The modular Hecke algebra and Steinberg representation of finite Chevalley groups (with an appendix by Peter Landrock) (Q1057361)

Let G be a finite group with split (B,N) pair of characteristic p. If \({\mathbb{F}}\) is a field, the Hecke algebra of G is \({\mathcal H}_{{\mathbb{F}}}=End_{{\mathbb{F}}[G]}(1^ G_ B)\). If char \({\mathbb{F}}=0\), there is an embedding of \({\mathcal H}_{{\mathbb{F}}}\) in \({\mathbb{F}}[G]\) and an idempotent in \({\mathbb{F}}[G]\) affording the Steinberg representation. The imbedding involves averaging over the elements of B and so does not carry over to \({\mathbb{F}}\) of characteristic p (which divides \(| B|).\) However in the characteristic p case, Steinberg gave a construction for an idempotent e in \({\mathbb{F}}[G]\), which affords the Steinberg representation. If G has rank n, there is a rank 1 subgroup \(G_ i\) corresponding to each simple root \(r_ i\), \(i=1,...,n\), and corresponding Steinberg idempotent \(e_ i\in {\mathbb{F}}[G_ i]\subset {\mathbb{F}}[G]\). The Iwahori relations of \({\mathcal H}_{{\mathbb{F}}}\) are used to show that these \(e_ i\) and 1 generate a copy of \({\mathcal H}_{{\mathbb{F}}}\) in \({\mathbb{F}}[G]\). The verification of these relations reduce to calculations in \({\mathbb{F}}[A_ 2(q)]\), \({\mathbb{F}}[B_ 2(q)]\) and \({\mathbb{F}}[G_ 2(q)].\) The author deals with the case \({\mathbb{F}}[A_ 2(q)]\). In an appendix, Landrock deals with the case \({\mathbb{F}}[B_ 2(q)]\). This already covers all finite Chevalley groups except \(G_ 2(q)\). It is implied that the calculations in this last case seem rather daunting. A plethora of idempotents is obtained explicitly in \({\mathcal H}_{{\mathbb{F}}}\), although this is still short of the \(2^ n\) such implied by the work of Curtis. The motivation and application of the results come from the work of Mitchell and Priddy in constructing stable splittings of various topological spaces.