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

§ 6. PIM's of the $u\text{-algebra}$ and Humphreys' numbers $d_{\lambda }$

6.1. The Lie algebra of an algebraic group in characteristic $p\ne 0$ possesses a $1\text{-ary}$ operation, called $p\text{-power}$ operation commonly denoted $x\to x^{\left[p\right]},$ under which it becomes a restricted Lie algebra (also called $p\text{-Lie}$ algebra), introduced and studied initially by Jacobson and Zassenhaus. Treating the elements of the Lie algebra as the right- or left-invariant derivations of the affine-ring, this operation arises directly from the fact that the $p^{\text{th}}$ power of a derivation is again a derivation. This operation is given explicitly for the basis elements $${\bar{h}}_i=h_i\otimes 1_K\in {𝔥}_{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }K={𝔥}_K\text{and}{\bar{e}}_{\alpha }=e_{\alpha }\otimes 1_K\in {𝔤}_{\mathbb{Z}}^{\alpha }\underset{\mathbb{Z}}{\otimes }K={𝔤}_K^{\alpha }$$ (for $1\leqq i\leqq l$ and $\alpha \in \Delta \text{)}$ of the Lie algebra ${𝔤}_K$ of our Chevalley group $G$ by $${\bar{h}}_i^p={\bar{h}}_i,{\bar{e}}_{\alpha }^p=0\text{.}$$ The quotient $\mathbf{u}$ of the universal enveloping algebra of ${𝔤}_K$ by the ideal generated by the elements of the form $x^p-x^{\left[p\right]}$ for $x\in {𝔤}_K$ is called the restricted enveloping algebra (also called the $u\text{-algebra)}$ of ${𝔤}_K\text{;}$ it is the universal object with respect to the restricted ${𝔤}_K\text{-modules,}$ viz. those representations for which the action of $x^{\left[p\right]}$ coincides with the $p^{\text{th}}$ power of the endomorphism representing $x\in {𝔤}_K\text{.}$ Thus restricted ${𝔤}_K\text{-modules}$ and $\mathbf{u}\text{-modules}$ are synonymous. We treat ${𝔤}_K$ as a subspace of $\mathbf{u}\text{.}$ By virtue of a Birkhoff—Witt type basis of $\mathbf{u}$ (with exponents in the monomials ranging from 0 to $p-1$ only), one has ${\text{dim}}_K\hspace{0.17em}\mathbf{u}=p^{\text{dim}\hspace{0.17em}G}\text{.}$

A most remarkable fact about $\mathbf{u}$ (which does not seem to have been noticed earlier) is that, in contrast with the above construction of $\mathbf{u}$ as a quotient of the ordinary enveloping algebra, it can also be identified with the subalgebra of $U_K=U_{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }K$ generated by ${𝔤}_K\text{;}$ this can be argued either by noticing that $$h_i^p-h_i\in p·U_{\mathbb{Z}}\text{and}{e}_{\alpha }^p\in p·U_{\mathbb{Z}}\text{for}1\leqq i\leqq l\text{and}\alpha \in \Delta$$ (of which the latter is trivial), or else through more sophisticated arguments which involve identifying $U_K$ with the algebra of all (right- or) left-invariant differential operators of the affine-ring $A_K$ of $G\text{.}$ We shall investigate this new object $U_K$ elsewhere (as we do not require it here), calling it the "hyperenveloping algebra for $G\text{",}$ since it is the noncommutative analog of the hyperalgebra introduced by Dieudonné for a commutative algebraic group in nonzero characteristic (which is again the algebra of invariant differential operators). Suffice it to say here that $U_K$ is a Hopf algebra in some sense dual to $A_K,$ and that though it is not finitely generated as an algebra, for representation-theoretic purposes it behaves infinitely better than either the ordinary or the restricted enveloping algebra of ${𝔤}_K,$ in that every finite-dimensional $U_K\text{-module}$ can be canonically lifted to a $G\text{-module}$ (just as a finite-dimensional $U_{ℂ}\text{-module}$ lifts to the connected simply-connected complex Lie group with Lie algebra ${𝔤}_C\text{).}$

At any rate, the infinitesimal of a $G\text{-module}$ is naturally defined as a restricted representation of ${𝔤}_K\text{.}$ But this correspondence from the isomorphism classes of $G\text{-modules}$ to those of (finite-dimensional) $\mathbf{u}\text{-modules}$ is neither one-one nor onto. Nevertheless, there are good reasons (even for someone interested purely in group representations, — for algebraic, or finite, Chevalley groups) to study the category of all $\mathbf{u}\text{-modules;}$ apart from the classical result of Curtis and Steinberg [Bor1970, § 7] giving a bijection between the isomorphism classes of the irreducible modules for algebras $\mathbf{u}$ and $K{G}_{{𝔽}_p},$ there have recently arisen some new indications in support of this view: in [Ver1972-3] and the forthcoming joint work [HVe1973] we provide certain structural relationships between the projective modules for the two associative algebras, whereas Humphreys has already demonstrated [Hum1971] how some of the structure of the indecomposable projective modules for $\mathbf{u}$ can be understood by analysing certain $\mathbf{u}\text{-modules}$ $Z_{\lambda }$ which cannot normally be lifted to the group $G,$ which are the analogues of the subject-matter of [Ver1966] and [BGG1971]. (Added in Proof: That is, in the terminology of Dixmier's book quoted in Footnote $\text{(}{}^5\text{)}$ in § 5.4, $Z_{\lambda }$ are the $u\text{-algebra}$ analogues of Verma modules.) We shall presently describe the modules $Z_{\lambda }$ and the said results of Humphreys in detail, one of the main objectives of this paper (Parts I and II combined) being the formulation of certain conjectures on the decomposition of these modules $Z_{\lambda }\text{.}$

It is easy to see that the infinitesimal action of ${𝔤}_K$ on the $G\text{-module}$ $M_{p\lambda }\cong M_{\lambda }^{\text{Fr}}$ is trivial for all $\lambda \in X^{+}\text{;}$ consequently, by Steinberg's theorem, for $\lambda \in X^{+}$ and $\mu \in X^{♯}$ one has $$\begin{array}{cc}{M}_{p\lambda +\mu }{\cong }_{\mathbf{u}}\left(\text{dim}\hspace{0.17em}{M}_{\lambda }\right)·M_{\mu }\text{.}& \text{(6.1)}\end{array}$$ It is a result of Curtis [Bor1970, 6.4-6.6] that $M_{\mu }$ is $\mathbf{u}\text{-irreducible}$ for every $\mu \in X^{♯},$ and that every irreducible $\mathbf{u}\text{-module}$ is isomorphic to a member of $\left\{M_{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in X^{♯}\}$ which are mutually nonisomorphic (as we shall see presently). Thus every irreducible $\mathbf{u}\text{-module}$ lifts uniquely to a $G\text{-module.}$ It is known that there exist indecomposable $\mathbf{u}\text{-modules}$ which do not arise as infinitesimal of any $G\text{-module;}$ for the case of type $A_1$ $\text{(}G\cong {SL}_2\text{)}$ it is not difficult to show that the indecomposable $\mathbf{u}\text{-module}$ $Z_{\lambda }$ to be defined presently for $\lambda \in X^{♯},$ has this property with one exception when (for $\lambda =(p-1)\delta \text{)}$ it is irreducible (giving the Steinberg module), and it is believed that the same is true for all types. In [Ver1972-3] we shall show that every projective $\mathbf{u}\text{-module}$ lifts to a $G\text{-module,}$ generalizing a phenomenon for the case of ${SL}_2$ that was very recently observed to follow from certain computations of A. Vincent Jeyakumar on $p\text{-modular}$ representations of the group ${SL}_2\left({𝔽}_p\right)\text{.}$ However, we do not know the answer to the crucial question: Do there exist nonisomorphic $G\text{-modules}$ such that the associated actions of.u are indecomposable and isomorphic? Clearly, indecomposability is essential, for the left and right sides in (6.1) are nonisomorphic $G\text{-modules.}$

6.2. There is a natural homomorphism from $X=X\left(T\right)$ into the dual ${𝔥}_K^{*}$ of the Lie algebra ${𝔥}_K$ of $T$ obtained by taking the infinitesimals of the irreducible representations (characters) of $T\text{.}$ Its kernel is $p·X$ and the image is the set of "restricted infinitesimal weights", viz. the set of all those elements of ${𝔥}_K^{*}$ which (considered as 1-dimensional representations of the Abelian Lie algebra ${𝔥}_K\text{)}$ afford restricted representations of ${𝔥}_K\text{.}$ Thus if ${𝔥}_{{𝔽}_p}=\sum _i{𝔽}_p·{\bar{h}}_i$ denotes the canonical ${𝔽}_p\text{-form}$ of ${𝔥}_K,$ this image is identified with the ${𝔽}_p\text{-dual}$ ${𝔥}_{{𝔽}_p}^{*}\text{;}$ following Humphreys we shall use the notation $\Lambda$ in place of ${𝔥}_{{𝔽}_p}^{*}\text{.}$ The map $X\to \Lambda ,$ which induces an isomorphism from $X|pX$ to $\Lambda ,$ gives a bisection $\pi :X^{♯}\to \Lambda$ on restriction to the subset $X^{♯}\text{.}$ We shall use letters $\lambda ,\mu$ to denote typical elements of $\Lambda$ as well as $X^{♯}\text{;}$ and for $\lambda \in \Lambda$ the $\mathbf{u}\text{-modules}$ $M_{{\pi }^{-1}\left(\lambda \right)}$ and ${\bar{V}}_{{\pi }^{-1}\left(\lambda \right)}$ shall be written simply $M_{\lambda }$ and ${\bar{V}}_{\lambda }$ (and in the same vein, for $\lambda \in X^{♯},$ the module $Z_{\pi \left(\lambda \right)}$ defined in the following paragraph may also be denoted by $Z_{\lambda }\text{)}\text{;}$ in spite of this notational abuse (which can hardly lead to confusion) it is important to maintain a conceptual distinction between the group $\Lambda \cong X/pX$ and the set $X^{♯}\text{.}$

Let us note that the action of the Borel subalgebra ${𝔟}_K={𝔥}_K+\sum _{\alpha \in {\Delta }_{+}}{𝔤}_K^{\alpha }$ on the $\mathbf{u}\text{-irreducible}$ module $M_{\lambda }$ for $\lambda \in \Lambda ,$ stabilizes a unique 1-dimensional subspace, viz. the highest-weight-space of this $G\text{-module}$ (stabilized by $B\text{),}$ on which the action of ${𝔥}_K$ is given by the linear function $\lambda \text{;}$ this shows that the irreducible $\mathbf{u}\text{-modules}$ $M_{\lambda }$ are mutually nonisomorphic for $\lambda \in \Lambda \text{.}$ Further, since each $G\text{-module}$ $M_{\lambda }$ is defined over ${𝔽}_p,$ these $\mathbf{u}\text{-irreducibles}$ possess ${𝔽}_p\text{-forms}$ with respect to the canonical ${𝔽}_p\text{-form}$ of $\mathbf{u},$ viz. the $u\text{-algebra}$ of the "form" ${𝔤}_{{𝔽}_p}={𝔤}_{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }{F}_p$ of ${𝔤}_K\text{.}$ It follows that the $\mathbf{u}\text{-modules}$ $M_{\lambda }$ are absolutely irreducible for $\lambda \in \Lambda$ (or $X^{♯}\text{).}$

In accordance with [Hum1971], for $\lambda \in \Lambda$ let $Z_{\lambda }$ denote the (right) $\mathbf{u}\text{-module}$ $1_{\lambda }\underset{\mathbf{b}}{\otimes }\mathbf{u}$ obtained by inducing from the 1-dimensional $\mathbf{b}\text{-module}$ $1_{\lambda }$ for the "Borel subalgebra" $\mathbf{b}$ of $\mathbf{u}$ generated by ${𝔟}_K,$ where on $1_{\lambda }$ the action of ${𝔥}_K$ is given by the restricted weight $\lambda$ and that of the radical $${𝔤}_K^{+}=[{𝔟}_K,{𝔟}_K]=\sum _{\alpha \in {\Delta }_{+}}{𝔤}_K^{\alpha }$$ of ${𝔟}_K$ is trivial. If ${\mathbf{u}}^{-}$ denotes the subalgebra of $\mathbf{u}$ generated by ${𝔤}_K^{-}=\sum _{\alpha \in {\Delta }_{-}}{𝔤}_K^{\alpha },$ one finds from the vector space isomorphism $\mathbf{u}\cong \mathbf{b}\otimes {\mathbf{u}}^{-}$ (Birkhoif—Witt theorem for $\mathbf{u}\text{)}$ that $Z_{\lambda }$ is a free ${\mathbf{u}}^{-}$ module of rank 1, and thus has dimension $p^r$ for $r=\left|{\Delta }_{+}\right|\text{.}$ The module $Z_{\lambda }$ is generated by a unique 1-dimensional subspace of the $\lambda \text{-weight-space}$ which is annihilated by ${𝔤}_K^{+}$ (is "standard cyclic with highest weight $\lambda \text{"}$ in the terminology of [Hum1971]), and is universal with this property, in that every $\mathbf{u}\text{-module}$ with this property is a quotient of $Z_{\lambda }\text{.}$ It is proved in [Hum1971, Proposition 1.1] that for $\lambda \in \Lambda$ the $\mathbf{u}\text{-module}$ ${\bar{V}}_{\lambda }$ has this property, so that one has the surjections $Z_{\lambda }\to {\bar{V}}_{\lambda }\to M_{\lambda }\text{.}$ (For $\lambda \in X^{+},$ $\lambda \notin X^{♯},$ the dimension of ${\bar{V}}_{\lambda }$ generally exceeds $p^r\text{;}$ however, it is easy to see that for no such $\lambda$ can the $\mathbf{u}\text{-module}$ ${\bar{V}}_{\lambda }$ be "standard cyclic".) It should be noted that though the $G\text{-module}$ $M_{\lambda }$ has the $\lambda \text{-weight-space}$ 1-dimensional for all $\lambda \in X^{+},$ in the infinitesimal situation normally the $\lambda \text{-weight-space}$ of the $\mathbf{u}\text{-irreducible}$ $M_{\lambda }$ for $\lambda \in \Lambda$ has dimension greater than one (and hence the same holds for ${\bar{V}}_{\lambda }$ and $Z_{\lambda }\text{),}$ on account of the merging of weights when one passes from the decomposition for the torus $T$ to that for the Lie algebra ${𝔤}_K\text{.}$ In fact, every weight-space of every $Z_{\lambda }$ has dimension $p^{r-l}\text{.}$ Considering the decomposition of ${\mathbf{u}}^{-}$ as a direct sum of the subspace of scalars and the augmentation ideal (generated by ${𝔤}_K^{-}\text{),}$ one finds that every proper submodule of $Z_{\lambda }$ is contained in the image.of the 1-dimensional generating subspace under this augmentation ideal. This shows that $Z_{\lambda }$ has a unique maximal proper submodule (the kernel of $Z_{\lambda }\to M_{\lambda }\text{)},$ and that $Z_{\lambda }$ has no irreducible module other than $M_{\lambda }$ as its quotient, and thus is indecomposable.

6.3. On account of the standard facts from the theory of Artinian rings (which applies to $\mathbf{u}$ because of its finite dimensionality), we know that (i) up to isomorphism there exists a unique projective $\mathbf{u}\text{-module}$ $Q_{\lambda }$ (called the projective cover of $M_{\lambda }\text{)}$ such that $M_{\lambda }$ is its only irreducible quotient, (ii) every indecomposable $\mathbf{u}\text{-projective}$ is isomorphic to some $Q_{\lambda }$ for $\lambda \in \Lambda ,$ which are clearly all indecomposable, (iii) the category of projective $\mathbf{u}\text{-modules}$ is semisimple, i.e. every projective module is isomorphic to a direct sum $\sum _{\lambda \in \Lambda }{n}_{\lambda }{Q}_{\lambda },$ and (iv) the (right-regular) representation of $\mathbf{u}$ on $\mathbf{u}$ (which is free, and therefore projective, over $\mathbf{u}\text{)}$ is equivalent to the direct sum $$\sum _{\lambda \in \Lambda }\left(\text{dim}\hspace{0.17em}{M}_{\lambda }\right)·Q_{\lambda },$$ so that every $Q_{\lambda }$ does occur in this representation. The modules $Q_{\lambda }$ are called PIM's (principal indecomposable modules). Two PIM's $Q_{\lambda }$ and $Q_{\mu }$ are called linked if there exists a chain of PIM's beginning at $Q_{\lambda }$ and ending at $Q_{\mu }$ such that the consecutive pairs have a common composition factor; we write $Q_{\lambda }\sim Q_{\mu }$ in that case. Further, one knows (from the theory of Artinian rings) that the sum of all PIM's belonging to one linkage class (i.e. equivalence class under $\sim \text{)}$ forms a 2-sided ideal of $\mathbf{u}$ generated by a primitive central idempotent, and thus is a minimal direct summand in the associative algebra $\mathbf{u},$ called a "block" of $\mathbf{u}\text{.}$ Since both $M_{\lambda }$ and $Q_{\lambda }$ are defined over the prime field, one finds that the semi-simple part of each block, i.e. the quotient by its radical, is "split" in the sense that it is a direct sum of matrix algebras, — exactly one of dimension ${\left(\text{dim}\hspace{0.17em}{M}_{\lambda }\right)}^2$ for each $\lambda$ corresponding to $Q_{\lambda }\text{'s}$ appearing in the linkage class. From #the decomposition of $\mathbf{u}$ as a direct sum (as $K\text{-algebra)}$ of blocks, it is clear that if $Q_{\lambda }$ and $Q_{\mu }$ are not linked then $M_{\lambda }$ and $M_{\mu }$ cannot be "linked" either (as $\mathbf{u}\text{-modules,}$ in a sense similar to that for $G\text{-modules:}$ see the beginning of § 5.3), the converse being trivial. It must be noted that if $M_{\lambda }$ and $M_{\mu }$ are linked as $G\text{-modules}$ (for $\lambda ,\mu \in X^{♯}\text{)}$ then these are also linked as $\mathbf{u}\text{-modules,}$ but not conversely: for in the former case one necessarily has $\lambda -\mu \in X\prime ,$ whereas this is far from necessary in the case of $\mathbf{u}\text{-modules}$ as we shall see presently.

The infinitesimal version of the Harish-Chandra principle (cf. § 5) conjectures that for $\lambda ,\mu \in \Lambda$ if $M_{\lambda }$ and $M_{\mu }$ occur as a composition factor of an indecomposable $\mathbf{u}\text{-module}$ (or more generally, if $M_{\lambda }$ and $M_{\mu }$ are linked as $\mathbf{u}\text{-modules)}$ then $\lambda +\bar{\delta }$ and $\mu +\bar{\delta }$ must be conjugate under the Weyl group $W$ (which acts canonically on $\Lambda \cong X/pX\text{)},$ where $\bar{\delta }$ stands for the image of $\delta \in X^{♯}$ under $\pi \text{.}$ When this conclusion on $\lambda ,\mu$ holds, let us introduce the terminology that $\lambda$ and $\mu$ are $W\text{-linked;}$ in the notation of [Hum1971] this is expressed as $\lambda \sim \mu \text{.}$ It must be observed that $\lambda ,\mu \in \Lambda$ are $W\text{-linked}$ if and only if the restricted dominant (global) weights ${\pi }^{-1}\left(\lambda \right),{\pi }^{-1}\left(\mu \right)\in X^{♯}$ are $\stackrel{\sim }{W}\text{-linked}$ (and not just $\stackrel{\sim }{W}\prime \text{-linked),}$ in the sense that ${\pi }^{-1}\left(\lambda \right)+\delta$ and ${\pi }^{-1}\left(\mu \right)+\delta$ are $\stackrel{\sim }{W}\text{-conjugate.}$ (The term $\text{"}\stackrel{\sim }{W}\prime \text{-linked"}$ was introduced in § 5.1.)

The conjecture on the linkage of $\mathbf{u}\text{-modules}$ that we just mentioned was proved to be true in [Hum1971], provided that $p>h_{\Delta }\text{.}$ In this (infinitesimal) case the converse is also easy, and follows from the fact that if a fundamental reflection $R_i\in W$ sends $\lambda +\bar{\delta }$ to $\mu +\bar{\delta }=\lambda +\bar{\delta }-k{\bar{\alpha }}_i$ with $1\leqq k\leqq p-1$ (where ${\bar{\alpha }}_i$ stands for the projection of the global root ${\alpha }_i\in \Delta \subset X$ under the natural projection $X\to \Lambda \text{)},$ then there is a nonzero homomorphism from $Z_{\mu }$ to $Z_{\lambda }$ (if $v$ and $v\prime$ are nonzero elements in the 1-dimensional generating subspaces of the highest-weight-spaces of $Z_{\lambda }$ and $Z_{\mu }$ respectively, then this homomorphism is determined by $v\prime \to v{\bar{f}}_i^k,$ which shows that the indecomposable $\mathbf{u}\text{-module}$ $Z_{\lambda }$ has both $M_{\lambda }$ and $M_{\mu }$ as composition factors. In fact, pursuing this argument further, Humphreys has found [Hum1971, Theorem 2.2] (without assuming the Harish-Chandra principle) that if $\lambda$ and $\mu$ are $W\text{-linked}$ elements of $\Lambda$ then $Z_{\lambda }{\sim }_{\mathbf{u}}{Z}_{\mu },$ where ${\sim }_{\mathbf{u}}$ denotes equivalence in the Grothendieck ring of the category of $\mathbf{u}\text{-modules,}$ i.e. $Z_{\lambda }$ and $Z_{\mu }$ have identical composition factors (multiplicity included). It follows that if $\lambda ,\mu \in X^{♯}$ are $\stackrel{\sim }{W}\text{-linked}$ then $M_{\lambda }$ and $M_{\mu }$ are linked as $\mathbf{u}\text{-modules,}$ which corroborates the statement immediately preceding the last paragraph.

6.4. Now let $\mathbf{b}\prime$ be the subalgebra of $\mathbf{u}$ generated by the Borel subalgebra ${𝔟}_K^{\prime }={𝔥}_K+{𝔤}_K^{-}$ opposite to ${𝔟}_K\text{.}$ It is readily seen, by looking at the regular representation of $\mathbf{b}\prime$ on $\mathbf{b}\prime ,$ that the modules $\left\{Z_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in \Lambda \}$ exhaust the (isomorphism classes of) projective $\mathbf{b}\prime \text{-modules}$ (upon restriction to $\mathbf{b}\prime \text{),}$ the unique irreducible quotient of $Z_{\lambda }$ being the 1-dimensional irreducible $\mathbf{b}\prime \text{-module}$ $1_{\lambda }^{\prime }$ on which ${𝔤}_K^{-}$ acts trivially and ${𝔥}_K$ acts through the restricted weight $\lambda \text{.}$ Since $\mathbf{u}$ is $\mathbf{b}\prime \text{-free}$ (under right-multiplication, as is clear from the Birkhoff—Witt theorem for $u\text{-algebras)}$ and $Q_{\lambda }$ is a direct summand in $\mathbf{u},$ we see that the restriction of $Q_{\lambda }$ to $\mathbf{b}\prime$ is $\mathbf{b}\prime \text{-projective.}$ Hence there is a decomposition in the semisimple category of projective $\mathbf{b}\prime \text{-modules:}$ $$Q_{\lambda }{\cong }_{\mathbf{b}\prime }\sum _{\mu }\oplus {[Q_{\lambda }:Z_{\mu }]}_{\mathbf{b}\prime \text{-proj}}·Z_{\mu },$$ where the meaning of $\text{"}\mathbf{b}\prime \text{-proj"}$ in subscript is clear.

We are now ready to state the most interesting consequence of the (infinitesimal) Harish-Chandra principle that Humphreys has derived [Hum1971, Theorem 4.5]:

If we define $d_{\mu }={[Z_{\mu }:M_{\mu }]}_{\mathbf{u}}$ $\text{(}={[Z_{\lambda }:M_{\mu }]}_{\mathbf{u}}$ for every $\lambda \in \Lambda$ which is $W\text{-linked}$ to $\mu \text{),}$ then $$Q_{\lambda }{\sim }_{\mathbf{b}\prime }{d}_{\lambda }·(\sum _{\mu \sim \lambda }\oplus Z_{\mu }),$$ i.e. the multiplicity ${[Q_{\lambda }:Z_{\mu }]}_{\mathbf{b}\prime \text{-proj}}$ is $d_{\lambda }$ or $0$ according as $Q_{\lambda }$ is linked to $Q_{\mu }$ or not.

It follows that if $a_{\lambda }$ is the cardinality of the $W\text{-linkage}$ class of $\lambda$ in $\Lambda$ then $$\text{dim}\hspace{0.17em}{Q}_{\lambda }=a_{\lambda }{d}_{\lambda }{p}^r\text{.}$$ In order to compute $a_{\lambda }$ one has only to know the stabilizer $W^{\lambda +\delta }$ of $\lambda +\delta$ in $W,$ for then one has $\left|W\right|=a_{\lambda }·\left|W^{\lambda +\delta }\right|\text{.}$ For $p\nmid [X:X\prime ]$ we claim that $W^{\lambda +\delta }$ is again a reflection subgroup. Since $\Lambda$ is the set of "integral points" on the torus $E/pX,$ this fact can be obtained from the study of Steinberg of the action of Weyl groups on tori [Ste1968-2, Lemma 4.1], but the following observation not only gives this fact directly but also shows how to compute the order of this stabilizer. Let $\mu ={\pi }^{-1}\left(\lambda \right)\in X^{♯},$ and denote by ${\stackrel{\sim }{W}}^{\mu +\delta }$ the stabilizer of $\mu +\delta$ in $\stackrel{\sim }{W}\text{.}$ Then $p\nmid [X:X\prime ]$ gives ${\stackrel{\sim }{W}}^{\mu +\delta }\subset \stackrel{\sim }{W}\prime$ by exactly the same type of argument as in the fourth paragraph of § 5. The subgroup ${\stackrel{\sim }{W}}^{\mu +\delta }$ is then generated by reflections; in fact, if $\nu \in X_J^{*}$ is the unique element of $X^{*}$ which is $\stackrel{\sim }{W}\prime \text{-linked}$ to $\mu$ then ${\stackrel{\sim }{W}}^{\mu +\delta }$ is conjugate to the stabilizer ${\stackrel{\sim }{W}}^J$ in $\stackrel{\sim }{W}\prime$ of $\nu +\delta$ $\text{(}{\stackrel{\sim }{W}}^J$ is defined in § 1.4 as the subgroup by the reflections $\left\{R_j\hspace{0.17em}\right|\hspace{0.17em}j\in J\}\text{).}$ Since ${\stackrel{\sim }{W}}^{\mu +\delta }$ contains no translations, the projection $\stackrel{\sim }{W}\prime \to W$ gives an isomorphism of ${\stackrel{\sim }{W}}^{\mu +\delta }$ onto $W^{\lambda +\delta }\text{.}$ Thus to calculate $a_{\lambda }$ (for $p\nmid [X:X\prime ]\text{)}$ one needs only determine $J⫋\{0,1,\dots ,l\},$ for the Coxeter relations on $\left\{R_j\hspace{0.17em}\right|\hspace{0.17em}j\in J\}$ can be then read off from the extended Dynkin diagram and the order of the group generated by them readily computed by standard methods. This effectively solves the problem of computing $a_{\lambda }$ in any given case. (Things could be accounted for even in the more general situation when $p$ is allowed to divide $[X:X\prime ],$ through a slightly more detailed analysis; see § 7.2.)

6.5. In the next section we offer a conjecture on the decomposition of the $\mathbf{u}\text{-module}$ $Z_{\lambda }$ which implies a neat formula ((7.2)) for Humphreys' numbers $d_{\lambda }\text{.}$ This formula involves the constants $b_{\sigma }$ of Conjecture I and the infinitesimal decomposition numbers $c_{\lambda \mu }^{\prime }={[{\bar{V}}_{\lambda }:M_{\mu }]}_{\mathbf{u}}$ analogous to the constants $c_{\lambda \mu },$ and thus forms the raison d'être for the two parts of this paper. The numbers $c_{\lambda \mu }^{\prime }$ for $\lambda ,\mu \in \Lambda$ can be readily derived from a knowledge of $c_{\lambda \mu }$ for $\lambda \in X^{♯}$ and $\mu \in X^{+}\text{;}$ we recall that the latter set of numbers were shown to be sufficient, in § 4, for determining the formal characters of all the irreducible $G\text{-modules}$ and their dimensions. Explicitly, $$\begin{array}{ccc}{\bar{V}}_{\lambda }& {\sim }_G& \sum _{\mu \in X^{+}}{c}_{\lambda \mu }{M}_{\mu }\\ & {\sim }_G& \sum _{\mu ″\in X^{♯}}(\sum _{\mu \prime \in Y_0}{c}_{\lambda ,\mu ″+p\mu ″}{M}_{\mu ″}\otimes M_{\mu \prime }^{\text{Fr}})\\ & {\sim }_{\mathbf{u}}& \sum _{\mu \in X^{♯}}(\sum _{\mu \prime \in Y_0}{c}_{\lambda ,\mu +p\mu \prime }·\text{dim}\hspace{0.17em}{M}_{\mu \prime })·M_{\mu },\end{array}$$ which gives $$\begin{array}{cc}{c}_{\lambda \mu }^{\prime }=\sum _{\mu \prime \in Y_0}{c}_{\lambda ,\mu +p\mu \prime }·\text{dim}\hspace{0.17em}{M}_{\mu \prime }=\text{trace}\hspace{0.17em}{C}_{\lambda \mu },& \text{(6.2)}\end{array}$$ where $C_{\lambda \mu }$ is as in (4.5), and the trace of an element $\sum _{\lambda }{n}_{\lambda }{e}^{\lambda }$ of $\mathbb{Z}·e^X$ means the sum $\sum _{\lambda }{n}_{\lambda }$ of all the coefficients (multiplicities). Here we have repeated the earlier abuse of identifying indices in $X^{♯}$ and $\Lambda$ via the bijection $\pi \text{.}$ In case $p$ is large enough to guarantee the irreducibility of ${\bar{V}}_{\mu \prime }$ for $\mu \prime \in Y_0$ (cf. Corollary to Proposition 3 in § 5), one substitutes $\text{dim}\hspace{0.17em}{M}_{\mu \prime }=D(\mu \prime +\delta )$ in (6.2) and obtains a cleaner formula for $c_{\lambda \mu }^{\prime }\text{.}$ Note that on has $c_{\lambda \mu }^{\prime }=c_{\lambda \mu }$ whenever $Y_0=\left\{0\right\}$ (as is the case for types $A_2$ and $B_2\text{;}$ cf. Example 2, § 5).

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).

page history