Polynomial functions on 0_(2λ+1)

Abstract

This thesis is an attempt to generalise to the odd orthogonal group Γ_K, over an Infinite field K not of characteristic two, the work of Schur [S], and more recently Green [G], on the general linear group G_K using the approach of Weyl [W] in characteristic zero. The special feature here is that we treat Γ_K as merely a group of matrices defined by the vanishing of polynomials in its coefficients (the classical view) rather than a group generated by elements derived from an associated Lie algebra, the approach used initially by Chevalley and adopted by most authors in recent times.
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\nAfter generalising Green's [G] Schur algebra for G_K to Γ_K in §0 we prove in §1 Chevalley's famous theorem on the 'Big Cell' in G_K and then, by an easy extension, prove it for the Big Cell in Γ_K. Chevalley's original proof uses representations of Lie algebras, ours requires nothing but a little knowledge of the coordinate ring K_+[G] of all 'polynomial' functions on G_K . We define K[Γ], the coordinate ring of Γ_K, to be the space of all polynomial functions on G_K restricted to Γ_K and in §2 give a generating set of the kernel of the restriction map ψ_K:K_+ [G]→K[Γ]. This generalises Weyl's result in characteristic zero. In §3 we use this result to show that the family, or 'scheme', of rings K[Γ] (K varying over all infinite fields not of characteristic two) is 'defined over Z' ; in fact K[Γ] is naturally isomorphic to K θ Z[Γ_Q], where Z[Γ_Q] is the subring of Q[Γ] spanned by 'monomial' functions. This enables us to formulate a 'modular' representation theory for Γ which connects polynomial representations of Γ_Q with those of Γ_K.
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\nIn §4 we investigate the Schur algebras of Γ_Q following Weyl [W] and in §5 find a complete set of irreducibles for each of them, once again following the lead of Weyl. In §6 we attempt to 'reduce' these modules modulo p to obtain 'Weyl' modules for Γ_K, a task only partially completed.