeld in Germany in 2007 and a series of lectures at the Winter School on Geometry and Physics in Cech Republic in 2010. The author is grateful to the organizers, in particular, Professors Abels and Souˇcek, for their warm hospitality. Notation: N = f0; 1; 2; · · · g, N+ = f1; 2; 3; · · · g. 2 Branching problem of Verma modules In general, Verma modules may not contain any simple g0-module when re- stricted to a reductive subalgebra g0. In this section, we use the geometry of 0 the double coset space NG(g )nG=P and clarify the problem in Theorem 2.1, which will then serve as a foundational setting of branching problems for the category Op in Theorem 4.1. 2.1 Generalized Verma modules We begin with a quick review of the (parabolic) BGG category Op and fix some notation. 4 Let g be a semisimple Lie algebra over C, and j a Cartan subalgebra. We _ write ∆ ≡ ∆(g; j) for the root system, gα (α 2 ∆) for the root space, and α for the coroot. We fix a positive system ∆+, and define a Borel subalgebra b = j + n with nilradical n := ⊕α2∆+ gα. The BGG category O is defined to be the full subcategory of g-modules whose objects are finitely generated g-modules X such that X are j-semisimple and locally n-finite [2]. Let p be a standard parabolic subalgebra, and p = l+u its Levi decomposi- + + tion with j ⊂ l. We set ∆ (l) := ∆ \∆(l; j), and define n−(l) := ⊕α2∆+(l)g−α. The parabolic BGG category Op is defined to be the full subcategory of O p whose objects X are locally n−(l)-finite. Then O is closed under submod- ules, quotients, and tensor products with finite dimensional representations. The set of λ for which λjj\[l;l] is dominant integral is denoted by Λ+(l) := fλ 2 j∗ : hλ, α_i 2 N for all α 2 ∆+(l)g: We write Fλ for the finite dimensional simple l-module with highest weight λ, inflate Fλ to a p-module via the projection p ! p=u ' l, and define the generalized Verma module by g g Mp (λ) ≡ Mp (Fλ) := U(g) ⊗U(p) Fλ: (2.1) g p p Then Mp (λ) 2 O , and any simple object in O is the quotient of some g g Mp (λ). We say Mp (λ) is of scalar type if Fλ is one-dimensional, or equiva- lently, if hλ, α_i = 0 for all α 2 ∆(l). Let ρ be half the sum of positive roots. If λ 2 Λ+(l) satisfies _ + hλ + ρ, β i 62 N+ for all β 2 ∆ n ∆(l); (2.2) g then Mp (λ) is simple, see [3].
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