Notes on Jantzen’s book

Induced modules and simple modules

• For \lambda \in X(T) write k_\lambda for the one-dimensional B-module with character e^\lambda.
• Define H^i(\lambda) = R^i \operatorname{ind}_B^G(k_\lambda), the right derived induction functor. (We could have defined this for any B-module). Note H^0(\lambda) = \operatorname{ind}_B^G(k_\lambda).
• The induced module H^0(\lambda), if it is nonzero, has a one-dimensional highest weight space of weight \lambda, and weights lying between w_0 \lambda and \lambda.
• If nonzero, it has a simple socle, which we call L(\lambda).
• If nonzero, then L(\lambda) \cong L(- w_0 \lambda)^*.
• H^0(\lambda) is nonzero precisely for the dominant weights \lambda \in X(T)_+.
• For \lambda dominant, the multiplicity of L(\lambda) in H^0(\lambda) is 1, and they share the highest-weight space.
• Define the Weyl module V(\lambda) = H^0(-w_0 \lambda)^*. Then V(\lambda) and H^0(\lambda) have the same character, and the V(\lambda) are the universal highest weight modules, as in they map onto any highest-weight module.

Irreducible representations of Frobenius kernels

• (3.16) Steinberg inductive rule: if \lambda \in X_r(T) and \mu \in X(T)_+, then L(\lambda + p^r \mu) \cong L(\lambda) \otimes L(\mu)^{[r]}.
• (3.17) Steinberg tensor product rule: write (non-uniquely) \lambda = \sum_{i \geq 0} p^i \lambda_i for \lambda_i \in X_1(T). Then L(\lambda) = L(\lambda_0) \otimes L(\lambda_1)^{[1]} \otimes L(\lambda_2)^{[2]} \otimes \cdots

Kempf’s vanishing theorem

• For \lambda dominant, H^i(\lambda) = 0 for i > 0.

BWB and Weyl Character Formula

• (1.5) Recall \rho is half the sum of the positive roots. It’s not clear that \rho is in the weight lattice X(T), however 2 \rho is a root and is therefore in the weight lattice. Since \langle \rho, \beta^\vee \rangle = 1 for all simple coroots \beta^\vee, we have that s_\beta \rho - \rho is again in the root lattice.
• (5.1) The dot action w \bullet \lambda = w(\lambda + \rho) - \rho, by the discussion above, maps X(T) into itself.
• (5.4a) For \lambda \in X(T), if \langle \lambda, \alpha^\vee \rangle = -1 for some simple \alpha \in S, then H^\bullet(\lambda) = 0.
• (5.5) If p = 0, set \overline{C}_\mathbb{Z} = \{ \lambda \in X(T) \mid 0 \leq \langle \lambda + \rho, \beta^\vee \rangle \text{ for all } \beta \in R^+\}
• (5.5) If p > 0, set \overline{C}_\mathbb{Z} = \{ \lambda \in X(T) \mid 0 \leq \langle \lambda + \rho, \beta^\vee \rangle \leq p \text{ for all } \beta \in R^+\}
• (5.5a) If \lambda \in \overline{C}_\mathbb{Z} and \lambda \notin X(T)_+, then H^\bullet(w \bullet \lambda) = 0 for all w \in W. This follows immediately from (5.4a) and the definition of \overline{C}_\mathbb{Z}.
• (5.5b) If \lambda \in \overline{C}_\mathbb{Z} \cap X(T)_+, then the Borel-Weil-Bott identity holds: H^i(w \bullet \lambda) \cong H^0(\lambda) if i = l(w) and H^i(w \bullet \lambda) = 0 otherwise. In characteristic zero this is precisely the Borel-Weil-Bott theorem.
• (5.6) For all \lambda \in X(T)_+ \cap \overline{C}_\mathbb{Z}, we have L(\lambda) = H^0(\lambda). We also have that there are no exts between L(\lambda) and L(\mu) for \lambda, \mu in the fundamental dominant alcove \overline{C}_\mathbb{Z} \cap X(T)_+, applying this in characteristic 0 gives semisimplicity of \mathsf{Rep}_G.
• (5.7) Define the Euler characteristic \chi(\lambda) = \sum_{i \geq 0} (-1)^i \operatorname{ch} H^i(\lambda) (we could have defined this for an arbitrary B-module rather than k_\lambda).
• (5.7) By Kempf’s vanishing theorem, \chi(\lambda) is the character of H^0(\lambda) whenever \lambda is dominant.
• (5.7) The long exact sequence of derived functors gives that if 0 \to M' \to M \to M'' \to 0 is exact, then \chi(M) = \chi(M') + \chi(M'').
• (5.7) The generalised tensor identity gives that \chi(V \otimes M) = \operatorname{ch}(V) \chi(M) for any G-module V.
• (5.8) After realising that the characters of the \{L(\lambda) \mid \lambda \in X(T)_+ \} or that the characters of the \{H^0(\lambda) \mid \lambda \in X(T)_+ \} form a basis for \mathbb{Z}[X(T)]^+, the generalised tensor identity then gives that for all \lambda \in X(T) and symmetric \sum_{\mu} a_\mu e^\mu \in \mathbb{Z}[X(T)]^W, we have that \chi(\lambda) \sum_\mu a_\mu e^\mu = \sum_\mu a_\mu \chi(\lambda + \mu).
• (5.9) We have \chi(w \bullet \lambda) = \det(w) \chi(\lambda) for all \lambda \in X(T) and w \in W.
• (5.9) For \lambda \in X(T) \otimes \mathbb{Q} define the formal character A(\lambda) = \sum_{w \in W} \det(w) e^{w \lambda}. Note w A(\lambda) = \det(w) A(\lambda).
• (5.10) The Weyl character formula: for all \lambda \in X(T) we have \chi(\lambda) = A(\lambda + \rho) / A(\rho).
• (5.11) Combining (5.10) with previous results gives \operatorname{ch} V(\lambda) = \operatorname{ch} H^0(\lambda) = \chi(\lambda) = A(\lambda + \rho) / A(\rho).

We have the following rules for working with the \chi(\lambda).

1. For any \lambda, we may write \chi(\lambda) in the standard basis by applying the Weyl character formula (5.10), or by using the Demazure character formula.
2. We can always replace a \chi(\lambda) by either 0 or, \chi(\mu) where \mu is dominant. To do this, we see that (5.4a) gives that if \langle \lambda + \rho, \alpha^\vee \rangle = 0 (i.e. \lambda is fixed by some \rho-shifted simple reflection) then \chi(\lambda) = 0. Otherwise, we have by (5.9) that \chi(\lambda) = \det(w) \chi(w \bullet \lambda) and we choose w such that w \bullet \lambda is dominant.
3. We can multiply \chi by any symmetric character in the standard basis using (5.8). Note the terms on the right side will not be linearly independent, and there will be cancellation once the \chi(\lambda + \mu) have been orbited under \bullet to make them dominant.

So we already have a way of computing tensor product characters for the Weyl/induced modules, e.g. to compute \operatorname{ch} (V(\lambda) \otimes V(\mu)), do the following:

1. Expand \chi(\lambda) = \sum_{\nu} a_\nu e^\nu using the Weyl/Demazure character formula.
2. Apply (5.8) to get \chi(\lambda) \chi(\mu) = \sum_\nu a_\nu \chi(\nu + \mu).
3. Rewrite the terms in the above sum to make all of the arguments to \chi dominant. This will add signs or kill terms, in the end we will have a linear combination of \chi(-) with only dominant arguments.

6. The Linkage Principle

• (6.1) Affine reflections: for \beta \in R and r \in \mathbb{Z}, set s_{\beta, r} (\lambda) = s_\beta(\lambda) + r \beta. Alternative formula is s_{\beta, r}(\lambda) = \lambda - (\langle \lambda, \beta^\vee \rangle - r) \beta.
• (6.2) W_p acts on X(T) \otimes \mathbb{R} and defines a system of “facets”. A facet F for W_p is defined by a decomposition R^+ = R_0^+(F) \sqcup R_1^+(F) and some integers n_\alpha, as the set of all \lambda \in X(T) \otimes \mathbb{R} satisfying \langle \lambda + \rho, \alpha^\vee \rangle = n_\alpha p for all of the \alpha \in R_0^+, and (n_\alpha - 1)p < \langle \lambda + \rho, \alpha^\vee \rangle < n_\alpha p for all \alpha \in R_1^+. If the set F \subseteq X(T) \otimes \mathbb{R} determined by this is empty, it’s not a facet.
• (6.3) \Sigma = \Sigma(C) the set of all reflections s_F in the p-dialated affine Weyl group where F is a wall of C, where C is the standard alcove
• (6.3) \Sigma can be described explicitly: it consists of s_\alpha for all simple roots \alpha \in S, and all s_{\beta, p} where \beta is the highest short root of an irreducible component of R.
• The closure of a facet is its usual closure in \mathbb{R}^n, replacing both the strict inequality signs above with \leq. The upper closure is what we get when we replace only the second inequality: (n_\alpha - 1)p < \langle \lambda + \rho, \alpha^\vee \rangle \leq n_\alpha p
• A facet which is open in X(T) \otimes \mathbb{R} is called an alcove.
• C = \{ \lambda \mid 0 < \langle \lambda + \rho, \alpha^\vee \rangle < p \} \subseteq X(T) \otimes \mathbb{R} is the standard alcove.

7. The Translation Functors

Assume that p > 0. We know our category breaks into blocks, where the block \mathcal{M}_\mu consists of all modules having composition factors in the same linkage class as \mu (in the orbit W_p \bullet \mu). Usually \overline{C}_\mathbb{Z} is the choice of representatives for the linkage classes.

• There are some functors T_\lambda^\mu: \mathcal{M}_\lambda \to \mathcal{M}_\mu.
• The T_\lambda^\mu are an equivalence of categories if \lambda, \mu belong to the same facet.
• (7.12) Recall \Sigma = \Sigma(C) from (6.3).
• (7.17a) Let \lambda, \mu \in \overline{C}_\mathbb{Z} and w \in W_p such that w \bullet \lambda is dominant. Suppose also that \mu belongs to the upper closure of the facet containing w \bullet \lambda. Then for all w_1 \in W_p and i \in \mathbb{N} we have [H^i(w_1 \bullet \lambda) : L(w \bullet \lambda)] = [H^i(w_1 \bullet \mu) : L(w \bullet \mu)].
• (7.17b) With the same conditions as the previous point, if \operatorname{ch} L(w \bullet \lambda) = \sum_{w' \in W_p} a_{w, w'} \chi(w' \bullet \lambda), then \operatorname{ch} L(w \bullet \mu) = \sum_{w' \in W_p} a_{w, w'} \chi(w' \bullet \mu).
• (7.18) For \lambda \in C \cap X(T) and w \in W_p and s \in \Sigma such that w \bullet \lambda < ws \bullet \lambda, then [H^i(w_1 \bullet \lambda) : L(w \bullet \lambda)] = [H^i(w_1 s \bullet \lambda : L(w \bullet \lambda))] for all w_1 \in W_p and i \in \bbN.

Filtrations of Weyl modules

Assume that p > 0. Throughout this chapter we are working with reduction modulo p, so we have some notation for dealing with that.

• Let A be a Dedekind domain, \Pi (A) the max spectrum, K its field of fractions.
• For \mathfrak{p} \in \Pi (A), let \nu_\mathfrak{p}: A \setminus \{0\} \to \mathbb{N} be the \mathfrak{p}-adic valuation, meaning that for nonzero a \in A, we have \nu_\mathfrak{p}(a) = r if a \in \mathfrak{p}^r and a \notin \mathfrak{p}^{r + 1}.
• Define \nu_p to be the valuation \nu_p = \nu_{p\mathbb{Z}} when A = \mathbb{Z}.
• For each dominant \lambda \in X(T)_+, there exists a filtration V(\lambda) = V(\lambda)^0 \supseteq V(\lambda)^1 \supseteq \cdots of the Weyl module V(\lambda), such that V(\lambda) / V(\lambda)^1 is the simple module L(\lambda) and \sum_{i > 0} \operatorname{ch} V(\lambda)^i = \sum_{\alpha \in R^+} \sum_{0 < mp < \langle \lambda + \rho, \alpha^\vee \rangle} \nu_p(mp) \chi(s_{\alpha, mp} \bullet \lambda).
• Note that there is no alternation in the sum on the left. So if the above sum is zero, we have V(\lambda) \cong L(\lambda). If the sum comes out as the character of a simple, that implies that V(\lambda)^2 = 0 and V(\lambda)^1 is that simple, and V(\lambda) has precisely two composition factors.

Computing simple characters in rank 2

Write \ell(\lambda) = \operatorname{ch} L(\lambda) from now on. The sets \{ \ell(\lambda) \mid \lambda \in X(T)_+ \}, \{\chi(\lambda) \mid \lambda \in X(T)_+ \} give different bases for the symmetric character ring \mathbb{Z}[X(T)]^W. Furthermore

• We can compute all of the \chi(\lambda) in the standard basis e^\lambda, via the Demazure character formula or the Weyl character formula.
• The change-of-basis between the \chi and \ell is triangular, since if L(\lambda) appears as a composition factor of H^0(\mu) then w_0 \mu \leq \lambda \leq \mu.
• It seems at first glance that the Jantzen filtration gives a formula of the form \chi(\lambda) - \ell(\lambda) = \sum_{\mu} a_\mu \chi(\mu), however this is not true because we will most likely be overcounting the composition factors V(\lambda)^2, V(\lambda)^3, \ldots.

Other assorted notes

• The coxeter number is 1 + \sum_i m_i, where \beta = \sum_i a_i \alpha_i is the highest root.