A Howe-type correspondence for the dual pair (sl2, sln) in sl2n

In this article, we study the decomposition of weight–sl2n–modules of degree 1 to a dual pair (sl2, sln). We show that in some generic cases we have an explicit branching rule leading to a Howe–type correspondence between simple highest weight modules. We also give a Howe–type correspondence in the non–generic case. This latter involves some (non simple) Verma modules. Let g denote a reductive Lie algebra over C. A dual pair in g is a pair (a, b) of reductive subalgebras of g which are the commutant of each other. Given a simple g–module M , one can try to solve the following branching problem: describe the restriction of M to the subalgebra a+ b. This problem and his group analogue have received particular attention since the late 80s. The first result concerning such a restriction was obtained by R. Howe in [2] and [3]. These articles were concerned with the Lie algebra sp2n (in fact the metaplectic group whose Lie algebra is the symplectic Lie algebra) and the so–called minimal (or Weil, or Shale–Segal–Weil, or oscillator) representation. From the infinitesimal point of view, the vector space of the representation is a polynomial algebra and the action is via differential operators. The restriction of this representation to the dual pair gives rise to a one–one correspondence between some simple representations of a and some simple representations of b. The correspondence from the point of view of Lie group, and for the Weil representation, is usually called θ–correspondence. In the case of Lie algebras we call such a correspondence a Howe–type correspondence or a dual pair correspondence. Other occurences of such a correspondence can be found in [6], [4], [5]. All these articles deal with the minimal representation of some Lie algebra or Lie group. The aim of this article is to prove a Howe–type correspondence for a new familly of representations of the Lie algebra sl2n, which was introduced by Benkart, Britten, and Lemire in [1]. The vector space of the representation is some kind of polynomial algebra and the action is via differential operators. Email address: [email protected] (Guillaume Tomasini) Preprint submitted to Journal of Algebra February 22, 2010 ha l-0 04 36 23 8, v er si on 2 22 F eb 2 01 0 The correspondence is completely explicit (see theorems 3.3, 3.7 and 3.8). In the first part of this article we give the construction of the representation and some of its properties. The second part is devoted to the description of the dual pair (sl2, sln) of sl2n and its action on the representation of the first part. In the last part we prove the Howe–type correspondence for this module with respect to our dual pair. Acknowledgements.– I thank gratefully Professor H. Rubenthaler for many helpful conversations and valuable comments concerning the writing of this article. 1. Simple weight–modules of degree 1 Let m be a positive integer greater than 1. Let g denote the complex Lie algebra slm of traceless m × m matrices. Let h denote its standard Cartan subalgebra, consisting of traceless diagonal matrices. In [1], Benkart, Britten and Lemire described all the simple infinite dimensional weight g–modules of degree 1. Recall that a weight module is a module for which the action of h is semisimple with finite multiplicities. A weight module is of degree 1 if all its non–trivial weight spaces are 1–dimensional. The definition of the representations which we are interested in uses the Weyl algebra Wm which is the associative algebra with generators qi and pi for i ∈ {1, . . . ,m} subject to the relations [qi, qj ] = 0 = [pi, pj ] and [pj , qi] = δi,j . Let a ∈ C. Set Pa = {b ∈ C m such that bi−ai ∈ Z for all i ∈ {1, . . . ,m} and bi < 0 ⇐⇒ ai < 0}. We consider the following vector space