H

Abstract

This is a detailed version of my 1992 short announcement [S3]. For prerequisites on Lie superalgebras seeAppendix 0 andAppendix1, which are mainly borrowed from Leites’s book [L3]. A draft of this paper was put on the net (math.RT/9 810113) “earlier and independently”, as Cheng and Wang referred to it in their papers [CW1; CW2], where they elucidate some of the results given here and also give an interpretation of a formula for projective symmetric functions. Still, a further elucidation will not hurt, and I intend to return to it elsewhere. Meanwhile, recall that Howe suggested a unified approach to the first and second theorems of the classical invariant theory: compare [Wy] with [H]. This approach becomes even more unified in [LSh], where Lie superalgebras that more or less implicitly linger in the background of [H] become the main characters. In this and a subsequent paper I consider analogs of these theorems for “classical” Lie superalgebras. Related are problems on description of the centers of U(g) (cf. [LS; S1; S5]). The pioneer here was Berezin [B1; B2; B3], who somewhat differently considered, to an extent, g = gl, sl, andosp. Scheunert [Sch1; Sch2; Sch3; Sch4] has reproduced some of my results. The reader should be aware of a totally different approach to invariant theory due to Shander [Sd1; Sd2], who justly observes that for Lie superalgebras it is possible not to restrict oneself to the study of polynomial functions and makes a step in this purely super direction.