An analog of the classical invariant theory for Lie superalgebras. I

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/9810113) "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.