Shapovalov determinant for restricted and quantized restricted enveloping algebras

Shrawan Kumar and Gail Letzter

Vol. 179 (1997), No. 1, 123–161

DOI: 10.2140/pjm.1997.179.123

Abstract

As is well known, the Shapovalov bilinear form and its determinant is an important tool in the representation theory of semisimple Lie algebras over char. 0. To our knowledge, the corresponding study of the Shapovalov bilinear form and its determinant is not available in the literature in char. p or the quantum case at roots of unity. The aim of this paper is to fully determine the Shapovalov determinant for both, the restricted enveloping algebra and its quantum analog.

More precisely, let g be a semisimple Lie algebra. Fix a prime p≠2 which also satisfies p≠3 whenever g contains a component of type G₂. This will be our tacit assumption on p through the paper. Let ξ be a primitive p^th root of unity. This paper is concerned with two algebras: a certain analog u_p of the restricted enveloping algebra (cf. Definition  3.1) and its quantized version u_ξ which is an algebra over the cyclotomic field ℚ_ξ (cf. Definition 3.3). The main results of this paper are complete descriptions of the Shapovalov determinant for both the algebras u_p and u_ξ (cf. Theorems 3.2 and 3.4).

Milestones

Received: 29 June 1995

Published: 1 May 1997