Modular representation theory of algebraic groups

introduction

\( \newcommand{\la}{\lambda} \newcommand{\ot}{\otimes} \DeclareMathOperator{\car}{char} \DeclareMathOperator{\rank}{rank} \newcommand{\Gt}{\widetilde{G}} \newcommand{\at}{\widetilde{\alpha}} \DeclareMathOperator{\Lie}{Lie} \DeclareMathOperator{\SO}{SO} \) Split semisimple linear algebraic groups over arbitrary fields can be viewed as a generalization of semisimple Lie algebras over the complex numbers, or even compact real Lie groups. As with Lie algebras, such algebraic groups are classified up to isogeny by their root system. Moreover, the set of irreducible representations of such a group is in bijection with the cone of dominant weights for the root system and the representation ring --- i.e., \(K_0\) of the category of finite-dimensional representations --- is a polynomial ring with generators corresponding to a basis of the cone.

• One way in which this analogy breaks down is that, for an algebraic group \(G\) over a field \(k\) of prime characteristic, in addition to the irreducible representation \(L(\la)\) corresponding to a dominant weight \(\la\), there are three other representations naturally associated with \(\lambda\), namely
  • the standard module \(H^0(\la)\),
  • the Weyl module \(V(\la)\),
  • and the tilting module \(T(\la)\).
• The definitions of these three modules make sense also when \(\car k = 0\), and in that case all four modules are isomorphic.
• The definition of \(H^0(\la)\) is particularly simple: view \(k\) as a one-dimensional representation of a Borel subgroup \(B\) of \(G\) where \(B\) acts via the character \(\la\), then define \(H^0(\la):=\text{ind}_{B}^{G}\lambda\) to be the induced \(G\)-module.
• The Weyl module \(V(\la)\) is the dual of \(H^0(-w_0\la)\) for \(w_0\) the longest element of the Weyl group and has head \(L(\la)\). Typical examples of Weyl modules are \(\Lie(G)\) for \(G\) semisimple simply connected (\(V(\la)\) for \(\la\) the highest root) and the natural module of \(\SO_n\).
• It turns out that if any two of the four representations \(L(\la)\), \(H^0(\la)\), \(V(\la)\), \(T(\la)\) are isomorphic over a given field \(k\), then all four are; our focus is on for which \(\la\) and which \(k\) this occurs. Consequently, it suffices to consider whether the Weyl module \(V(\la)\) --- which for any \(k\) is obtained by base change from \(\ZZ\) --- equals the irreducible module \(L(\la)\).

articles

• Skip Garibaldi, Robert M. Guralnick, Daniel K. Nakano, Irreducibility of Weyl Modules over Fields of Arbitrary Characteristic, arXiv:1604.08911 [math.RT], April 29 2016, http://arxiv.org/abs/1604.08911
• Lusztig, George, and Geordie Williamson. ‘On the Character of Certain Tilting Modules’. arXiv:1502.04904 [math], 17 February 2015. http://arxiv.org/abs/1502.04904.