The Steinberg representation, also called the Steinberg module, is a distinguished irreducible representation associated with finite groups of Lie type. It was introduced by Robert Steinberg in the early 1950s and plays a central role in the representation theory of these groups.
Definition and construction
Let $G$ be a connected reductive algebraic group defined over a finite field $\mathbb{F}_q$, and let $G(\mathbb{F}_q)$ denote the corresponding finite group of Lie type. Choose a Borel subgroup $B$ of $G$ (defined over $\mathbb{F}_q$) with unipotent radical $U$. The Steinberg representation $\operatorname{St}_G$ of $G(\mathbb{F}_q)$ can be realized in several equivalent ways:
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Induced representation:
$$ \operatorname{St}G ;=; (-1)^{\operatorname{rank}(G)}; \operatorname{Ind}{B}^{G(\mathbb{F}_q)} \mathbf{1}, $$ where $\mathbf{1}$ denotes the trivial one‑dimensional representation of $B$ and the induction is taken in the sense of virtual characters. After cancellation of lower‑dimensional constituents, the resulting genuine representation is irreducible. -
Deligne–Lusztig theory:
The Steinberg representation arises as the (alternating) sum of the $\ell$-adic cohomology groups of the Deligne–Lusztig variety attached to the maximal torus $T$ that is split over $\mathbb{F}_q$. It is the unique irreducible constituent occurring with multiplicity one in the top‑degree cohomology. -
Highest‑weight description:
For the corresponding algebraic group $G$ over an algebraically closed field of characteristic $p$ (the characteristic of $\mathbb{F}_q$), the Steinberg module is the simple rational $G$-module with highest weight $(p-1)\rho$, where $\rho$ is the half‑sum of positive roots.
Key properties
| Property | Description |
|---|---|
| Dimension | $\dim \operatorname{St}_G = q^{N}$, where $N$ is the number of positive roots of $G$. Equivalently, $\dim \operatorname{St}_G = |
| Irreducibility | Over fields whose characteristic does not divide $ |
| eq p$), the Steinberg representation is irreducible. In characteristic $p$ it remains simple as a rational $G$-module. | |
| Self‑duality | The Steinberg representation is self‑dual; its character satisfies $\chi_{\operatorname{St}}(g)=\chi_{\operatorname{St}}(g^{-1})$. |
| Unipotent support | Its character takes the value $(-1)^{\operatorname{rank}(G)}$ on regular unipotent elements and vanishes on many other conjugacy classes, making it a prototypical unipotent character. |
| Tensor product | For any irreducible representation $V$ of $G(\mathbb{F}_q)$, the tensor product $V\otimes \operatorname{St}_G$ is again an irreducible representation if and only if $V$ is a “cuspidal” unipotent representation. |
| Compatibility with Frobenius | Under the Frobenius endomorphism defining the finite field structure, the Steinberg module is invariant, reflecting its construction from the whole algebraic group. |
Historical significance
The Steinberg representation was the first example of a “large” irreducible representation of a finite group of Lie type whose dimension is a power of the field size $q$. Its discovery led to the development of the theory of unipotent representations and influenced later work on Deligne–Lusztig characters, Lusztig’s classification of irreducible characters, and the theory of modular representations of algebraic groups.
Applications
- Cohomology of flag varieties: The top cohomology group of the flag variety $G/B$ with coefficients in a suitable field realizes the Steinberg representation.
- Algebraic combinatorics: The character values of the Steinberg representation appear in formulas for counting points on varieties over finite fields.
- Number theory: In the context of automorphic forms, Steinberg representations of $p$-adic groups correspond to certain highly ramified local components of automorphic representations.
References (selected)
- R. Steinberg, Lectures on Chevalley Groups, Yale University, 1968.
- P. Deligne and G. Lusztig, “Representations of Reductive Groups over Finite Fields,” Annals of Mathematics 103 (1976), 103–161.
- J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., American Mathematical Society, 2003.
The Steinberg representation remains a fundamental object in modern representation theory, linking algebraic groups, finite groups of Lie type, and geometric methods.