RATIONAL REPRESENTATIONS OF GL₂

Abstract

Abstract. Let F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra C ← − p (F), with a transparent combinatorial structure, which controls the rational representation theory of GL 2 (F).

Homotopy, homology, and GL2

“…We have now seen that the algebras A n , B n and D n are all isomorphic. As a corollary, we obtain the following result (conjectured in [19]). Lemma 6.…”

“…Proof. As we noted in a previous article, every block of representations of GL 2 is equivalent to the category of lim n B n -modules [19]. By Theorem 1 and Corollary 21, we have B n ∼ = E n .…”

“…The projective indecomposable B n -module P Bn (a) is a quotient of the projective indecomposable P An (a), by Lemmas 3 and 4. We now recall some properties of B n , from a previous article of ours [19], and from papers of Erdmann and Henke [12], [10].…”

“…. , a n−1 , c ± 1) in P Bn (a) ( [19], Proposition 29), and isomorphic to the indecomposable tilting module T B c±1 n−1 (ã) indexed by vertexã = (p−1−a i ) (i=0,...,n−1) ([10], Proposition 25).…”

“…By Proposition 5, B n is graded in the same way. We established previously that the graded ring grB n of B n , taken with respect to a certain filtration, is isomorphic to C n ( [19], Corollary 33). The Z n + -grading here is compatible with this filtration, which means to say that B n ∼ = grB n .…”