Steinberg module

Let $G = \operatorname{GL} _ { n } ( \mathbf{F} _ { q } )$, the group of all invertible $( n \times n )$-matrices over the finite field $\mathbf{F} _ { q }$ with $q$ elements and characteristic $p$, let $B$ be the subgroup of all superdiagonal elements, let $U$ be the subgroup of elements of $B$ whose diagonal entries are all $1$, and let $W$ be the subgroup of permutation matrices. In the group algebra $k [ G ]$ of $G$ over any field $k$ of characteristic $0$ or $p$, the element

\begin{equation*} e = \frac { | U | } { | G | } \left( \sum _ { b \in B } b \right) \left( \sum _ { w \in W } \operatorname { sign } ( w ) w \right) \end{equation*}

is an idempotent, called the Steinberg idempotent, and the $G$-module that it generates in $k [ G ]$ by right multiplication is called the Steinberg module (see [a8]) and is commonly denoted $\operatorname{St}$ (as are all modules isomorphic to it). A similar construction holds for any finite group $G$ of Lie type (and for any $B N$-pair, which is an axiomatic generalization due to J. Tits) defined over a field of characteristic $p$ with $B$ replaced by a Borel subgroup (which is a certain kind of solvable subgroup), $U$ by a maximal unipotent subgroup (cf. Unipotent group) of $B$ (which is also a Sylow $p$-subgroup of $G$; cf. also Sylow subgroup; $p$-group) and $W$ by the corresponding Weyl group. $\operatorname{St}$ is always irreducible and it has $\{ e u : u \in U \}$ as a basis, so that its dimension is $| U |$ (see [a8]). Its character values are given as follows [a3]. If $x \in G$ has order prime to $p$, then $\chi ( x )$ equals, up to a sign which can be determined, the order of a Sylow $p$-subgroup of the centralizer of $x$; otherwise it equals $0$.

In case the characteristic of $k$ equals $p$, $\operatorname{St}$ has the following further properties [a5]. It is the only module (for $G$) which is both irreducible and projective. As an irreducible module it is the largest (in dimension), and as a projective module it is the smallest since it is a tensor factor of every projective module. It follows that it is also self-dual and that every projective module is also injective and vice versa. Because of these remarkable properties, $\operatorname{St}$ plays a prominent role in ongoing work in the still (2000) unresolved problem of determining all of the irreducible $G$-modules (with characteristic $k$ still equal to $p$), or equivalently, as it turns out, of determining all of the irreducible rational $\overline { G }$-modules, where $\overline { G }$ is the algebraic group obtained from $G$ by replacing $\mathbf{F} _ { q }$ by its algebraic closure $\mathbf{F} _ { q }$, i.e., where $\overline { G }$ is any simple affine algebraic group of characteristic $p$ (see [a6]). This equivalence comes from the fact that every irreducible $G$-module extends to a rational $\overline { G }$-module. In particular, $\operatorname{St}$ extends to the $\overline { G }$-module with highest weight $q - 1$ times the sum of the fundamental weights, which is accordingly also denoted $\operatorname{St}$, or $\operatorname{St} _ { q }$ since there is one such $\overline { G }$-module for each $q = p , p ^ { 2 } , p ^ { 3 } , . .$. These modules are ubiquitous in the module theory of $\overline { G }$ and figure prominently, for example, in the proofs of many cohomological vanishing theorems and in W. Haboush's proof of the Mumford hypothesis (see [a4]).

Back in the finite case, some other constructions of $\operatorname{St}$, with the characteristic of $k$ now equal to $0$, are as follows. According to C.W. Curtis [a2]

\begin{equation*} \text{St} = \sum _ { P } \pm 1 _ { P } ^ { G }, \end{equation*}

in which $P$ runs through the $2 ^ { r }$ ($r$ equal to the rank of $G$) (parabolic) subgroups of $G$ containing $B$, $1 ^{ G } _ { P }$ is the $G$-module induced by the trivial $P$-module, and the $+$ or is used according as the rank $r _ { P }$ of $P$ is even or odd. For $G = \operatorname{GL} _ { n } ( \mathbf{F} _ { q } )$, for example, there is one $P$ for each solution of $n = a _ { 1 } + \ldots + a _ { s }$ ($1 \leq s \leq n$, each $a _ { i } \geq 1$); it consists of all of the elements of $G$ that are superdiagonal in the corresponding block matrix form. A third construction, due to L. Solomon and Tits [a7], yields $\operatorname{St}$ as the top homology space $H _ { r - 1 } ( C )$ for the Tits simplicial complex or Tits building $C$ of $G$, formed as follows: corresponding to each parabolic subgroup $P$ there exists an $( r - r _ { P } - 1 )$-simplex $S _ { P }$ in $C$, and $S _ { P }$ is a facet of $S _ { Q }$ just when $P$ contains $Q$. These three constructions are, in fact, closely related to each other (see [a9]). In particular, the idempotent $e$ used at the start can be identified with an $( r - 1 )$-sphere in the Tits building, the sum over $W$ corresponding to a decomposition of the sphere into simplexes: in the usual action of $W$ on $S ^ { r - 1 } \subset \mathbf{R} ^ { r }$ the reflecting hyperplanes divide $S ^ { r - 1}$ into $| W |$ oriented spherical simplexes, each of which is a fundamental domain for $W$. Finally, $\operatorname{St}$ has a simple presentation (as a linear space). It is generated by the Borel subgroups of $G$ subject only to the relations that for every parabolic subgroup of rank $1$ the sum of the Borel subgroups that it contains is $0$.

There are also infinite-dimensional versions of the above constructions, usually for reductive Lie groups — real, complex or $p$-adic — such as $\operatorname { GL} _ { n }$. The $p$-adic case most closely resembles the finite case. There, the affine Weyl group and a certain compact-open subgroup, called an Iwahori subgroup, come into play (in place of $W$ and $B$), and the three constructions agree. In [a1] several types of buildings, Curtis' formula and the Steinberg idempotent, in the guise of a homology cycle, all appear. In the infinite case the constructed object is sometimes called the Steinberg representation, sometimes the special representation.

References [a5] and [a9] are essays on $\operatorname{St}$.

References

[a1]	A. Borel, J-P. Serre, "Cohomologie d'immeubles et de groupes $S$-arithmétiques" Topology , 15 (1976) pp. 211–232 MR447474
[a2]	C.W. Curtis, "The Steinberg character of a finite group with $B N$-pair" J. Algebra , 4 (1966) pp. 433–441 MR201524
[a3]	C.W. Curtis, G.I. Lehrer, J. Tits, "Spherical buildings and the character of the Steinberg representation" Invent. Math. , 58 (1980) pp. 201–220 MR0571572 Zbl 0435.20024
[a4]	W. Haboush, "Reductive groups are geometrically reductive" Ann. of Math. , 102 (1975) pp. 67–83 MR0382294 Zbl 0316.14016
[a5]	J.E. Humphreys, "The Steinberg representation" Bull. Amer. Math. Soc. (N.S.) , 16 (1987) pp. 237–263 MR0876960 Zbl 0627.20024
[a6]	J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) MR0899071 Zbl 0654.20039
[a7]	L. Solomon, "The Steinberg character of a finite group with $B N$-pair" , Theory of Finite Groups (Harvard Symp.) , Benjamin (1969) pp. 213–221
[a8]	R. Steinberg, "Prime power representations of finite linear groups II" Canad. J. Math. , 9 (1957) pp. 347–351 MR0087659 Zbl 0079.25601
[a9]	R. Steinberg, "Collected Papers" , Amer. Math. Soc. (1997) pp. 580–586 Zbl 0878.20002