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: 2 April 2014

§ 4. Fundamental constants $c_{\lambda \mu }$ and the character formula

As in Part I, let $\Delta$ be an indecomposable reduced root-system, and ${𝔤}_{ℂ}$ be the simple Lie algebra over $ℂ$ (determined up to isomorphism) possessing $\Delta$ as its root-system.

4.1. Let $G$ be an algebraic Chevalley group, i.e. a connected semisimple (affine) algebraic group, of type ${𝔤}_{ℂ}$ (or of type $\Delta \text{)}$ defined over a field $K$ of characteristic $p\ne 0\text{;}$ one denotes by $G_L$ the group of $L\text{-rational}$ points of $G$ for $L⫆K\text{.}$ Thus, depending on one's approach, $G$ itself may be identified either with $G_K$ for the algebraic closure $\bar{K}$ of $K,$ or with $G_L$ for a very large algebraically closed (universal) field $L,$ or else with the functor $L\to G_L\text{;}$ $G$ is characterised by its affine-ring $A=A_K$ (being $K\left[G\right]$ in the notation of [Bor1970], our canonical reference for standard facts summarized here too briefly), which is a Hopf algebra over $K,$ such that $G_L$ is identified with the maximal-ideal spectrum of $A_L=A\underset{K}{\otimes }L$ for every field $L⫆K,$ the group-operation on $G_L$ coming from the coproduct in $A_L$ in the standard manner. By a $G\text{-module}$ we shall always mean an algebraic $G\text{-module}$ defined over $K,$ using the words "module" and "representation" interchangeably. Thus $K$ is the underlying field of every representation space $V,$ and concretely the representation affords a homomorphism of $G_L$ into ${\text{Aut}}_L\hspace{0.17em}{V}_L$ for every overfield $L$ of $K,$ where $V_L=V\underset{K}{\otimes }L\text{.}$

Let us assume that $G$ is simply-connected, and split over $K\text{.}$ The former assumption means that every projective representation (i.e. algebraic homomorphism into $PGL\left(n\right)\text{)}$ of $G$ comes from a $G\text{-module}$ (i.e. factors through $GL\left(n\right)\text{);}$ this is particularly suitable for representation-theoretic purposes, and there is no loss in this assumption because of the existence and uniqueness of simply-connected (universal) coverings. The latter assumption means that $G$ contains a maximal torus $T$ which is defined and split over $K,$ i.e. is isomorphic over $K$ to ${\left({𝔾}_m\right)}^l$ where ${𝔾}_m=GL\left(1\right)$ is the "multiplicative group" (which, as a group-functor, assigns to $L⫆K$ the multiplicative group of $L\text{).}$ One knows that under these assumptions $G$ is uniquely determined (up to isomorphism) by $K$ and $\Delta ,$ and that $K$ may be taken to be the prime field ${𝔽}_p\text{;}$ this means existence of an ${𝔽}_p\text{-form}$ of the affine-ring, which was described explicitly by Kostant purely in terms of the root-system $\Delta$ (via the $\mathbb{Z}\text{-form}$ $U_{\mathbb{Z}}$ of the enveloping algebra $U_{ℂ}$ of ${𝔤}_{ℂ}$ which we shall describe below in some detail, without discussing the affine-ring further as it is not required by us here); cf. [Bor1970, § 4]. We find it convenient not to specify $K,$ but treat it as a floating (working) field. Thus the Lie algebra ${𝔤}_K$ of $G$ is a vector space over $K$ of dimension $\text{dim}\hspace{0.17em}G=\text{dim}\hspace{0.17em}{𝔤}_{ℂ}=\left|\Delta \right|+l\text{.}$ The elements of ${𝔤}_K$ may be identified with either the right or the left-invariant derivations of $A_K\text{.}$ It may be remarked that unlike the situation in characteristic 0, the object ${𝔤}_K$ is not entirely "local"; in particular, the Lie algebra of two isogenous groups (in the sense of algebraic groups) may not be isomorphic. In fact, one knows that this pathology arises if and only if $p$ divides the "connexion number" $[X:X\prime ]$ of $\Delta$ where $X,X\prime$ are as in § 1; in this case ${𝔤}_K$ is nonsimple, and the only time that happens again is for types $F_4$ with $p=2$ and $G_2$ with $p=3\text{;}$ but we shall not require this fact. An explicit description of ${𝔤}_K$ in terms of $\Delta ,$ essentially due to Chevalley, is as follows.

Let ${𝔤}_{\mathbb{Z}}={𝔥}_{\mathbb{Z}}+\sum _{\alpha \in \Delta }{𝔤}_{\mathbb{Z}}^{\alpha }$ be a Chevalley lattice in ${𝔤}_{ℂ}\text{.}$ This means the following: (i) ${𝔥}_{ℂ}={𝔥}_{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }ℂ$ is a Cartan subalgebra of ${𝔤}_{ℂ}$ on which elements of $\Delta$ are considered as linear functions and ${𝔥}_{\mathbb{Z}}$ consists of all $h\in {𝔥}_{ℂ}$ such that $\lambda \left(h\right)\in \mathbb{Z}$ for all $\lambda$ in the weight-lattice $X$ of $\Delta$ (as defined in § 1, with $\Delta \subset X\subseteq {𝔥}_{ℂ}^{*}=$ vector space dual of ${𝔥}_{ℂ}\text{);}$ thus if $h_{\alpha }\in {𝔥}_{ℂ}$ is defined by $\lambda \left(h_{\alpha }\right)=⟨\lambda ,{\alpha }^{\vee }⟩$ for all $\lambda \in {𝔥}_{ℂ}^{*},$ then putting $h_i=h_{{\alpha }_i}$ one has ${𝔥}_{\mathbb{Z}}=\sum _{i=1}^l\mathbb{Z}{h}_i\text{;}$ (ii) ${𝔤}_{ℂ}^{\alpha }={𝔤}_{\mathbb{Z}}^{\alpha }\underset{\mathbb{Z}}{\otimes }ℂ$ is the $\alpha \text{-rootspace}$ of ${𝔤}_{ℂ}$ with respect to ${𝔥}_{ℂ},$ and if $e_{\alpha }$ denotes a generator of the (additive) infinite-cyclic group ${𝔤}_{\mathbb{Z}}^{\alpha },$ then $[e_{\alpha },e_{-\alpha }]=\pm 2h_{\alpha }\text{;}$ and (iii) if $U_{\mathbb{Z}}$ denotes the subring of the universal enveloping algebra $U_{ℂ}$ of ${𝔤}_{ℂ}$ generated by $\left\{\frac{e_{\alpha }^n}{n!}\hspace{0.17em}\right|\hspace{0.17em}\alpha \in \Delta ,n\in {\mathbb{Z}}^{+}\},$ then ${𝔤}_{\mathbb{Z}}$ is stable under the adjoint action of $U_{\mathbb{Z}}$ on ${𝔤}_{ℂ}\text{.}$ It is customary to choose $e_{\alpha }\text{'s}$ such that $[e_{\alpha },e_{-\alpha }]=2h_{\alpha },$ and we write $e_i$ for $e_{{\alpha }_i}$ and $f_i$ for $e_{-{\alpha }_i},$ for $1\leqq i\leqq l\text{.}$ It may be remarked that the set $$\{\frac{e_i^n}{n!},\frac{f_i^n}{n!}\hspace{0.17em}|\hspace{0.17em}1\leqq i\leqq l,\hspace{0.17em}n\in {\mathbb{Z}}^{+}\}$$ generates $U_{\mathbb{Z}}$ as a ring [Ver1972-2].

The Lie algebra ${𝔤}_K$ of $G$ can be identified with ${𝔤}_Z\underset{\mathbb{Z}}{\otimes }K$ in such a way that ${𝔥}_K={𝔥}_{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }K$ the Lie algebra of the maximal torus $T,$ and identifying $X$ with the character group $X\left(T\right)$ of $T$ (i.e. the group of algebraic homomorphisms of $T$ into ${𝔾}_m$ with the group-operation given by tensoring these 1-dimensional $T\text{-modules,}$ which is written additively), the subspace ${𝔤}_K={𝔤}_{\mathbb{Z}}^{\alpha }\underset{\mathbb{Z}}{\otimes }K$ of ${𝔤}_K$ becomes its $\alpha \text{-rootspace}$ with respect to the adjoint action of $T\text{.}$ Thus, we are identifying $\Delta$ with the root-system of $G$ with respect to $T,$ as is common practice, and this should lead to no confusion. In general, for any (algebraic) $T\text{-module}$ $M$ and for $\lambda \in X\left(T\right)=X$ we denote by $M_{\left(\lambda \right)}$ its $\lambda \text{-weight-space}$ consisting of all vectors in $M$ on which $T$ acts by $\lambda ,$ so that one has $M=\sum _{\lambda \in X}{M}_{\left(\lambda \right)}$ in view of the complete reductibility of $T\text{-modules;}$ thus ${𝔤}_{K\left(\alpha \right)}$ (for the adjoint action of $T$ on ${𝔤}_K\text{)}$ is the same as ${𝔤}_K^{\alpha }\text{.}$ The structure of a $T\text{-module}$ is determined by the dimensions of the various weight-spaces, i.e. by the element $\text{char}\hspace{0.17em}M$ of the group-ring of $X$ over $\mathbb{Z},$ called the formal character of $M,$ which assigns $\lambda \to \text{dim}\hspace{0.17em}{M}_{\left(\lambda \right)}\text{.}$ More precisely, denoting by $\mathbb{Z}·e^X$ the set of all $\mathbb{Z}\text{-linear}$ combinations $\sum _{\lambda \in X}{n}_{\lambda }·e^{\lambda }$ of formal symbols $\left\{e^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in X\}$ where all but a finite number of $n_{\lambda }\text{'s}$ are zero, and making it into a ring under the usual coordinatewise addition and convolution multiplication with $e^{\lambda }·e^{\mu }=e^{\lambda +\mu },$ we have $$\text{char}\hspace{0.17em}M=\sum _{\lambda \in X}\left(\text{dim}\hspace{0.17em}{M}_{\left(\lambda \right)}\right)·e^{\lambda }\in \mathbb{Z}·e^X\text{.}$$ The formal character of a $G\text{-module}$ $M$ shall mean simply that of $M$ as a $T\text{-module}$ upon restriction. From the standard fact that the Weyl group $W$ of $\Delta$ can be identified with the quotient $N/T$ in a canonical manner, where $N$ is the normalizer of $T$ in $G,$ it is immediate that the formal character of a $G\text{-module}$ is a $W\text{-invariant}$ element of $\mathbb{Z}·e^X$ under the natural action of $W\text{.}$

Let $B$ be the Borel subgroup of $G$ having as its Lie algebra ${𝔟}_K={𝔥}_K+\sum _{\alpha \in {\Delta }_{+}}{𝔤}_K^{\alpha },$ where ${\Delta }_{+}$ is a given choice of positive roots in $\Delta \text{.}$ One knows that every irreducible $G\text{-module}$ contains a unique 1-dimensional highest-weight-space, in the sense that it is a weight-space for $T$ whose elements remain fixed under the unipotent radical $[B,B]$ of $B\text{.}$ If this weightspace corresponds to the character $\lambda \in X,$ we say $\lambda$ is the highest weight of the given module. Necessarily $\lambda \in X^{+}$ (the dominant elements of $X$ with respect to the fixed choice of positive roots), and if $\mu \in X$ is a "weight" of the given $G\text{-module,}$ in the sense that the $\mu \text{-weight-space}$ is nonzero, then $\lambda -\mu \in \mathbb{Z}·{\Delta }_{+},$ which we rewrite as $\mu \leqq \lambda$ by setting up the canonical partial-order $\leqq$ on $X$ compatible with the addition under which ${\mathbb{Z}}^{+}·{\Delta }_{+}$ is the set of all elements $\geqq 0\text{.}$ Conversely, to every $\lambda \in X^{+}$ there corresponds an irreducible $G\text{-module}$ $M_{\lambda }$ with highest weight $\lambda ,$ which is unique up to isomorphism. As stated in the Introduction, one of the long-standing unsolved problems in the study of $G\text{-modules}$ is to give a formula for $\text{char}\hspace{0.17em}{M}_{\lambda }$ in terms of $\lambda \in X^{+}\text{.}$

4.2. A famous theorem of Steinberg reduces the study of all irreducible $G\text{-modules}$ $M_{\lambda }$ for $\lambda \in X^{+},$ to that of the $p^l$ basic ones with $\lambda$ lying in the set $$X^{♯}=\{\sum _i{m}_i{\lambda }_i\in X\hspace{0.17em}|\hspace{0.17em}0\leqq m_i\leqq p-1\hspace{0.17em}\text{for}\hspace{0.17em}1\leqq i\leqq l\}\text{;}$$ the elements of $X^{♯}$ will be called the "restricted dominant characters". The formulation of this theorem hinges on the fact that $G$ is defined over ${𝔽}_p$ (as stated earlier) and that all the $M_{\lambda }\text{'s}$ are definable over ${𝔽}_p\text{;}$ we will be able to see the latter fact from the construction of $M_{\lambda }$ by the standard technique of "reduction $\text{mod}\hspace{0.17em}p\text{",}$ which we shall need to describe presently in some detail. Now if $M$ is any $G\text{-module}$ defined over the prime field, one defines its Frobenius-twist $M^{\text{Fr}}$ to be the $G\text{-module}$ whose underlying space is $M$ itself, such that the new action of $g\in G_L$ on $M_L=M\underset{K}{\otimes }L$ is obtained as follows: take a basis of the ${𝔽}_p\text{-form}$ $M_{{𝔽}_p}$ of $M=M_K$ and write the matrix of (the old action of) $g$ with respect to this basis of $M_L\text{;}$ then the new matrix obtained by replacing each entry by its $p^{\text{th}}$ power gives the new (Frobenius-twisted) action of $g$ on $M_L$ with respect to the same basis. It is immediate that the $\mu \text{-weight-space}$ of $M$ coincides with the $p\mu \text{-weight-space}$ of $M^{\text{Fr}},$ which gives $\text{char}\hspace{0.17em}{M}^{\text{Fr}}={\left(\text{char}\hspace{0.17em}M\right)}^{\text{Fr}},$ where $$\text{for}\chi =\sum _{\lambda }{n}_{\lambda }{e}^{\lambda }\in \mathbb{Z}{e}^X,\text{we define}{\chi }^{\text{Fr}}=\sum _{\lambda }{n}_{\lambda }{e}^{p^{\lambda }}\text{.}$$ Steinberg's tensor-product theorem then essentially says that for $\lambda \in X^{+}$ and $\mu \in X^{♯}$ the module $M_{\lambda }^{\text{Fr}}\otimes M_{\mu }$ is irreducible and hence isomorphic to $M_{p\lambda +\mu }\text{;}$ in particular, $M_{\lambda }^{\text{Fr}}\cong M_{p\lambda }\text{.}$ It follows that if $\lambda \in X^{+}$ has the $p\text{-expansion}$ $$\lambda ={\mu }_0+{\mu }_{1}p+\dots +{\mu }_k{p}^k,\text{with}{\mu }_j\in X^{♯}\text{for}0\leqq j\leqq k,$$ then $$M_{\lambda }\cong M_{{\mu }_0}\otimes M_{{\mu }_1}^{\text{Fr}}\otimes \dots \otimes M_{{\mu }_k}^{{\text{Fr}}^k},$$ where ${\text{Fr}}^j$ denotes the $j\text{-fold}$ iteration of Fr, and consequently, $$\begin{array}{cc}\text{char}\hspace{0.17em}{M}_{\lambda }=\prod _{j=0}^k{\left(\text{char}\hspace{0.17em}{M}_{{\mu }_j}\right)}^{{\text{Fr}}^j}\text{.}& \text{(4.1)}\end{array}$$

4.3. In order to define the constants $c_{\lambda \mu }$ in the title of this section, we need to quickly introduce certain $G\text{-modules}$ $V_{\lambda ,K}$ which are modelled after the irreducible ${𝔤}_{ℂ}\text{-module}$ $V_{\lambda ,ℂ}$ with highest weight $\lambda \in X^{+},$ so that the formal character of $V_{\lambda ,K}$ coincides with that of $V_{\lambda ,ℂ}\text{.}$ We caution in advance that $V_{\lambda ,K}$ is not uniquely determined by $\lambda$ alone; however, there are only a finite number of choices (up to isomorphism). The formal character of $V_{\lambda ,K}$ is determined by $\lambda ,$ and is given by Weyl's character formula as in characteristic 0; thus $\text{char}\hspace{0.17em}{V}_{\lambda ,K}$ is independent of $p\text{.}$ Also (by Proposition 2 below) the composition factors of $V_{\lambda ,K}$ are determined by $\lambda$ alone; these are far from being independent of $p$ though, and in fact for every $\lambda$ one can find a bound for $p$ beyond which $V_{\lambda ,K}$ is irreducible. This is not a new result, and follows from the "discriminant criterion" described below.

Let $V_{\lambda ,\mathbb{Z}}$ be a $U_{\mathbb{Z}}\text{-stable}$ lattice in $V_{\lambda ,ℂ}$ ("admissible $\mathbb{Z}\text{-form"}$ in the terminology of [Bor1970]). One sees that up to dilatation, i.e. multiplication by a nonzero scalar, there are only a finite number of choices for $V_{\lambda ,\mathbb{Z}}\text{;}$ this follows from the fact that if an arbitrary nonzero element $v_0$ in the highest-weightspace of $V_{\lambda ,ℂ}$ is required to be a primitive $\text{(}=$ nondivisible) vector in $V_{\lambda ,\mathbb{Z}}$ (as can always be ensured upon dilatation, as the highest-weight-space is 1-dimensional), then there exist two uniquely determined extremes for $V_{\lambda ,\mathbb{Z}}\text{:}$ $$V_{\lambda ,\mathbb{Z}}^{\text{min}}\leqq V_{\lambda ,\mathbb{Z}}\leqq V_{\lambda ,\mathbb{Z}}^{\text{max}}\text{.}$$ It is clear that the minimal lattice $V_{\lambda ,\mathbb{Z}}^{\text{min}}$ has to be simply the image $v_0{U}_{\mathbb{Z}}$ of $v_0$ under the action of $U_{\mathbb{Z}}$ on $V_{\lambda ,ℂ}\text{.}$ Let $v_0^{*}\in {\left(V_{\lambda ,ℂ}\right)}^{*}=$ the vector space dual to $V_{\lambda ,ℂ},$ be the linear function having value 1 at $v_0$ and vanishing at all the weight-spaces of $V_{\lambda ,ℂ}$ except the highest one. Then $v_0^{*}$ is a lowest-weight-vector of the contragredient ${𝔤}_{ℂ}\text{-module}$ ${\left(V_{\lambda ,ℂ}\right)}^{*}=V_{{\lambda }^{*},ℂ}$ with highest weight ${\lambda }^{*}={\lambda }^{{\tau }_0}=-{\lambda }^{{\sigma }_0}\text{.}$ Then, setting $V_{{\lambda }^{*},\mathbb{Z}}^{\text{min}}=v_0^{*}{U}_{\mathbb{Z}},$ one has $$V_{\lambda ,\mathbb{Z}}^{\text{max}}=\{x\in V_{\lambda ,ℂ}\hspace{0.17em}|\hspace{0.17em}y\left(x\right)\in \mathbb{Z}\hspace{0.17em}\text{for all}\hspace{0.17em}y\in V_{{\lambda }^{*},\mathbb{Z}}^{\text{min}}\}\text{.}$$ An alternate way of describing this is in terms of the "contragredient bilinear form" introduced by W. J. Wong [Won1971]:

Let $\theta$ denote the (unique) anti-automorphism of $U_{ℂ}$ which interchanges $e_i$ and $f_i$ for $1\leqq i\leqq l$ (and hence leaves the $h_i\text{'s}$ fixed); the existence of $\theta$ is classical and follows, for example, from the now well-known Harish-Chandra—Jacobson—Serre presentation of the Lie algebra ${𝔤}_{ℂ}$ (equivalently, of the associative algebra $U_{ℂ}\text{)}$ in terms of the generators $\{e_1,e_2,\dots ,e_l,f_1,f_2,\dots ,f_l\}\text{.}$ Then $V_{\lambda ,ℂ}$ possesses a unique symmetric bilinear form $(,)$ such that $(v_0,v_0)=1$ and $(x·e_i,y)=(x,y·f_i)$ for $1\leqq i\leqq l$ and $x,y\in V_{\lambda ,ℂ}\text{;}$ more generally $(x·u,y)=(x,y·u^{\theta })$ for $u\in U_{ℂ},$ and $(,)$ is determined by the fact that $(v_0·u,v_0·u\prime )=c$ if the projection of $v_{0}u\prime u^{\theta }$ on the highest-weight-space along the sum of all other weight-spaces is $c{v}_0\text{.}$ The restriction of $(,)$ to $V_{\lambda ,\mathbb{Z}}^{\text{min}}$ takes integer values, and one has $$V_{\lambda ,\mathbb{Z}}^{\text{max}}=\{x\in V_{\lambda ,ℂ}\hspace{0.17em}|\hspace{0.17em}(x,y)\in \mathbb{Z}\hspace{0.17em}\text{for all}\hspace{0.17em}y\in V_{\lambda ,\mathbb{Z}}^{\text{min}}\}\text{.}$$

The "reduction mod $p\text{"}$ $V_{\lambda ,K}=V_{\lambda ,\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }K$ of any $U_{\mathbb{Z}}\text{-stable}$ lattice $V_{\lambda ,\mathbb{Z}}$ in $V_{\lambda ,ℂ},$ becomes a $G\text{-module}$ in a natural manner [Bor1970] (here simply-connectedness of $G$ is required, just as in characteristic 0), such that the associated infinitesimal action of ${𝔤}_K$ coincides with that of ${𝔤}_{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }K$ (which acts naturally on $V_{\lambda ,\mathbb{Z}}\underset{\mathbb{Z}}{\otimes }K,$ since ${𝔤}_{\mathbb{Z}}\subset U_{\mathbb{Z}}\text{)},$ with which ${𝔤}_K$ has already been identified. The module $V_{\lambda ,K}$ need not be irreducible as we shall see presently, and its internal structure may vary with the choice of $V_{\lambda ,\mathbb{Z}}\text{.}$ Perhaps $V_{\lambda ,K}$ is always indecomposable; this is certainly true of the two extreme choices of $V_{\lambda ,\mathbb{Z}}\text{.}$ In fact $V_{\lambda ,K}^{\text{min}}=V_{\lambda ,\mathbb{Z}}^{\text{min}}\underset{\mathbb{Z}}{\otimes }K$ has a unique maximal proper $G\text{-submodule}$ (contained in the sum of all the weight-spaces except the highest one), the quotient by which is the irreducible module $M_{\lambda }\text{;}$ and on the other extreme $$V_{\lambda ,\mathbb{Z}}^{\text{max}}\underset{\mathbb{Z}}{\otimes }K=V_{\lambda ,K}^{\text{max}}={\left(V_{{\lambda }^{*},K}^{\text{min}}\right)}^{*}$$ has $M_{\lambda }$ as its unique irreducible submodule, on account of the duality given by the last equality. The injection $$\begin{array}{cccc}{i}_{\lambda ,\mathbb{Z}}:& V_{\lambda ,\mathbb{Z}}^{\text{min}}& \hookrightarrow & V_{\lambda ,\mathbb{Z}}^{\text{max}}\end{array}$$ gives, on reduction mod $p,$ only a homomorphism $$\begin{array}{cccc}{i}_{\lambda ,K}:& V_{\lambda ,K}^{\text{min}}& ⟶& V_{\lambda ,K}^{\text{max}},\end{array}$$ whose kernel is the maximal proper submodule of $V_{\lambda ,\mathbb{Z}}^{\text{min}}$ and image is the unique copy of $M_{\lambda }$ in $V_{\lambda ,K}^{\text{max}}\text{;}$ in other words, $i_{\lambda ,K}$ factors through $M_{\lambda }\text{.}$ Hence we have the "discriminant criterion": $V_{\lambda ,K}$ is irreducible if and only if $p$ does not divide the index $[V_{\lambda ,\mathbb{Z}}^{\text{max}}:V_{\lambda ,\mathbb{Z}}^{\text{min}}]\text{;}$ this index equals the discriminant of Wong's contragredient form $(,)$ on $V_{\lambda ,\mathbb{Z}}^{\text{min}}$ described above (cf. [Bor1970, Theorem 2J). It does not seem easy to give a formula for this number. (See ${\text{Footnote}}^3\text{.)}$ It would be particularly interesting to determine all "small" prime divisors of this index for each $\lambda \text{;}$ prime divisors not exceeding the Coxeter number $h_{\Delta }$ of $\Delta$ are of special interest, since the ideas of the next section provide an alternate method towards deciding the irreducibility of $V_{\lambda ,K}$ particularly when $p>h_{\Delta }$ (cf. Proposition 3).

It may be mentioned at this point that for the case of type $A_1,$ i.e. for $G\cong {SL}_2,$ all $V_{\lambda ,K}$ for $\lambda \in X^{♯}$ are irreducible. But this is an exceptional situation; it will be shown in a later paper that when $\Delta$ is not of type $A_1,$ there are values of $\lambda \in X^{♯}$ (in fact, for large $p,$ a sizeable percentage of values exist) such that $V_{\lambda ,K}$ is not irreducible. The first example of a non-irreducible $V_{\lambda ,K}$ with $\lambda \in X^{♯}$ was given by Curtis [Cur1960, p. 859], for type $A_2$ $(G\cong {SL}_3)\text{.}$

4.4. Let us write ${\bar{V}}_{\lambda }=V_{\lambda ,K}^{\text{min}},$ and for $\mu \in X^{+}$ define $c_{\lambda \mu }$ to be the multiplicity ${[{\bar{V}}_{\lambda }:M_{\mu }]}_G,$ of the occurrence of $M_{\mu }$ as a composition factor of the $G\text{-module}$ ${\bar{V}}_{\lambda }\text{.}$ (See ${\text{Footnote}}^4\text{.)}$ It is clear that $c_{\lambda \lambda }=1,$ and that $c_{\lambda \mu }\ne 0$ implies $\text{(}\mu$ is a weight of ${\bar{V}}_{\lambda }$ and therefore) $\mu \leqq \lambda \text{;}$ in particular, $\lambda -\mu \in X\prime$ is a necessary condition for $c_{\lambda \mu }\ne 0\text{.}$ We have ${\bar{V}}_{\lambda }{\sim }_G\sum _{\mu \in X^{+}}{c}_{\lambda \mu }{M}_{\mu },$ where ${\sim }_G$ denotes equivalence in the Grothendieck ring of the category of $G\text{-modules,}$ and thus $$\begin{array}{cc}\sum _{\mu }{c}_{\lambda \mu }·\text{char}\hspace{0.17em}{M}_{\mu }=\text{char}\hspace{0.17em}{\bar{V}}_{\lambda }=\frac{\sum _{\sigma \in W}\text{det}\hspace{0.17em}\sigma ·e^{\left({(\lambda +\delta )}^{\sigma }\right)}}{\sum _{\sigma \in W}\text{det}\hspace{0.17em}\sigma ·e^{\left({\lambda }^{\sigma }\right)}}& \text{(4.2)}\end{array}$$ where the second equality comes from the Weyl's character formula for $V_{\lambda ,ℂ}\text{.}$

The linear equations (4.2) for the "unknowns" ${\chi }_{\mu }=\text{char}\hspace{0.17em}{M}_{\mu }$ can be solved in terms of the "known" quantities ${\bar{\chi }}_{\mu }=\text{char}\hspace{0.17em}{\bar{V}}_{\mu },$ in the form ${\chi }_{\lambda }=\sum _{\mu }{\gamma }_{\lambda \mu }{\bar{\chi }}_{\mu },$ where the matrix ${\left[{\gamma }_{\lambda \mu }\right]}_{X^{+}}$ is the inverse of the infinite matrix ${\left[c_{\lambda \mu }\right]}_{X^{+}},$ the outer subscript $X^{+}$ denoting the set over which the matrix indices $\lambda ,\mu$ run. With respect to any enumeration of the set $X^{+}$ which is stronger than the partial order $\leqq$ (definition at the end of § 4.1), the matrix ${\left[c_{\lambda \mu }\right]}_{X^{+}}$ is unipotent in the sense that it is triangular with diagonal entries all 1. As a matter of fact, in order to compute ${\chi }_{{\mu }_0}$ for given ${\mu }_0\in X^{+},$ one needs to consider (4.2) only for $\lambda$ ranging over the finite set $Y=Y\left({\mu }_0\right)$ consisting of those $\lambda \in X^{+}$ such that there exists a sequence ${\mu }_1,{\mu }_2,\dots ,{\mu }_{k-1},{\mu }_k=\lambda$ with $c_{{\mu }_{j-1}{\mu }_j}\ne 0$ for $1\leqq j\leqq k\text{.}$ Since for $\lambda \in Y,$ $\mu \in X^{+},$ $$c_{\lambda \mu }\ne 0\text{implies}\mu \in Y\left(\lambda \right)⫅Y\left({\mu }_0\right)=Y,$$ we have for $\lambda \in Y,$ $$\sum _{\mu \in Y}{c}_{\lambda \mu }{\chi }_{\mu }={\bar{\chi }}_{\lambda },$$ i.e. only the finite segment ${\left[c_{\lambda \mu }\right]}_Y$ of ${\left[c_{\lambda \mu }\right]}_{X^{+}}$ needs to be inverted.

Proposition 1 (Character formula). There exist unique integers ${\gamma }_{\lambda \mu }$ for $\lambda ,\mu \in X^{+},$ such that for given $\lambda$ only finitely many ${\gamma }_{\lambda \mu }$ are nonzero and $$\begin{array}{cc}\text{char}\hspace{0.17em}{M}_{\lambda }=\frac{\sum _{\mu }(\sum _{\sigma \in W}\text{det}\hspace{0.17em}\sigma ·{\gamma }_{\lambda \mu }·e^{\left({(\mu +\delta )}^{\sigma }\right)})}{\sum _{\sigma \in W}\text{det}\hspace{0.17em}\sigma ·e^{\left({\delta }^{\sigma }\right)}}\text{.}& \text{(4.3)}\end{array}$$

By virtue of the standard fact that the set $\left\{{\bar{\chi }}_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in X^{+}\}$ is a $\mathbb{Z}\text{-free}$ basis of the $W\text{-invariants}$ in the group-ring $\mathbb{Z}·e^X$ (see [Bou1968]: VI. 3.4, Proposition 3 and Example 2, on p. 187-188), and the fact that the matrix ${\left[{\gamma }_{\lambda \mu }\right]}_{X^{+}}$ in $\sum _{\mu }{\gamma }_{\lambda \mu }{\bar{\chi }}_{\mu }={\chi }_{\lambda }$ is unipotent, one finds that $\left\{{\chi }_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in X^{+}\}$ is also a $\mathbb{Z}\text{-free}$ basis of the formal characters of $G\text{-modules.}$ Hence we have

Proposition 2. The composition factors of a $G\text{-module}$ are determined by its formal character.

Because of (4.1) one is primarily interested in computing the formal characters of the $p^l$ basic irreducible $G\text{-modules}$ with highest weight in $X^{♯}\text{.}$ We claim that these are completely determined by our knowledge of the "decomposition" $${\bar{V}}_{\lambda }{\sim }_G\sum _{\mu \in X^{+}}{c}_{\lambda \mu }{M}_{\mu }$$ for $\lambda$ ranging in $X^{♯}$ alone. To see this, let us impose the new partial-order ${\leqq }^{♯}$ on $X^{♯},$ defined to be the weakest one such that whenever ${\mu }_0+{\mu }_{1}p+\dots +{\mu }_k{p}^k\leqq \lambda$ with $\lambda ,{\mu }_0,{\mu }_1,\dots ,{\mu }_k$ all in $X^{♯},$ one has ${\mu }_i{\leqq }^{♯}\lambda$ for $0\leqq i\leqq k\text{;}$ and let $\leqq \prime$ denote any linear order on $X^{♯}$ stronger than ${\leqq }^{♯}\text{.}$ Then ${\chi }_{\lambda }$ can be computed, recursively with respect to $\leqq \prime ,$ i.e. assuming that ${\chi }_{\mu }$ is known for all $\mu \leqq \prime \lambda ,$ simply by substitution in $${\chi }_{\lambda }={\bar{\chi }}_{\lambda }-\sum _{\underset{\mu \ne \lambda }{\mu \in X^{+}}}{c}_{\lambda \mu }{\chi }_{\mu },$$ where the ${\chi }_{\mu }\text{'s}$ appearing on the right-hand side are known by virtue of (4.1) and the definition of ${\leqq }^{♯}\text{.}$ Our claim is proved; and as a corollary we note that a knowledge of ${\left[c_{\lambda \mu }\right]}_{\lambda \in X^{♯},\mu \in X^{+}}$ determines, by virtue of Proposition 2, the entire matrix ${\left[c_{\lambda \mu }\right]}_{X^{+}}\text{.}$ It would be very interesting to devise a formula (short of a computer algorithm) which enables one to write down, efficiently and explicitly, the decompositions of ${\bar{V}}_{\lambda }$ for all $\lambda \in X^{+}$ in terms of those for $\lambda \in X^{♯}$ alone. (Here and elsewhere in this paper, by the "decomposition" of a module we mean that in the Grothendieck group of the category of modules at hand.)

4.5. When we substitute ${\chi }_{\mu }=\prod _j{\left({\chi }_{{\mu }_j}\right)}^{{\text{Fr}}^j}$ for $\mu =\sum _j{\mu }_j{p}^j,$ with all ${\mu }_j\text{'s}$ in $X^{♯},$ in $\sum _{\mu }{c}_{\lambda \mu }{\chi }_{\mu }={\bar{\chi }}_{\lambda },$ what we get is a set of nonlinear equations in $\left\{{\chi }_{\nu }\hspace{0.17em}\right|\hspace{0.17em}\nu \in X^{♯}\}\text{.}$ However, under certain circumstances, it is possible to replace these by a set of linear equations $$\begin{array}{cc}\sum _{\mu \in X^{♯}}{C}_{\lambda \mu }{\chi }_{\mu }={\chi }_{\lambda }\text{for}\lambda \in X^{♯},& \text{(4.4)}\end{array}$$ where the coefficients $C_{\lambda \mu }$ are no longer integers, but instead lie in the group-ring $\mathbb{Z}·e^X$ itself; in fact $C_{\lambda \mu }$ is a $W\text{-invariant}$ element of the subring $\mathbb{Z}·e^{pX}$ consisting of $\left\{{\chi }^{\text{Fr}}\hspace{0.17em}\right|\hspace{0.17em}\chi \in e^X\}\text{.}$ To this end let us define $$Y_0=\{\nu \in X^{+}\hspace{0.17em}|\hspace{0.17em}{c}_{\lambda ,\mu +\nu p}\ne 0\text{for some}\lambda ,\mu \in X^{♯}\};$$ then $$\begin{array}{ccc}{\chi }_{\lambda }& =& \sum _{\underset{\nu \in Y_0}{\mu \in X^{♯}}}{c}_{\lambda ,\mu +\nu p}·{\chi }_{\mu +\nu p}\\ & =& \sum _{\mu \in X^{♯}}(\sum _{\nu \in Y_0}{c}_{\lambda ,\mu +\nu p}·{\chi }_{\nu }^{\text{Fr}}){\chi }_{\mu },\end{array}$$ which gives (4.4) with $$\begin{array}{cc}{C}_{\lambda \mu }={(\sum _{\nu \in Y_0}{c}_{\lambda ,\mu +\nu p}·{\chi }_{\nu })}^{\text{Fr}}\text{.}& \text{(4.5)}\end{array}$$ Our "coefficients" in (4.4) till now are themselves polynomials in the unknowns $\left\{{\chi }_{\nu }\hspace{0.17em}\right|\hspace{0.17em}\nu \in X\}\text{.}$ If for a given value of $p$ it can be shown that ${\bar{V}}_{\nu }$ is irreducible for $\nu \in Y_0,$ one can replace ${\chi }_{\nu }$ in (4.5) by the known quantity ${\bar{\chi }}_{\nu }\text{;}$ in Corollary to Proposition 3 in § 5, we shall see that such is indeed the case at least for $p\geqq 2h_{\Delta }-3,$ using the fact that $$\begin{array}{cc}{Y}_0⫅\{\sum _i{m}_i{\lambda }_i\in X^{+}\hspace{0.17em}|\hspace{0.17em}\sum _{i=1}^l{m}_i{n}_i\leqq h_{\Delta }-2\},& \text{(4.6)}\end{array}$$ where $h_{\Delta }=\left(\sum _i{n}_i\right)+1$ is the Coxeter number as in § 1.

To see (4.6) we note that $$\begin{array}{ccc}{Y}_0& ⫅& \{\nu \in X^{+}\hspace{0.17em}|\hspace{0.17em}\mu +p\nu \leqq \lambda \text{for some}\lambda ,\mu \in X^{♯}\}\\ & ⫅& \{\nu \in X^{+}\hspace{0.17em}|\hspace{0.17em}⟨\mu +p\nu ,-{\alpha }_0^{\vee }⟩\leqq ⟨\lambda ,-{\alpha }_0^{\vee }⟩\text{for some}\lambda ,\mu \in X^{♯}\}\\ & ⫅& \{\nu \in X^{+}\hspace{0.17em}|\hspace{0.17em}⟨p\nu ,-{\alpha }_0^{\vee }⟩\leqq ⟨(p-1)\delta ,-{\alpha }_0^{\vee }⟩\}\\ & =& \{\nu \in X^{+}\hspace{0.17em}|\hspace{0.17em}⟨\nu ,-{\alpha }_0^{\vee }⟩\leqq \frac{p-1}{p}(h_{\Delta }-1)\},\end{array}$$ where the second (resp. the third) inclusion follows in view of the fact that $-{\alpha }_0$ has non-negative inner product with all the fundamental roots (resp. all the fundamental weights), and hence also with all elements of ${\mathbb{Z}}_{+}·{\Delta }_{+}$ (resp. all elements of $X^{♯}\subset X^{+},$ and so also all elements of the form $(p-1)\delta -\lambda$ for $\lambda \in X^{♯}\text{).}$ The inclusion (4.6) follows in view of the fact that $h_{\Delta }=\frac{2}{r}l$ an integer. This also shows that $Y_0⫅X^{♯}$ if $p\geqq h_{\Delta }-2\text{.}$ It is shown in § 5.4 (Example 2) that $Y_0=\left\{0\right\}$ for types $A_2$ and $B_2$ (and of course also for type $A_1\text{).}$ It seems likely that for all other types $Y_0⫋\left\{0\right\}\text{.}$

Once again, the linear equations (4.4) can be solved (in those cases where $C_{\lambda \mu }\text{'s}$ are known) in the same way as (4.2) earlier, because the matrix ${\left[C_{\lambda \mu }\right]}_{X^{♯}}$ is unipotent with respect to $\leqq \prime$ (any linear-order on $X^{♯}$ stronger than ${\leqq }^{♯}\text{):}$ $$\{\begin{array}{c}{C}_{\lambda \lambda }=1=\text{the identity element of}\hspace{0.17em}\mathbb{Z}·e^X\text{, and}\\ C_{\lambda \mu }\ne 0\hspace{0.17em}\text{implies}\hspace{0.17em}\text{(}\mu +\nu p\leqq \lambda \hspace{0.17em}\text{for some}\hspace{0.17em}\nu \in X^{+}\hspace{0.17em}\text{and hence)}\hspace{0.17em}\mu {\leqq }^{♯}\lambda \text{.}\end{array}$$ The fact that $C_{\lambda \mu }\text{'s}$ are themselves formal characters is no obstacle to inverting the unipotent matrix as the inverse can be explicitly written down as a polynomial in the $C_{\lambda \mu }\text{'s.}$ In fact there is no need to handle the entire matrix of size as large as $\left|X^{♯}\right|=p^l\text{;}$ in order to compute ${\chi }_{{\mu }_0}$ for ${\mu }_0\in X^{♯}$ one need consider (4.4) simply for $\lambda$ ranging in the set $Y\prime =Y\prime \left({\mu }_0\right)$ consisting of $\lambda \in X^{♯}$ such that there exists a sequence ${\mu }_1,{\mu }_2,\dots ,{\mu }_k$ with $C_{{\mu }_{j-1}{\mu }_j}\ne 0$ for $1\leqq j\leqq k\text{.}$ The validity of the "Harish-Chandra principle" to be discussed in § 5 implies that the cardinality of $Y\prime$ never exceeds $\left|W\right|,$ which is quite comfortable for large $p\text{.}$ Note that $Y\left(\mu \right)⫅Y\prime \left({\mu }_0\right)+p{Y}_0\text{;}$ the set $Y_0$ can also be "relativized", i.e. replaced by a subset $Y_0\left({\mu }_0\right)$ of cardinality bounded by a number independent of $p,$ which would show that $\left|Y\right|$ is also bounded by a number independent of $p\text{.}$

${}^3$ (Added in Proof.) This problem does not seem intractable now. First note that the required index $N_{\lambda }=[V_{\lambda ,\mathbb{Z}}^{\text{max}}:V_{\lambda ,\mathbb{Z}}^{\text{min}}]$ for $\lambda \in X^{+},$ is the product of the partial indices $N_{\lambda ,\mu }=[V_{\lambda ,\mathbb{Z}\left(\mu \right)}^{\text{max}}:V_{\lambda ,\mathbb{Z}\left(\mu \right)}^{\text{min}}]$ as $\mu$ ranges over all the weights (since every $U_{\mathbb{Z}}\text{-stable}$ $\mathbb{Z}\text{-form}$ is compatible with the weight-space decomposition); clearly it suffices to determine $N_{\lambda ,\mu }$ for $\mu \in X^{+}\text{.}$ I have solved this problem only when $\mu$ is "sufficiently close" to $\lambda$ in this special sense: the dimension of the $\mu \text{-weightspace}$ of $V_{\lambda ,ℂ}$ equals $P(\lambda -\mu )$ (which is the leading term in Kostant's multiplicity formula) where $P\left(\right)$ is Kostant's partition function. Based on that result it seems plausible that the value of $N_{\lambda ,\mu }$ is given in general by $$N_{\lambda ,\mu }=\prod _{\alpha \in {\Delta }^{+}}\prod _{j\in {\mathbb{Z}}^{+}}{\left(\genfrac{}{}{0ex}{}{(\lambda +\delta )\left(h_{\alpha }\right)-1}{j}\right)}^{m_j\left(\alpha \right)}\text{(}\left(\genfrac{}{}{0ex}{}{·}{·}\right)\hspace{0.17em}\text{being the usual binomial symbol)}$$ where $m_j\left(\alpha \right)=m_j^{\lambda \text{;}\mu }\left(\alpha \right)$ is the dimension of the subspace of $V_{\lambda ,ℂ\left(\mu \right)}$ annihilated by $e_{\alpha }^{j+1}$ but not by $e_{\alpha }^j\text{;}$ when $\mu$ is sufficiently close to $\lambda$ it is easily seen that $m_j\left(\alpha \right)=P(\lambda -\mu -j\alpha )-P(\lambda -\mu -(j+1)\alpha )\text{.}$ The suggested expression for $N_{\lambda ,\mu }$ is of particular interest for $\mu =0,$ since it appears in a crucial manner in the work of K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan (Ann. Math. 85 (1967), Theorem 4.2 on page 424) on the principal series representations of complex semi-simple Lie groups; cf. § 8.5.10(b) in Dixmier's book cited in Footnote ${}^5\text{.}$ My proof of this formula (for $\mu$ sufficiently close to $\lambda \text{)}$ is obtained by refining certain ideas of N. N. Sapovalov (cf. § 7.8.23 in Dixmier's book) that are not unrelated to the computations of the 3 authors quoted above; it will be published in due course.

In this connection, reader's attention may also be drawn to N. Burgoyne's ideas in § 1 of "Modular representations of some finite groups" in Proc. of Symposia in Pure Mathematics, Vol. XXI, AMS, Providence, 1971.

${}^4$ (Added in Proof.) In a recent preprint (titled 'On the modular representations of the general linear and symmetric groups') R. W. Carter and G. Lusztig have introduced the imaginative term "Weyl Module" for ${\bar{V}}_{\lambda },$ albeit in the special situation of the general (or the special) linear groups. The allusion to Weyl is to emphasize the fact that the dimension is given by Weyl's formula as in characteristic zero: $\text{dim}\hspace{0.17em}{\bar{V}}_{\lambda }=D(\lambda +\delta )\text{;}$ in the special situation of Carter and Lusztig it has an additional association with the treatment given by H. Weyl (in his classic monograph "Classical groups") to the problem of decomposing the tensor spaces in accordance with certain symmetries. Our notation ${\bar{V}}_{\lambda }$ conforms that in [Hum1971].

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