数理解析研究所講究録
第 1795 巻 2012 年 83-87

83

Weyl modules and principal series modules
奈良工業高等専門学校

吉井豊 (Yutaka Yoshii)

Nara National College of Technology1Introduction

be a connected, simply connected and semisimple algebraic groupover an algebraically closed field of characteristic $p>0$ , which is definedand split over , and set $q=p^{n}$ . We fix a maximal split torus and a Borel
subgroup $B$ . We shall use the following standard notation:

Let$G$$k$$T$$F_{p}$$X=X(T)$ : character group of $T$ ;: root system relative to the pair $(G, T)$ ;;: coroot corresponding to$W=N_{G}(T)/T$ : Weyl group;: W-invariant inner product on;: the longest element of $W$ ;: Borel subgroup opposite to $B$ ;: set of positive roots corresponding to;: set of simple roots;: Coxeter number;$\Phi=\Phi(G, T)$$\alpha\in\Phi$$\alpha^{\vee}$$\{\cdot,$$E=X\otimes \mathbb{R}$$\cdot\}$$w_{0}$$B^{+}$$B^{+}$$\Phi^{+}$$\Delta$$h$$\rho=(\sum_{\alpha\in\Phi+}\alpha)/2$;$X^{+}=\{\lambda\in X|\{\lambda, \alpha^{\vee}\}\geq 0, \forall\alpha\in\triangle\}$: set of dominant weights;: set of q-restricted weights;$X_{n}=\{\lambda\in X^{+}|\langle\lambda, \alpha^{\vee}\rangle<q, \forall\alpha\in\Delta\}$: fundamental weight for;
$F:Garrow G$ : Frobenius map relative to$\alpha\in\triangle$$\omega_{\alpha}$$F_{p}$.

The simple (rational) G-modules are parametrized by the elements of, and they are denoted by, let. For
befor$V(\lambda)=$the one-dimensional T-(B-or)module with weight and we set
and call it the Weyl module with highest weight . The Weylmodule
is generated by an element of weight , which is unique up toscalar multiple and called the highest weight vector.
$X^{+}$$L(\lambda)$$\lambda\in X^{+}$$\lambda\in X^{+}$$B^{+_{-}}$$\lambda$$(Ind_{B}^{G}k_{-w_{0}\lambda})^{*}$$V(\lambda)$$\lambda$$\lambda$$k_{\lambda}$

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Let $G(n)=G^{F^{n}}$ be the finite Chevalley group corresponding to , and, the simple G-module is also simple as aset $B^{+}(n)=B^{+F^{n}}$ For$kG(n)$ -module and any simple $kG(n)$ -module can be obtained in this way.
and call it a principal series module.
, we setFor
Pillen has given a kind of relation between Weyl modules and principal$G$$\lambda\in X_{n}$$\lambda\in X_{n}$$M_{n}(\lambda)=Ind_{B^{+}(n)}^{G(n)}k_{\lambda}$series modules:

Theorem 1.1 ([3, Theorem 1.2]) Suppose that $q>2h-1$ . Let
and let be the highest weight vector of $V(\lambda+(q-1)\rho)$ . Then the $kG(n)-$submodule genemted by is isomorphic toif and only if.
for any$\lambda\in X_{n}$

$v$

$v$

$M_{n}(\lambda)$$\langle\lambda,$$\alpha^{\vee}\}>0$$\alpha\in\triangle$$\langle\lambda,$

In this article, we report that this theorem can be extended to the case.
for some$\alpha^{\vee}\}=0$2$\alpha\in\Delta$

Main result

Without loss of generality, we assume that is simple for the rest of thisarticle.
We shall introduce some further notation to describe the main result. For.
and set, let
be the complement of ina subset, set
For$G$$I\subseteq\Delta$$I$$I^{c}$$\triangle$$\rho_{I}=\sum_{\alpha\in I}\omega_{\alpha}$$\lambda\in X_{n}$$I_{0}(\lambda)=\{\alpha\in\Delta I \langle\lambda, \alpha^{\vee}\}=0\}$

and$I_{q-1}(\lambda)=\{\alpha\in\Delta|\{\lambda, \alpha^{\vee}\rangle=q-1\}$

It is known that

$M_{n}(\lambda)$.

can be decomposed as$M_{n}( \lambda)\cong\bigoplus_{J\subseteq I_{0}(\lambda)}$$\bigoplus_{J’\subseteq I_{q-1}(\lambda)}Y(\lambda+(q-1)\rho_{J}-(q-1)\rho_{J’})$,

where each $Y(\mu)$ has a simple $G(n)$ -head which is isomorphic to $L(\mu)$ (see[2, 4.6 (1)] and [4, \S 3] .
Now we can state the main result:$)$Theorem 2.1 ([5, Theorem 2.1]) Suppose that $q>h+1$ . Let
and let be the highest weight vector of $V(\lambda+(q-1)\rho)$ . Then the $kG(n)-$submodule genemted by is isomorphic to$\lambda\in X_{n}$

$v$

$v$

$\bigoplus_{J\subseteq I_{q-1}(\lambda)}Y(\lambda+(q-1)\rho_{I_{0}(\lambda)}-(q-1)\rho_{J})$.

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Remarks (1) Actually Pillen’s original proof in [3] contains an error and theassumption $q>2h-1$ is not appropriate. However, after modifying it, thisgeneralized theorem holds under the weaker assumption $q>h+1$ .
(2) Ifis empty, then the resulting direct sum is isomorphic toand so this theorem is certainly a generalization of Theorem 1.1.
$I_{0}(\lambda)$$M_{n}(\lambda)$

Example Consider the case $G=SL_{5}(k)$ and $q=7$ . Set
whereare in the standard numbering of type
as in [1, 11.4]. Let be
the fundamental weight corresponding to . Then any dominant weight is ofthe form
with, which is usually abbreviated.
Now we take $\lambda=(0,0,2,6)$ . Thenand.
The principal series module $M_{n}(0,0,2,6)$ is decomposed as$\triangle=\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\}$$A_{4}$$\alpha_{i}$$\omega_{i}$$\alpha_{i}$$\sum_{i=1}^{4}c_{i}\omega_{i}$$(c_{1}, c_{2}, c_{3}, c_{4})$$c_{i}\in Z_{\geq 0}$$I_{0}(\lambda)=\{\alpha_{1}, \alpha_{2}\}$$I_{q-1}(\lambda)=\{\alpha_{4}\}$$Y(0,0,2,0)\oplus Y(0,0,2,6)\oplus Y(0,6,2,0)\oplus Y(0,6,2,6)$$\oplus Y(6,0,2,0)\oplus Y(6,0,2,6)\oplus Y(6,6,2,0)\oplus Y(6,6,2,6)$(the entries of whose values are or $q-1$ ‘split’ into and $q-1$ ), and thehighest weight vector of the Weyl module $V(\lambda+(q-1)\rho)=V(6,6,8,12)$
generates a kG(n)-submodule which is isomorphic to$0$$\lambda$

$0$

$v$

$Y(0,0,2,6)\oplus Y(0,6,2,6)\oplus Y(6,0,2,6)\oplus Y(6,6,2,6)$(the entries of3$\lambda$

whose values are$0$‘split’ into$0$

and $q-1$ ).

Strategy of the proof

The method of proof of the main theorem is essentially similar to Pillen
original proof. But we need to use the following two generalized lemmas
instead of Lemmas 1.5 and 1.6 in [3]:$s$

Lemma 3.1 [5, Lemma 2.4] Let. Suppose that
and$\mu\in
X_{n}$$\mu\geq(q-1)\rho+w_{0}\rho_{I}$$L(\mu)$that. Then the multiplicity ofin thesatisfies
composition factors of the $kG(n)$ -module
is one if $\mu=(q-1)\rho+w_{0}\rho_{I}$and zero otherwise.
$I\subseteq\Delta$$q>\langle\rho_{I},$$\alpha_{0}^{\vee}\rangle+2$$M_{n}(\rho_{I})$

Lemma 3.2 [5, Lemma 2.5] Let, and suppose that
Then the kG(n)-submodule genemted by the highest weight vector of$(q-1)\rho)$ is isomorphic to $Y(\rho_{I}+(q-1)\rho_{I^{c}})$ .
$I\subseteq\Delta$$q>\{\rho_{I}, \alpha_{0}^{\vee}\}+2$.

$V(\rho_{I}+$86Lemma 3.1 is used to prove Lemma 3.2. Now we outline the proof of Theorem
1.1.
andandbe the generators of the kG(n)-modulesLet
respectively, and let and be the highest weight vectors of the$m_{0}$$M_{n}(\lambda)$$m_{1}$$M_{n}(\rho_{I_{0}(\lambda)^{c}})$$v_{2}$$v_{1}$

respectively. To beginand
Weyl modules
with, consider the composite map of two $kG(n)$ -module homomorphisms:$V((q-1)\rho+\rho_{I_{0}(\lambda)^{c}})$$V(\lambda-\rho_{l_{0}(\lambda)^{c}})$,$f\otimes id:M_{n}(\rho_{I_{0}(\lambda)^{c}})\otimes V(\lambda-\rho_{I_{0}(\lambda)^{c}})arrow V((q-1)\rho+\rho_{I_{0}(\lambda)^{c}})\otimes V(\lambda-\rho_{I_{0}(\lambda)^{c}})$$\varphi:M_{n}(\lambda)arrow M_{n}(\rho_{I_{0}(\lambda)^{c}})\otimes V(\lambda-\rho_{I_{0}(\lambda)^{c}})$,and is defined byis defined bywhere(and is injective). It is enough to show that the image of the composite map
generates
is isomorphic to the desired $kG(n)$ -module since$V(\lambda+(q-1)\rho)$ as a G-module.
, letbe the Levi subgroup relative to and let
For a subset$G_{I}(n)$ be the corresponding finite group. An analogous notation will be used. Now consider theand, for example,for-module embedding$m_{1}\otimes v_{2}\mapsto v_{1}\otimes v_{2}$$f\otimes id$$m_{0}\mapsto m_{1}\otimes v_{2}$$\varphi$$(f\otimes id)\circ\varphi$$v_{1}\otimes v_{2}$$I\subseteq\Delta$$V_{I}(\lambda),$$L_{I}(\lambda),$$G_{I}$$I$$G_{I}$$Y_{I}(\mu)$$M_{n,I}(\lambda)$$kG_{I_{0}(\lambda)}(n)$$\varphi_{I_{0}(\lambda)}:M_{n,I_{0}(\lambda)}(\lambda)arrow M_{n,I_{0}(\lambda)}(\rho_{I_{0}(\lambda)^{c}})\otimes V_{I_{0}(\lambda)}(\lambda-\rho_{I_{0}(\lambda)^{c}})$is one-dimensional
which is analogous to . Since theis bijective and maps the summand. We shallfor anyonto. Taking Harish-Chandra induction
denote this restriction map by
we haveHCIn$V_{I_{0}(\lambda)}(\lambda-\rho_{I_{0}(\lambda)^{c}})$$\varphi$$(=k_{\lambda-\rho_{I_{0}(\lambda)^{C}}}),$$Y_{I_{0}(\lambda)}(\lambda+(q-1)\rho_{J})$$\varphi_{I_{0}(\lambda)}$$Y_{I_{0}(\lambda)}(\rho_{I_{0}(\lambda)^{c}}+(q-1)\rho_{J})\otimes V_{I_{O}(\lambda)}(\lambda-\rho_{I_{O}(\lambda)^{c}})$$J\subseteq\Delta$$\varphi_{I_{0}(\lambda),J}$$d^{}$$G_{I_{0}(\lambda)}(n)$$\varphi$$=$

HCInd$=$$\bigoplus_{J\subseteq I_{0}(\lambda)}$$(\varphi_{I_{0}(\lambda)})=$

HCInd

HCInd

$(\varphi I_{0}(\lambda),J)$$( \bigoplus_{J\subseteq I_{0}(\lambda)}\varphi_{I_{0}(\lambda),J})$.

Moreover, one can prove that
$HCInd_{G_{I_{0}(\lambda)}(n)}^{G(n)}Y_{I_{0}(\lambda)}(\lambda+(q-1)\rho_{J})=\bigoplus_{J’\subseteq I_{q-1(\lambda)}}Y(\lambda+(q-1)\rho_{J}-(q-1)\rho_{J’})$and$HCInd_{G_{I_{0}(\lambda)}(n)}^{G(n)}Y_{I_{0}(\lambda)}(\rho_{I_{0}(\lambda)^{c}}+(q-1)\rho_{J})=Y(\rho_{I_{0}(\lambda)^{c}}+(q-1)\rho_{J})$,87
by using Frobenius reciprocity. These three formulas imply that maps theright-hand side of the second formula toinjectively. Moreover, Lemma 3.2 implies that the restriction of $f\otimes id$ on
the tensor product is injective for $J=I_{0}(\lambda)$ , and zero otherwise. Therefore,
the theorem follows.
$\varphi$$Y(\rho_{I_{0}(\lambda)^{c}}+(q-1)\rho_{J})\otimes V(\lambda-\rho_{I_{0}(\lambda)^{c}})$References[1] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, Springer, 1972.
[2] J. C. Jantzen, Filtrierungen der Darstellungen in der Hauptserie
endlicher Chevalley-Gruppen, Proc. London Math. Soc. (3) 49 (1984)

445-482.

[3] C. Pillen, Loewy series for principal series representations of finite
Chevalley groups, J. Algebra 189 (1997) 101-124.
[4] H. Sawada, A characterization of the modular representations of finitegroups with split $(B,N)$ -pairs, Math. Z. 155 (1977) 29-41.
[5] Y. Yoshii, A generalization of Pillen’s theorem for principal series modules, to appear.