If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over $\mathrm{Spec}(\mathbf{Z})$? Or is life not so easy? I feel like if I actually knew the construction of the irreps of $G_{\mathbf{C}}$ (rather than treating the entire "highest weight" theory as a black box, which I've done up to this point in my life) I should be able to answer this.

For $\mathrm{SL}(2)$ the answer is going to be yes, because the only irreps are symmetric powers, which can be realised over $\mathbf{Z}$. But for groups of rank greater than 1 I realise I am a bit unclear about the relationship between irreps (of the algebraic group) in char 0 and in char $p$; for example can the dimension of the irrep with some given highest weight jump as one moves from char 0 to char $p$? Even if it can, this doesn't mean my question has a negative answer.


2 Answers 2


This is an application of the geometric method of constructing highest-weight representations via line bundles on the "variety of Borel subgroups", together with the theory of split semisimple Chevalley groups (existence of Borel $\mathbf{Z}$-subgroups containing a split maximal $\mathbf{Z}$-torus, existence of $G/B$ as a projective $\mathbf{Z}$-scheme, etc.).

More precisely, let $T$ be a split maximal $\mathbf{Z}$-torus and $B$ a Borel $\mathbf{Z}$-subgroup of $G$ containing $T$, so $B/U=T$ for the "relative unipotent radical" $U \subset B$ over $\mathbf{Z}$. Use $B$ to define a positive system of roots in $\Phi(G,T) = \Phi(G_{\mathbf{C}}, T_{\mathbf{C}})$, and let $\chi:T \rightarrow \mathbf{G}_m$ be a dominant weight for this positive system of roots. View $\chi$ as a character on $B$ via the identification $B/U=T$, and let $V$ be a rank-1 $\mathbf{Z}$-lattice on which $B$ acts through $1/\chi$. Let $L(\chi) = G \times^B V$ be the associated line bundle on the projective $\mathbf{Z}$-scheme $G/B$. The $G_{\mathbf{C}}$-representation space ${\rm{H}}^0(G/B, L(\chi))_{\mathbf{C}} = {\rm{H}}^0(G_{\mathbf{C}}/B_{\mathbf{C}}, L(\chi)_{\mathbf{C}})$ is the irreducible highest-weight representation for the weight $\chi$ (by the geometric theory in characteristic 0), and the finitely generated $\mathbf{Z}$-module ${\rm{H}}^0(G/B, L(\chi))$ admits a natural action by the $\mathbf{Z}$-flat $G$; the image of this $\mathbf{Z}$-module in the $\mathbf{C}$-version is the integral model that you seek.

  • $\begingroup$ Thanks grghxy. I should have thought to use the geometric construction. $\endgroup$
    – slider
    Jun 22, 2015 at 19:51

Let me add a couple of things to what grghxy has said.

1) The study of these groups over $\mathbb{Z}$ has been complicated, going back to Chevalley's work and Borel's lecture notes (which aren't quite in focus). The most comprehensive account is given in a more recent paper by Lusztig here. In any case, the gist of the matter is that all goes well for split semisimple groups over $\mathbb{Z}$. But at some point you have to get into the scheme language of SGA3.

2) Concerning your second paragraph, there is much more to be said (and to be done) to make the transition from characteristic 0 to prime characteristic. Much of the known theory is contained in Jantzen's treatise Representations of Algebraic Groups (first published by Academic Press in 1987, then published by AMS in 2003 with considerable added material). Briefly put, the more familiar irreducible finite dimensional representations in characteristic 0 yield "Weyl modules". These turn out to be universal highest weight modules in characteristic $p$ but are typically more complicated to study than infinite dimensional Verma modules are in characteristic 0. Affine Weyl groups of Langlands dual type (relative to the powers of $p$) enter into the picture along the lines of the Kazhdan-Lusztig conjecture in characteristic 0 using the finite Weyl group and Hecke algebra, but with many added complications and uncertainties. At any rate, it's been clear for a long time that the finite dimensional irreducibles are in general smaller in characteristic $p$ and occur as top composition factors of Weyl modules (or socles in the dual geometric construction).

  • $\begingroup$ Yes, my attempt to read Jantzen before I posted this question left me with the general feeling that the dimension could drop in char p. I initially thought that this would spike the question but when I realised it didn't I thought I'd just ask. Thanks very much for the comments! Jantzen still looks much too thick for me to read :-/ (in contrast to your book, which was much easier going!) $\endgroup$
    – slider
    Jun 22, 2015 at 20:39
  • $\begingroup$ @slider: It may (or may not) help to look at my short notes on Weyl modules posted at my homepage: people.math.umass.edu/~jeh/pub/weyl.pdf. By now the notation and terminology have shifted a lot, so the history gets a bit lost. But the notion of "tilting module" adds to the complexity. I should emphasize that higher sheaf cohomology modules attached to line bundles are still quite hard to study, though I've proposed some approaches using inverse Kazhdan-Lusztig polynomials for affine Weyl groups. $\endgroup$ Jun 22, 2015 at 22:09
  • $\begingroup$ @slider A good way to see that in fact the dimensions can be arbitrarily smaller is to use (a consequence of) Steinberg's tensor product formula, which says that the dimension of the simple module of highest weight $\lambda$ is the same as the dimension of the one with highest weight $p\lambda$ when $p$ is the characteristic of the field. Then compare this to Weyl's dimension formula and how it increases when one multiplies the weight with a constant. $\endgroup$ Jun 23, 2015 at 6:46
  • $\begingroup$ Oh! I'm just being completely stupid. Symm^2 of the standard representation is reducible in char 2. $\endgroup$
    – slider
    Jun 23, 2015 at 7:19

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