Rational representation

of an algebraic group $ G $ over an algebraically closed field $ k $

A linear representation of $ G $ on a finite-dimensional vector space $ V $ over $ k $ which is a rational homomorphism of $ G $ into $ \mathop{\rm GL}\nolimits (V) $. One also says that $ V $ is a rational $ G $- module. Direct sums and tensor products of a finite number of rational representations of $ G $ are rational representations. Subrepresentations and quotient representations of any rational representation are rational representations. Symmetric and exterior powers of any rational representation are rational representations. The representation contragredient to a rational representation is a rational representation.

If $ G $ is finite, then each of its linear representations will be a rational representation, and the theory of rational representations coincides with the theory of representations of finite groups (cf. Representation of a group). To a large extent, specific methods of the theory of linear algebraic groups are used to study rational representations in case the group under consideration is connected, and the most thoroughly developed theory is that of rational representations of connected semi-simple algebraic groups. Let $ G $ be such a group, $ T $ a maximal torus, $ X (T) $ its group of rational characters (written additively), $ \Sigma $ the root system of $ G $ with respect to $ T $, $ W $ its Weyl group, and $ ( \ ,\ ) $ a $ W $- invariant positive-definite non-degenerate scalar product on $ X(T) \otimes \mathbf R $. Now let $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ be a rational representation. The restriction of $ \phi $ to $ T $ decomposes into a direct sum of one-dimensional representations; more precisely, $$ V = \oplus _ {\chi \in P _ \phi } V ( \chi ) , $$ where $ P _ \phi \subset X (T) $ is some set of characters of $ T $, called the weights of the representation, and $$ V ( \chi ) = \{ {v \in V} : {\phi (t) v = \chi (t) v \forall t \in T} \} \neq 0 . $$ The set of weights $ P _ \phi $ is invariant under the action of $ W $.

If $ \mathop{\rm char}\nolimits \ k = 0 $, then every rational representation of $ G $ is completely reducible, but if $ \mathop{\rm char}\nolimits \ k > 0 $, then this is not so (see Mumford hypothesis). Whatever the characteristic of $ k $, however, there is a complete description of the irreducible rational representations.

Let $ B $ be a Borel subgroup in $ G $ containing $ T $ and let $ \Delta $ be the set of simple roots in $ \Sigma $ defined by $ B $. Identify the group $ X(B) $ of rational characters of $ B $ with $ X (T) $. In the space $ V $, for any irreducible rational representation $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ there is a unique one-dimensional weight subspace $ V ( \delta _ \phi ) $, $ \delta _ \phi \in P _ \phi $, invariant under $ B $. The character $ \delta _ \phi $ is called the highest weight of the irreducible rational representation $ \phi $; it is dominant, i.e. $ ( \delta _ \phi ,\ \alpha ) \geq 0 $ for any $ \alpha \in \Delta $, and every other weight $ \chi \in P _ \phi $ has the form $$ \chi = \delta _ \phi - \sum _ {\alpha \in \Delta} m _ \alpha \alpha , m _ \alpha \in \mathbf Z , m _ \alpha \geq 0 . $$ The mapping $ \phi \mapsto \delta _ \phi $ defines a bijection between the classes of equivalent irreducible rational representations and the dominant elements of $ X (T) $. An explicit construction of all irreducible rational representations can be obtained in the following way. Let $ k [ G ] $ be the algebra of regular functions on $ G $. Given any $ \chi \in X (T) = X (B) $, consider the subspace $$ k [ G ] _ \chi = \{ {f \in k [ G ]} : {f ( g b ) = \chi (b) f (g) \forall b \in B , g \in G} \} . $$ It is finite-dimensional and is a rational $ G $- module under the action of $ G $ by left translation. The geometric meaning of this space is as follows: it can be canonically identified with the set of regular sections of the one-dimensional homogeneous vector bundle over $ G / B $ determined by the character $ - \chi $. Let $ w _{0} \in W $ be the element mapping positive roots into negative ones. If $ k [ G ] _{ {-} w _{0} ( \chi )} \neq 0 $, then $ \chi $ is a dominant character and the minimal non-zero $ G $- submodule in $ k [ G ] _{ {-} w _{0} ( \chi )} $ is an irreducible rational $ G $- module with highest weight $ \chi $. Every irreducible rational $ G $- module can be obtained in this way. If $ \mathop{\rm char}\nolimits \ k = 0 $, then the $ G $- module $ k [ G ] _{ {-} w _{0} ( \chi )} $ is itself irreducible.

To obtain irreducible rational representations, one often applies the above-mentioned operations to given rational representations. For example, if $ \phi _{i} $ is an irreducible rational representation with highest weight $ \chi _{i} $, $ i = 1 \dots d $, then some quotient representation of $ \phi _{1} \otimes \dots \otimes \phi _{d} $ is an irreducible rational representation with highest weight $ \chi _{1} + \dots + \chi _{d} $( it is called the Cartan product of $ \phi _{1} \dots \phi _{d} $). If $ \phi $ is an irreducible rational representation with highest weight $ \chi $, then some quotient representation of $ S _ \phi ^{d} $ is an irreducible rational representation with highest weight $ d \chi $. Moreover $ \phi ^{*} $ is irreducible and its highest weight is $ - w _{0} ( \chi ) $.

Let $ \mathfrak g $ be the Lie algebra of $ G $( cf. Lie algebra of an algebraic group). If $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ is a rational representation, then its differential $ d \phi $ is a representation of the Lie algebra $ \mathfrak g $. A rational representation $ \phi $ is called infinitesimally irreducible if $ d \phi $ is an irreducible representation of the algebra $ \mathfrak g $. An infinitesimally-irreducible rational representation is irreducible, and when $ \mathop{\rm char}\nolimits \ k = 0 $, the converse is also true (which largely reduces the theory of rational representations of a group to the theory of representations of its Lie algebra). But when $ \mathop{\rm char}\nolimits \ k =p > 0 $, this is not so; the infinitesimally-irreducible rational representations in this case are just those irreducible rational representations with highest weight $ \chi $ for which $$ 0 \leq \frac{2 ( \chi ,\ \alpha )}{( \alpha ,\ \alpha )} < p \textrm{ for all } \alpha \in \Delta . $$ Moreover, all the irreducible rational representations can be constructed using the infinitesimally-irreducible ones. More precisely, if $ G $ is simply connected, that is, if $ X (T) $ coincides with the lattice of weights of the root system $ \Sigma $, then every irreducible rational representation factors uniquely into a tensor product of the form $$ \phi _{0} \otimes \phi _{1} ^{ \textrm Fr} \otimes \dots \otimes \phi _{d} ^{ \textrm Fr ^{d}} , $$ where $ \phi _{0} \dots \phi _{d} $ are infinitesimally irreducible, and $ \phi _{i} ^{ \textrm Fr ^{i}} $ is the representation obtained by applying the Frobenius automorphism $ a \mapsto a ^{ {p} ^{i}} $( $ a \in k $, $ p = \mathop{\rm char}\nolimits \ k $) to the matrix entries of the representation $ \phi _{i} $.

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See [a1], especially for complications in $ \mathop{\rm char}\nolimits \ k > 0 $.

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