The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras
Part II
Representations and decompositions in characteristic $p\ne 0$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 3 April 2014

§ 7. Conjecture on the decomposition of induced modules $Z_{\lambda }$

7.1. The following is independent of the conjectures in § 5.

Conjecture IV. Let $\lambda \in X^{*}$ (i.e. $\lambda \in X$ and $p^{-1}(\lambda +\delta )\in E^{*}\text{)},$ and for $\sigma \in W$ let us denote ${(\lambda +\delta )}^{w_{\sigma }}-\delta$ by ${\lambda }_{\left(\sigma \right)}\text{.}$ Then for every $\nu \in \Lambda$ which is $W\text{-linked}$ (definition in § 6.3) to the image of $\lambda$ in $\Lambda$ (under the canonical projection $X\to \Lambda \text{)}$ one must have $$\begin{array}{cc}{Z}_{\nu }{\sim }_{\mathbf{u}}\sum _{\sigma \in W}{b}_{\sigma }·{\bar{V}}_{{\lambda }_{\left(\sigma \right)}}\text{.}& \text{(7.1)}\end{array}$$ It should be noted that when $\lambda +\delta$ is not $\stackrel{\sim }{W}\text{-regular}$ (in particular, when it is not $\stackrel{\sim }{W}\prime \text{-regular,}$ i.e. when $\lambda \in X_J^{*}$ for $J$ nonempty), there may be several values of $\sigma$ which give rise to same ${\lambda }_{\left(\sigma \right)},$ and that some of the values of ${\lambda }_{\left(\sigma \right)}+\delta$ could even be $W\text{-singular}$ (since the image of $X^{*}+\delta$ under $w_{\sigma }\in {\stackrel{\sim }{W}}^{♯}\subset {\stackrel{\sim }{W}}^{+}$ lies only in $X^{+}$ and not always in $X^{+}+\delta \text{)}$ in which case we use the convention of treating ${\bar{V}}_{{\lambda }_{\left(\sigma \right)}}$ as the zero-module in consonance with the formula $\text{dim}\hspace{0.17em}{\bar{V}}_{{\lambda }_{\left(\sigma \right)}}=D({\lambda }_{\left(\sigma \right)}+\delta )\text{.}$ (See ${\text{Footnote}}^6\text{.)}$

Note that the equality of the dimensions of the left and right-hand sides of (7.1) is already asserted by Conjecture I; one sees this by recalling the modification in the action of $\stackrel{\sim }{W}$ on $X$ that has been introduced in Part II (third paragraph of § 5.1) under which (2.1) becomes $$\sum _{\sigma \in W}{b}_{\sigma }·D\left(p^{-1}{(\lambda +\delta )}^{w_{\sigma }}\right)=1,$$ whence $$\sum _{\sigma \in W}{b}_{\sigma }·D({\lambda }_{\left(\sigma \right)}+\delta )=p^r=\text{dim}\hspace{0.17em}{Z}_{\nu },$$ as $D\left(\right)$ is homogeneous of degree $r\text{.}$ As a matter of fact, the validity of Conjecture IV for infinitely many values of $p$ implies that of Conjecture I itself, for if (2.1) holds for $\lambda \in E^{*}\cap p^{-1}X$ for the given values of $p$ then it must hold for all $\lambda ,$ as the union of $E^{*}\cap p^{-1}X$ over those values is (topologically dense and hence) Zariski-dense.

Substituting ${\bar{V}}_{{\lambda }_{\left(\sigma \right)}}=\sum _{\mu \in X^{♯}}{c}_{{\lambda }_{\left(\sigma \right)}\mu }^{\prime }{M}_{\mu }$ in (7.1), we have by (6.2) the following equivalent formulation of (7.1): $$\begin{array}{cc}{d}_{\mu }=\sum _{\mu \prime \in Y_0}(\sum _{\sigma \in W}{b}_{\sigma }·c_{{\lambda }_{\left(\sigma \right)},\mu +p\mu \prime })\text{dim}\hspace{0.17em}{M}_{\mu \prime }& \text{(7.2)}\end{array}$$ for all $\mu \in X^{♯}$ that are $\stackrel{\sim }{W}\text{-linked}$ to $\lambda \in X^{*},$ where we write $d_{\mu }$ for $d_{\pi \left(\mu \right)}$ $\text{(}\mu \in X^{♯}\text{)}$ in keeping with the similar notational abuses earlier. (Once again we recall that for $\lambda ,\mu$ in $X^{♯},$ the infinitesimal weights $\pi \left(\lambda \right)$ and $\pi \left(\mu \right)$ are $W\text{-linked}$ if and only if $\lambda$ and $\mu$ are $\stackrel{\sim }{W}\text{-linked.)}$

The following statement for $d_{\mu }$ is analogous to the statement in Conjecture III for the fundamental constant $c_{\lambda \mu }\text{.}$

Conjecture III' (Corollary to Conjectures III and TV). Let $J⫋\{0,1,\dots ,l\},$ ${\nu }_1,{\nu }_2\in X_J^{*},$ and $w\in \Omega \text{.}$ ${\stackrel{\sim }{W}}^{♯}=\left\{w_{\sigma }\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}$ be all given, and write ${\mu }_i={({\nu }_i+\delta )}^w-\delta$ for $1\leqq i\leqq 2\text{.}$ Then for $p\nmid [X:X\prime ]$ one must have $d_{{\mu }_1}=d_{{\mu }_2}\text{.}$

Due to (7.2), to prove this corollary it would suffice to show that for $\lambda \in X_J^{*},$ $w\in {\stackrel{\sim }{W}}_J^{++},$ $\mu ={(\lambda +\delta )}^w-\delta ,$ the sum $\sum _{\sigma \in W}{b}_{\sigma }·c_{{\lambda }_{\left(\sigma \right)}\mu }$ depends only on $J$ and $w$ (and not on $\lambda \text{).}$ But this is immediate from Conjecture III which gives $c_{{\lambda }_{\left(\sigma \right)}\mu }=c_{w_{\sigma }w}^J$ in the notation introduced towards the end of § 5. The role of the hypothesis $p\nmid [X:X\prime ]$ is to guarantee that (for $i=1,2\text{)}$ $v_i$ is the unique element of $X_J^{*}$ which is $\stackrel{\sim }{W}\text{-linked}$ to ${\mu }_i$ and satisfies ${\nu }_i\equiv {\mu }_i$ (mod $X\prime \text{);}$ cf. the fourth paragraph of § 5.

7.2. In this paragraph we briefly comment on the modification one needs to introduce in order to remove the assumption $p\nmid [X:X\prime ]$ in the preceding. For $\lambda \in X^{*},$ let ${\Omega }^{\lambda }$ denote the stabilizer of $\lambda$ in $\Omega$ (defined in § 1), and for an element $\mu$ of $X^{♯}$ which is $\stackrel{\sim }{W}\text{-linked}$ to $\lambda$ define the "virtual Humphreys number" $d_{\mu }^{\prime }$ to be $d_{\mu }·{\left|{\Omega }^{\lambda +\delta }\right|}^{-1}\text{.}$ With this definition, we might strengthen Conjecture III' above, to assert that the value of $d_{\mu }^{\prime }$ for $\mu ={(\lambda +\delta )}^w-\delta ,$ $\lambda \in X_J^{*},$ depends only on $J$ and $w\text{.}$ This would have a very pleasant repercussion on the formula in § 6.4 for the dimension of $Q_{\mu }=Q_{\pi \left(\mu \right)}$ for $\mu \in X^{♯}\text{.}$ Setting $$a_{\mu }^{\prime }=\left|W\right|·{\left|{\stackrel{\sim }{W}\prime }^{\mu +\delta }\right|}^{-1}$$ where ${\stackrel{\sim }{W}\prime }^{\mu +\delta }$ is the stabilizer of $\mu +\delta$ in $\stackrel{\sim }{W}\prime ,$ one finds $$a_{\mu }=\left|W\right|·{\left|{\stackrel{\sim }{W}}^{\mu +\delta }\right|}^{-1}=a_{\mu }^{\prime }·{\left|{\Omega }^{\lambda +\delta }\right|}^{-1}$$ where $\lambda$ is any element of $X^{*}$ that is $\stackrel{\sim }{W}\text{-linked}$ to $\mu \text{;}$ and one has $$\text{dim}\hspace{0.17em}{Q}_{\mu }=a_{\mu }^{\prime }{d}_{\mu }^{\prime }{p}^r\text{.}$$ Since the value of $a_{\mu }^{\prime }$ for $\mu ={(\lambda +\delta )}^w-\delta ,$ $\lambda \in X_J^{*},$ depends only on $J$ and $w$ (the order of the stabilizer of $\mu +\delta$ or $\lambda +\delta$ in $\stackrel{\sim }{W}\prime$ being simply the order of the Coxeter group ${\stackrel{\sim }{W}}^J$ generated by $\left\{R_j\hspace{0.17em}\right|\hspace{0.17em}j\in J\}\text{)},$ it follows (from the preceding unlabelled extension of Conjecture III') that the dimension of $Q_{\mu }$ depends only on $J$ and $w,$ even when $p$ divides $[X:X\prime ]\text{.}$ We shall prove this for the case of type $A_2$ in a later paper, calling the preceding (suggested) phenomenon, the "ramification" phenomenon (term suggested by Humphreys) for the constants $a_{\mu }$ and $d_{\mu },$ the number $\left|{\Omega }^{\lambda +\delta }\right|$ being the measure of this ramification.

7.3. To get ready for the concluding remarks in § 8, let us make the following notation: If $\mu \in X^{♯}$ is $\stackrel{\sim }{W}\prime \text{-linked}$ to $\lambda \in X_J^{*}$ by $\mu ={(\lambda +\delta )}^w-\delta$ for $w\in {\stackrel{\sim }{W}}_J^{♯},$ then assuming the truth of Conjecture III', we write $d_w^J$ in place of $d_{\mu }\text{.}$ Since an element $\omega$ of $\Omega$ sends $X_J^{*}$ to some $X_{J\prime }^{*},$ $\text{(}J\prime$ being precisely the image of $J$ under the permutation ${\theta }_j$ of § 1.2, in case $\omega ={\omega }_j$ for $j\in J_0\text{)},$ it follows from Conjecture III' that $d_w^J=d_{w\prime }^{J\prime },$ where $w\prime ={\omega }^{-1}w\omega \in {\stackrel{\sim }{W}}_{J\prime }^{♯}\text{.}$

${}^6$ (Added in Proof.) I have been able to prove Conjecture IV rather recently by obtaining an explicit formula for the formal character of the $G\text{-module}$ $Q_{\mu }$ for $\mu$ of the form $\mu =(p-1)\delta -{\lambda }^{*}\in p·E^{*}$ (* denoting the Dynkin outer automorphism) with $\lambda \in X_J^{*}+\delta$ $\text{(}J⫋\{0,1,\dots ,l\}\text{)}$ which guarantees $$\left|W\right|·Z_{\lambda -\delta }\sim \mathbf{u}\left|{\stackrel{\sim }{W}}^j\right|·Q_{\mu }{\sim }_G\sum _{\sigma \in W}{\bar{V}}_{(p-1)\delta +{\lambda }^{\sigma }},$$ where the second equivalence was known for a while. For years I was unable to figure out exactly how to replace (in the Grothendieck ring of $G\text{-modules)}$ an arbitrary Weyl module (e.g. one of the $\left|W\right|$ terms in the last sum) in terms of Weyl modules whose highest weights are restricted (i.e. lie in $X^{♯}\text{),}$ though this was evident from the intuition gained by various computations (this means that a suitable analogue of Steinberg's tensor-product theorem holds with Weyl modules in place of irreducible modules). My first breakthrough came on seeing the recursive argument needed to justify this; though explicit formulae were lacking, it was enough to yield the existence of virtual ${𝔤}_{ℂ}\text{-modules}$ with formal characters $B_{\sigma }\in {(\mathbb{Z}·e^X)}^W$ (independent of $\lambda$ as well as $p\text{)}$ such that $$\left|{\stackrel{\sim }{W}}^J\right|·\text{char}\hspace{0.17em}{Q}_{\mu }=\sum _{\sigma \in W}{\left({\bar{B}}_{\sigma }\right)}^{\text{Fr}}·\text{char}\hspace{0.17em}{\bar{V}}_{{\lambda }^{\sigma }+p{\delta }_{\sigma }-\delta }$$ which gives $Z_{\lambda -\delta }{\sim }_{\mathbf{u}}\sum _{\sigma \in W}({\left|W\right|}^{-1}·\text{deg}\hspace{0.17em}{B}_{\sigma })·{\bar{V}}_{{\lambda }^{\left(\sigma \right)}-\delta }$ where ${\lambda }^{\left(\sigma \right)}={\lambda }^{w_{\sigma }}={(\lambda -\delta )}_{\left(\sigma \right)}+\delta \text{.}$ From this Conjectures IV and I follow, except for the integrality of $b_{\sigma }={\left|W\right|}^{-1}·\text{deg}\hspace{0.17em}{B}_{\sigma }\text{.}$ (Here "deg" means the virtual degree, and is the same as "trace" used in formula (6.2)). The next breakthrough came only a few weeks later, when I realized that the ideas at the end of ${\text{Footnote}}^2$ can be used for a quick proof of the preceding as well as a closed expression for $B_{\sigma }$ in terms of the virtual $T\text{-modules}$ ${\beta }_{\sigma }$ (explicitly, if ${\beta }_{\sigma }$ is written as $\sum s_{\lambda }·e^{\lambda },$ one has $B_{\sigma }=\sum s_{\lambda }·\left(\sum _{\tau \in W}{e}^{{\lambda }^{\tau }}\right),$ in consonance with ${\left|W\right|}^{-1}·\text{deg}\hspace{0.17em}{B}_{\sigma }=\text{deg}\hspace{0.17em}{\beta }_{\sigma }=b_{\sigma }\text{).}$ The details will appear in due course.

Notes and References

This is an excerpt of the paper The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras by Daya-Nand Verma. It appeared in Lie Groups and their Representations, ed. I.M.Gelfand, Halsted, New York, pp. 653–705, (1975).

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