Steinberg module

Let , the group of all invertible -matrices over the finite field with elements and characteristic , let be the subgroup of all superdiagonal elements, let be the subgroup of elements of whose diagonal entries are all , and let be the subgroup of permutation matrices. In the group algebra of over any field of characteristic or , the element

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

In case the characteristic of equals , has the following further properties [a5]. It is the only module (for ) 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, plays a prominent role in ongoing work in the still (2000) unresolved problem of determining all of the irreducible -modules (with characteristic still equal to ), or equivalently, as it turns out, of determining all of the irreducible rational -modules, where is the algebraic group obtained from by replacing by its algebraic closure , i.e., where is any simple affine algebraic group of characteristic (see [a6]). This equivalence comes from the fact that every irreducible -module extends to a rational -module. In particular, extends to the -module with highest weight times the sum of the fundamental weights, which is accordingly also denoted , or since there is one such -module for each . These modules are ubiquitous in the module theory of 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 , with the characteristic of now equal to , are as follows. According to C.W. Curtis [a2]

in which runs through the ( equal to the rank of ) (parabolic) subgroups of containing , is the -module induced by the trivial -module, and the or is used according as the rank of is even or odd. For , for example, there is one for each solution of (, each ); it consists of all of the elements of that are superdiagonal in the corresponding block matrix form. A third construction, due to L. Solomon and Tits [a7], yields as the top homology space for the Tits simplicial complex or Tits building of , formed as follows: corresponding to each parabolic subgroup there exists an -simplex in , and is a facet of just when contains . These three constructions are, in fact, closely related to each other (see [a9]). In particular, the idempotent used at the start can be identified with an -sphere in the Tits building, the sum over corresponding to a decomposition of the sphere into simplexes: in the usual action of on the reflecting hyperplanes divide into oriented spherical simplexes, each of which is a fundamental domain for . Finally, has a simple presentation (as a linear space). It is generated by the Borel subgroups of subject only to the relations that for every parabolic subgroup of rank the sum of the Borel subgroups that it contains is .

There are also infinite-dimensional versions of the above constructions, usually for reductive Lie groups — real, complex or -adic — such as . The -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 and ), 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 .

References

[a1]	A. Borel, J-P. Serre, "Cohomologie d'immeubles et de groupes -arithmétiques" Topology , 15 (1976) pp. 211–232 MR447474
[a2]	C.W. Curtis, "The Steinberg character of a finite group with -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 -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