Irreducible Representations of the Generalized Jacobson-Witt Algebras

Abstract

Let L be the generalized Jacobson-Witt algebra W(m;n) over an algebraically closed field F of characteristic p > 3, which consists of special derivations on the divided power algebra R= 𝔄(m;n). Then L is a so-called generalized restricted Lie algebra. In such a setting, we can reformulate the description of simple modules of L with the generalized p-character χ when ht (χ) < min {pni-pni-1| 1 ≤ i ≤ m} for n=(n1,…,nm), which was obtained by Skryabin. This is done by introducing a modified induced module structure and thereby endowing it with a so-called (R,L)-module structure in the generalized χ-reduced module category, which enables us to apply Skryabin's argument to our case. Simple exceptional-weight modules are precisely constructed via a complex of modified induced modules, and their dimensions are also obtained. The results for type W are extended to the ones for types S and H.

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

“…Skryabin's C-module category has been extended to the case of special Lie algebras of Cartan type by the authors (see [24]). This paper is a continuation of our previous work (see [17,24]). Recall that Skryabin first introduced the category C for the generalized Jacobson-Witt algebra W(m; n) in [18].…”

“…. , m. In the generalized restricted Lie algebra setup, the 'modified' induced modules for W(m; n) (induced from 'twist' modules of the distinguished maximal subalgebra W(m; n) 0 ) turn out to be objects of the category C (see [17]). The category C is described based on the understanding that Cartan type Lie algebras are Lie algebras of differential operators on the divided power algebras A(m; n).…”

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.…”

On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures

“…Roughly speaking, Lie(G) is actually the maximal distinguished filtered subalgebra L 0 . To a large extent, the study of irreducible representations of L can be reduced to those of L 0 (see [28], [31], [32], etc.). Such a G of the form G ⋉U is called semi-reductive in [25].…”