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Dimensions of the simple restricted modules for the restricted contact Lie algebra

By Randall R. Holmes

Abstract

It is shown that, with a few exceptions, the simple restricted modules for a restricted contact Lie algebra are induced from those for the homogeneous component of degree zero. In [3], Shen constructed the simple restricted modules for the restricted Witt, special and hamiltonian Lie algebras which comprise three of the four classes of restricted Lie algebras of Cartan type. His methods, however, do not apply to the algebras in the fourth class, namely, the contact algebras. Here, this remaining case is considered and it is shown that, with a few exceptions, the simple restricted modules are induced from simple restricted modules (extended trivially to positive components) for the homogeneous component of degree zero in the usual grading of the algebra. As this component is isomorphic to the direct sum of a symplectic algebra and the trivial algebra, the problem of determining, say, the dimensions of the simple restricted modules is then reduced to the classical situation for which Lusztig has a conjecture (see [3], p. 294). The author would like to thank Dan Nakano for introducing him to Lie algebras of Cartan type and for the infectious enthusiasm with which he discusses their properties. 1 Notation and Statement of Main Theorem The notation will be, for the most part, the same as that in [4] and this reference can also be consulted for the precise definition and fundamental properties of the contact Lie algebras. Let F be an algebraically closed field of characteristic p> 2 and let n = 2r + 1 with r ∈ N. For 1 ≤ k ≤ n let εk be the n-tuple with jth component δjk (Kronecker delta). Set A = {a = ∑ k akεk | 0 ≤ ak < p} ⊂ Z n. 1 The underlying vector space of the restricted contact Lie algebra K (denoted K(2r + 1, 1) in [4]) has as basis {x(a) | a ∈ A} if n + 3 ≡ 0 (mod p) and {x(a) | a ∈ A, a = ∑ (p − 1)εk} otherwise. (The x (a) are standard basis vectors for a divided power algebra.

Year: 1994
OAI identifier: oai:CiteSeerX.psu:10.1.1.352.4611
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