The Steinberg representation is a remarkable irreducible representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1. The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation.

**Question:** What are the analogs of the Steinberg representation for sporadic simple groups ? (If there are any ideas for other finite groups beyond standard - it also welcome).

Wikipedia suggests: Some of the sporadic simple groups act as doubly transitive permutation groups so have a BN-pair for which one can define a Steinberg representation, but for most of the sporadic groups there is no known analogue of it.

However googling does not lead me to an answer even about that "some of sporadic ... " .

*Further question (cohomological construction) :*

At Mathoverflow D.Pasechnik answering S.Lentner question: (weak?) BN-Pair / Tits System for Sporadic Groups ? writes:
*Instead of starting from a weak BN-pair, one can weaken Tits' axioms from his "Local approach to buildings" to develop a theory dealing with sporadics.*

That may give a way to answer my question since it is known that Steinberg representation is realized in the cohomology group of the Bruhat–Tits building.

**Question 2** Are there similar cohomological constructions of "Steinberg" representation for sporadic groups ?

*Further question (mod-p reduction):*

In the remarkable (must&pleasure to read - imho) survey "The Steinberg representation" BAMS 1987 J. E. Humphreys takes modular representation point view on the Steinberg representation.
The point is: for G(F_p) "St" can be reduced mod "p".
Moreover the reduction is quite remarkble - it preserves irreducibility and some other properties (quote: *it is also a "principal
indecomposable" representation determining a "block" by itself. Moreover, the
character of the representation vanishes at all elements of G having order divisible by p)* - that follows from Brauer-Nesbit theory.

**Question 3** If there any mod-p properties of "St" for sporadic ?
The positive answer would have a strange conclusion - that some sporadics
might be considered as kind of groups defined over "F_p" for some "p".

**Motivation (one of):**

As described here: MO271067 I would hope to have some bijection(s) between irreducible representations of sporadic groups (and other groups also) and their conjugacy classes. Distinguishing some representations as "Steinberg like" would be quite helpful.

For example for Mathiew group M11 looking on the character table (e.g. here page 3 or Magma):

Class | 1 2 3 4 5 6 7 8 9 10

Size | 1 165 440 990 1584 1320 990 990 720 720

Order | 1 2 3 4 5 6 8 8 11 11

X.1 + 1 1 1 1 1 1 1 1 1 1

X.2 + 10 2 1 2 0 -1 0 0 -1 -1

X.3 0 10 -2 1 0 0 1 Z1 -Z1 -1 -1

X.4 0 10 -2 1 0 0 1 -Z1 Z1 -1 -1

X.5 + 11 3 2 -1 1 0 -1 -1 0 0

X.6 0 16 0 -2 0 1 0 0 0 Z2 Z2#2

X.7 0 16 0 -2 0 1 0 0 0 Z2#2 Z2

X.8 + 44 4 -1 0 -1 1 0 0 0 0

X.9 + 45 -3 0 1 0 0 -1 -1 1 1

X.10 + 55 -1 1 -1 0 -1 1 1 0 0

Z1 = i * sqrt(2) Z2 = (-1+i * sqrt(11))/2

One can easily guess to map pair of conjugacy classes of order 8 to pair of irreps X3,X4 and classes of order 11 to X.6 X.7 because

1) that are only 4 complex classes/irreps - that distiguishes 4 classes and 4 irreps

2) orders 8, 11 of classes correspond to degree of rationality of those irreps - thus we get: two classes of order 8 <-> two irreps X3,X4; two classes of order 11 <-> two irreps X6,X7;

**Bonus Question** What irrep is Steinberg for M11 and what
conjugacy class might correspond to it ?

More generally one may ask is there an analog of Alvis-Curtis duality for sporadics ? Is there splitting of classes and characters to semisimple/unipotent for sporadics ? Analogs of parabolic subgroups ? Etc... (i.e. can one extend properties from Lie groups to sporadics ? )

EDIT: From the comments it becomes clear that there is lot of research generalizing the Steinberg representation. Still it is not clear for the direction suggested in Wikipedia - from double transitive action to BN-pair , and hence to Steinberg reprsentation.

Subgroup complexesbyPeter Webb, pp. 349-365 in: ed. P. Fong, The Arcata Conference on Representations of Finite Groups, AMS Proceedings of Symposia in Pure Mathematics 47 (1987). This leads to the generalized Steinberg characters mentioned by Geoff in his last comment. BTW for the general linear groups and a prime $p$ NOT its characteristic, one should look at the Benson complex, i.e., the elementary abelian $p$-subgroups that contain ALL central elements of order $p$ of their centralizers. $\endgroup$6more comments