Let p be a prime and q=p^g. We show that the Grothendieck ring of finitely generated F_{ q}[SL(2,F_{ q})]-modules is naturally isomorphic to the quotient of the polynomial algebra Z[x] by the ideal generated by f^[g](x)-x, where f(x)=sum_{ j=0}^{ floor(p/2)}(-1)^{ j}(p/(p-j))((p-j); j)x^{ p-2j}, and the superscript [g] denotes g-fold composition of polynomials. We conjecture that a similar result holds for simply connected semisimple algebraic groups defined and split over a finite field.
Categories: math.RT