On restricted simple modules of contact Lie superalgebras of odd type

Abstract

For contact Lie superalgebras of odd type over an algebraically closed field of characteristic > 3, the composition factors of Kac modules and the character formulas for all the restricted simple modules are explicitly determined. And the dimensions of simple modules in the restricted supermodule category are precisely calculated, thereby dimensions of all the restricted simple modules are given. K E Y W O R D SRestricted Lie superalgebras, restricted simple modules, composition factors, character formulas M S C ( 2 0 1 0 ) 17B10, 17B35, 17B50 INTRODUCTIONFinite dimensional simple Lie algebras over an algebraically closed field of characteristic > 5 fall into two types of infinite series: classical type and Cartan type, which was formulated by Kostrikin and Shafarevich in the 1960s and was proved by Block and Wilson in the 1980s. This is also true for the case of finite dimensional complex simple Lie superalgebras by Kac's classification theorem. Until now, the classification problem for finite dimensional simple Lie superalgebras over an algebraically closed field of prime characteristic remains open. However, eight families of -graded Lie superalgebras of Cartan-type, the Witt type, the special type, the Hamiltonian type, the contact type, the Hamiltonian Lie superalgebras of odd type, the contact Lie superalgebras of odd type, the special Hamiltonian Lie superalgebras of odd type and the special contact Lie superalgebras of odd type, were constructed over a field of characteristic > 3 (cf. [1,2,5,6,18]). These Lie superalgebras are subalgebras of the full superderivation algebras of the associative superalgebras-tensor products of the divided power algebras and the exterior superalgebras. Restricted representations have been playing an key role in the theory of modular Lie superalgebras. The first four types are analogous to the corresponding four series of finite-dimensional graded simple modular Lie algebras of Cartan type. Shu, Yao and Zhang studied restricted representations of the these four types in [9][10][11][14][15][16]. Note that the latter four series of Lie superalgebras have no analogues in Lie algebras and each of them possesses a more complicated structure. Liu, Wang and Yuan studied restricted representations of the first two types in the latter four series in [13,17]. This paper is sequel to an early paper [13], in which a sufficient and necessary condition is provided in terms of typical or atypical weights for a Kac module of contact Lie superalgebras of odd type over an algebraically closed field of characteristic > 3 to be simple. Kac modules, which themselves encapsulate rich information on structure of the restricted representations, have been playing extremely active roles in the restricted representation theory of modular Lie superalgebras (for example, see [12]). For characteristic zero case, V. Serganova studied the finite-dimensional simple modules over Lie superalgebras of Cartan type in [8]. Let be a contact Lie superalgebra of odd type over an algebraically closed field of charac...