The annihilators of irreducible Harish-Chandra modules for \(SU(p,q)\) and other type \(A_{n-1}\) groups. (English) Zbl 0786.22023

Garfinkle, Devra

Am. J. Math. 115, No. 2, 305-369 (1993).

This paper shows how to compute annihilators of irreducible Harish- Chandra modules for \(G:=SU(p,q)\). Such modules \(M\) are parametrized by D. Vogan’s \(\mathbb{Z}/2\mathbb{Z}\) character data [Representations of real reductive Lie groups (Birkhäuser, Boston 1981; Zbl 0469.22012)], of which the author gives a nice combinatorial reformulation for \(G\). A typical annihilator \(\text{Ann} M\) is then a primitive ideal in the enveloping algebra of \({\mathfrak {sl}}(p+q,\mathbb{C})\) and as such corresponds to a standard Young \((p+q)\)-tableau [A. Joseph, Comput. Math. 40, 69-78 (1980; Zbl 0424.17005)]. Thus the main task of the paper is to show how to attach a standard Young tableau to a \(\mathbb{Z}/2\mathbb{Z}\) character datum \(D\) for \(G\).
Actually, the author attaches not one but two Young tableaux to \(D\); the second one is signed rather than standard. The first one is built up by an inductive procedure similar to the Robinson-Schensted algorithm (followed by a twist), but now the inductive step uses the second tableau. It is conjectured that this second tableau parametrizes the wavefront set of the module \(M\) corresponding to \(D\). Using D. Vogan’s character-multiplicity duality [Duke Math. J. 49, 943-1073 (1982; Zbl 0536.22022)], the author extends her recipes to all real Lie groups of type \(A\).
The author proves her recipes via a careful study of wall crossing on both character data and primitive ideals. The main point is that once the generalized \(\tau\)-invariant of a typical annihilator is known, so is the annihilator itself, thanks to an old result of Vogan.

Reviewer: W.M.McGovern (Seattle)

MSC:

22E47	Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17B35	Universal enveloping (super)algebras

Keywords:

signed tableau; primitive ideal; standard Young tableau; character- multiplicity duality

Citations:

Zbl 0469.22012; Zbl 0424.17005; Zbl 0536.22022