Representations of Rank 2 Reductive Groups

Weyl characters

Fix a pinning T \subseteq B \subseteq G of a reductive group G so that we can talk about the weight lattice X(T) = \Hom(T, \bbG_m). For each weight \lambda \in X(T) there is a natural way of constructing a G-module via induction: let k be the base field and denote by k_\lambda the one-dimensional B-representation with weight \lambda, then we say that H^0(\lambda) := \ind_B^G(k_\lambda) is the induced module for weight \lambda. (Ignore the strange notation H^0 for a moment). It turns out that the induced module is nonzero precisely when \lambda is a dominant weight, and in this case we define the Weyl module V(\lambda) = H^0(- w_0 \lambda)^* and the Weyl character \chi(\lambda) = \ch H^0(\lambda) = \ch V(\lambda).

There is a way to extend the definition of \chi(\lambda) to non-dominant \lambda. Since the induction functor \ind_B^G \colon B\dashmod \to G\dashmod is left exact, we can form its right derived functors H^i(\lambda) := R^i \ind_B^G(k_\lambda). Note that H^0(\lambda) = \ind_B^G(k_\lambda) is the induced module, explaining the strange notation choice from the previous paragraph. We then may define \chi(\lambda) for any \lambda \in X(T) by taking an alternating sum across all of the derived functors: \chi(\lambda) = \sum_{i \geq 0} (-1)^i \ch H^i(\lambda). This agrees with the previous definition by Kempf’s vanishing theorem, which states that H^i(\lambda) = 0 for i > 0 and dominant \lambda.

The following visualisation shows the Weyl character of the selected weight. The blue dots show positive contributions, while the orange dots show negative contributions. The larger the dot, the larger the magnitude of the contribution. You can pan by clicking and dragging, and zoom by holding shift and scrolling. You can click once to freeze a weight in place, and double-click anywhere else to unfreeze.

The computation of each Weyl character is done using the Demazure character formula (which works without modification for non-dominant weights). For each simple root \alpha \in S, the isobaric Demazure operator \pi_\alpha is a linear endomorphism on the space \bbZ[X(T)] of characters, defined as the following geometric series: \pi_\alpha(e^\lambda) = \frac{e^\lambda - e^{s_\alpha \lambda - \alpha}}{1 - e^{-\alpha}}. There are essentially three cases in the expansion, depending on whether \innprod{\lambda, \alpha^\vee} is greater than, less than, or equal to -1. You can look at the effect of the \pi_\alpha operator by switching the visualisation above to show Demazure characters. It might be handy to also turn on the “Show Weyl orbit” option…

Orbits under the (Affine) Weyl group

The Weyl group W associated to a root system \Phi consists of all reflections over the hyperplanes H_\alpha = \ker \innprod{-, \alpha^\vee} where \alpha is a root. Another group which often arises is the p-dialated affine Weyl group, consisting of all reflections over the hyperplanes H_{\alpha, p} = \ker(\innprod{-, \alpha^\vee} - p). There are two different actions of the (affine) Weyl group of interest to us: the usual action just stated, and the \rho-shifted action w \bullet \lambda = w(\lambda + \rho) - \rho, where \rho is half the sum of the positive roots. Note that the \bullet action depends on a choice of simple system.

In the visualisation, the groups acting are all of the form W_p = p \bbZ \Phi \rtimes W, where \bbZ \Phi is the root lattice and W is the Weyl group of the root system. Note that W_0 is the usual Weyl group, and W_1 is the affine Weyl group for the dual root system (using the Bourbaki definition of the affine Weyl group).

The thin grey lines mark the hyperplanes defined by those weights \mu such that \innprod{\mu, \alpha^\vee} \in \mathbb{Z} for all coroots \alpha^\vee. The thick black lines mark the weights where \innprod{\mu, \alpha^\vee} = 0, and the blue lines mark the weights where \innprod{\mu, \alpha^\vee} \in p \bbZ. When the \rho-shifted action is selected, the blue lines instead mark the weights \mu where \innprod{\mu + \rho, \alpha^\vee} \in p \bbZ. Therefore when the shifted action is selected, the small blue chambers are the facets of the W_p \bullet action.

Move your mouse over the diagram to select a weight \lambda (shown in green). The red points will be the orbit of the green point under the group W_p. Clicking a weight will freeze the selection, and double-clicking anywhere else will un-freeze the selection.

The Jantzen Filtration

The Jantzen filtration is a certain filtration of a Weyl module V(\lambda) into G-modules. There is a sum which overcounts the character of the first proper submodule occuring in this filtration, and by “getting lucky” with this sum (for example, when the sum just happens to be multiplicity-free), we can compute some simple representations in positive characteristic.

Fix a dominant weight \lambda \in X(T)_+. Then there exists a filtration V(\lambda) = V(\lambda)^0 \supseteq V(\lambda)^1 \supseteq V(\lambda)^2 \supseteq \cdots of the Weyl module V(\lambda), such that V(\lambda)^0 / V(\lambda)^1 \cong L(\lambda), and furthermore such that the sum of the characters of the filtered pieces is \sum_{i > 0} \ch V(\lambda)^i = \sum_{\alpha \in R^+} \sum_{0 < mp < \innprod{\lambda + \rho, \alpha^\vee}} \nu_p(mp) \chi(s_{\alpha, mp} \bullet \lambda). The notation here is:

1. The characteristic of the underlying field is p > 0.
2. L(\lambda) is the simple module with highest weight \lambda.
3. \chi(\lambda) is the Weyl character for the weight \lambda. Even though the \lambda appearing in the sum are not dominant, we can use \chi(w \bullet \lambda) = \det(w) \chi(\lambda) to restate the sum in terms of dominant \lambda – see the reflect to dominant option in the visualisation.
4. \nu_p(a) is the largest n such that p^n \mid a.
5. s_{\alpha, mp} is the affine reflection in the hyperplane \innprod{-, \alpha^\vee} = mp, or alternatively \lambda \mapsto s_\alpha(\lambda) + mp \alpha. (Note that this appears in the formula with the \bullet action, not as just written).

The visualisation below shows the various terms \chi(\mu) appearing in the sum \sum_{i > 0} \ch V(\lambda)^i. You can pan by clicking and dragging, and zoom by holding shift and scrolling. You can click once to freeze a weight in place, and double-click anywhere else to unfreeze.

Since V(\lambda) / V(\lambda)^1 \cong L(\lambda) and \ch V(\lambda) is known, a formula for the character of the first proper submodule V^1(\lambda) would give a formula for the character of L(\lambda). Unfortunately, the sum above overcounts: \begin{aligned} \ch V(\lambda)^1 &= \ch V^1 / V^2 + \ch V^2 / V^3 + \ch V^3 / V^4 + \cdots \\ \sum_{i > 0} \ch V(\lambda)^i &= \ch V^1 / V^2 + 2 \ch V^2 / V^3 + 3 \ch V^3 / V^4 + \cdots \end{aligned} However, if (for example) the sum \sum_{i > 0} \ch V(\lambda)^i happened to be multiplicity-free when decomposed into sums characters of simple G-modules, then we would know that everything is concentrated in the first term \ch V^1 / V^2, and hence multiplicity-free sums give the actual character of V^1(\lambda). This works to determine all simple characters in types A_2 and B_2, and most in type G_2, as is shown in the next section. (See the section on sticking points for some examples of where I can’t yet pull enough information out of the Jantzen filtration to determine characters.

Characters of simple modules

Let T \subseteq B \subseteq G be a pinned reductive group defined over a field k of characteristic p > 0, and \lambda \in X(T)_+ a dominant weight. A major open problem is to determine a character formula for the simple module L(\lambda). For the groups of type A_2 and B_2, all simple characters can be computed using standard facts and the Jantzen filtration, and the same tactic almost works for G_2. The following visualisation shows various ways of looking at the simple characters for these groups.

Selecting simples in terms of Weyls will show the coefficients of the expansion L(\lambda) = \sum_{\mu} c_\mu \chi(\mu), which is the most performant as there is the least amount of work to do computationally. Selecting Weyls in terms of simples will show the coefficients the other way around, of the expansion \chi(\lambda) = \sum_{\mu} b_\mu \ch L(\mu). This has some nasty behaviour as the weights get large (inverting the upper triangular matrix requires incredibly large numbers), and so this mode peters out after a while. Finally, selecting simples in the standard basis will show the honest character L(\lambda) = \sum_{\mu} a_\mu e^\mu inside the group algebra \bbZ[X(T)]. The performance of this mode is dependent on the number of terms in the “simples in terms of Weyls” expansion.

You can pan by clicking and dragging, and zoom by holding shift and scrolling. You can click once to freeze a weight in place, and double-click anywhere else to unfreeze. If you select an incredibly large weight while in a “slow” mode, you might have to close your browser tab and re-open it. Sorry!

I’ve checked my program’s output on p-restricted weights for A_2, B_2, and G_2 against tables published by Frank Luebeck, available here. Any weight which is not p-restricted computed via the Steinberg tensor product theorem, and hopefully I have not made an error while programming that part. Any suggestions for how to check my results there would be very welcome.

Future work

This is a todo list for myself in rough order of priority, for when I have time.

1. Add a more in-depth section on how the code works (also as a reference for myself). This would start of well paired with the positive and simple systems section, since we can explicitly examine a whole bunch of stuff to do with embeddings, Cartan matrices, Dynkin diagrams, etc.
2. Clean up the visualisation code and the algebra code so that it’s ready to be shown to people.
3. Add better support for groups like GL_2 whose rank is not equal to its semisimple rank. At the moment the code responsible for the Steinberg tensor product gives up on GL_2, because it doesn’t know how decompose a weight into a p-restricted one.
4. The simple “multiplicity-free” tactic with the Jantzen filtration gives all the simple characters for A_1, A_2, A_3, and B_2, however it is not enough to get all the simple characters of G_2. Is there some other tactic that leverages the Jantzen filtration for all weights of G_2?
5. Are tilting characters easy to implement?

Sticking points

• Computing the decomposition of \chi(1, 4) in G_2 with p = 5. The Jantzen sum evaluates as L(0, 5) + 2 L(1, 3) + L(2, 0) and hence the multiplicity-free tactic does not apply. However, the formal character \chi(1, 4) - (\ch L(0, 5) + 2 L(1, 3) + L(2, 0)) has some negative terms, and so we know that actually the simple L(1, 3) should only occur once, rather than twice, as a composition factor of the Weyl module. This strategy is now programmed in, and gets a few more G2 weights than previously.
• Computing the decomposition of \chi(1, 1) in G_2 with p = 3. The Jantzen sum is \chi(0, 0) + \chi(0, 1) + \chi(1, 0) = L(0, 0) + 2 L(0, 1) + L(1, 0), but unfortunately the previous strategy can’t proceed further here..
• Similarly, Computing the decomposition of \chi(1, 0, 1) in A_3 with p = 2. In this case, the Jantzen sum gives 2 \chi(0, 0, 0), and so we have two possibilities: the Weyl module V(1, 0, 1) has either one or two trivial composition factors (the answer should be 1, since the simple L(1, 0, 1) has a two-dimensional zero weight space according to Luebeck’s tables).