Mathematische Zeitschrift

, Volume 267, Issue 1, pp 185–219

Support varieties, AR-components, and good filtrations


    • Department of MathematicsUniversity of Kiel
  • Gerhard Röhrle
    • Department of MathematicsUniversity of Bochum

DOI: 10.1007/s00209-009-0616-6

Cite this article as:
Farnsteiner, R. & Röhrle, G. Math. Z. (2011) 267: 185. doi:10.1007/s00209-009-0616-6


Let G be a reductive group, defined over the Galois field \({\mathbb{F}_p}\) with p being good for G. Using support varieties and covering techniques based on G r T-modules, we determine the position of simple modules and baby Verma modules within the stable Auslander–Reiten quiver Γ s (G r ) of the rth Frobenius kernel of G. In particular, we show that the almost split sequences terminating in these modules usually have an indecomposable middle term. Concerning support varieties, we introduce a reduction technique leading to isomorphisms
$$\mathcal{V}_{G_r}(Z_r(\lambda)) \cong \mathcal{V}_{G_{r-d}}(Z_{r-d}(\mu))$$
for baby Verma modules of certain highest weights \({\lambda, \mu \in X(T)}\), which are related by the notion of depth.

Mathematics Subject Classification (2000)

Primary 16G70 Secondary 17B50

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© Springer-Verlag 2009