IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

Abstract

Let L = H(2r; n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristic p > 2. In the generalized restricted Lie algebra setup, any irreducible representation of L corresponds uniquely to a (generalized) p-character χ. When the height of χ is no more than min{p ni − p ni−1 | i = 1, 2, . . . , 2r} − 2, the corresponding irreducible representations are proved to be induced from irreducible representations of the distinguished maximal subalgebra L 0 with the aid of an analogy of Skryabin's category C for the generalized Jacobson-Witt algebras and modulo finitely many exceptional cases. Since the exceptional simple modules have been classified, we can then give a full description of the irreducible representations with p-characters of height below this number.2010 Mathematics subject classification: primary 17B10; secondary 17B50, 17B70.

Support varieties of semisimple-character representations for Cartan type Lie algebras

“…Then we have the following direct corollary. Proof When the character χ satisfies the condition stated in the corollary, all irreducible non-exceptional U χ (L)-modules are baby Kac modules by [5][6][7]13,20] (see also [15,18,19]). Therefore, the statement is a direct consequence of Corollary 1.…”