## Modular representations that look ordinary

Let $G$ be a finite group and let $(K, \mathcal{O}, k)$ be a $p$-modular system for $G$. Thus $K$ is a field of characteristic zero sufficiently large that if $V$ is any irreducible representation of $G$ then $\mathrm{End}_{KG}(V) \cong K$ and $K$ contains all the eigenvalues of the matrices representing elements of $G$ in their action on $V$, $\mathcal{O}$ is an integrally closed subring of $K$ containing all these eigenvalues, and $k$ is a field of prime characteristic $p$ obtained by quotienting $\mathcal{O}$ by a maximal ideal $\mathfrak{m}$, necessarily lying above $\langle p \rangle \unlhd \mathcal{O}$. By localizing if necessary, we may assume that all elements of $\mathcal{O}$ not in $\mathfrak{m}$ are invertible. So $\mathcal{O}$ is a local ring with unique maximal ideal $\mathfrak{m}$.

This is a standard setup for modular representation theory. The main purpose of this proof is to prove Theorem 2 below, which gives a precise description of a family of representations that look substantially the same, whether defined over $K$, $\mathcal{O}$ or $k$.

As a warm-up we prove the following result on ordinary characters.

Proposition 1. Let $\chi$ be an irreducible character of $G$ such that $\chi(1)$ is divisible by the highest power of $p$ dividing $|G|$. If $g \in G$ is a $p$-element then $\chi(g) = 0$.

We need the following fact: the central primitive idempotent in $KG$ corresponding to $\chi$ is

$e_\chi = \frac{\chi(1)}{|G|} \sum_{g \in G} \chi(g^{-1}) g.$

Note that the coefficients of $e_\chi$ are contained in $\mathcal{O}$.

Proof of Proposition 1. Let $P$ be a Sylow $p$-subgroup of $G$ containing $g$. Since

$(1) \qquad \mathcal{O}G = e_\chi \mathcal{O}G \oplus (1-e_\chi) \mathcal{O}G,$

considered as a right $\mathcal{O}G$-module, $e_\chi \mathcal{O}G$ is projective. Hence the restriction of $e_\chi \mathcal{O}G$ to $\mathcal{O}P$ is also projective. Now $\mathcal{O}P$ is indecomposable as a right $\mathcal{O}P$-module, since any direct sum decomposition induces a direct sum decomposition of the indecomposable $kP$-module $kP$ by reduction modulo $\mathfrak{m}$. Hence the unique projective indecomposable $\mathcal{O}P$-module is $\mathcal{O}P$ itself and $e_\chi \mathcal{O}G\downarrow_P \cong \mathcal{O}P \oplus \cdots \oplus \mathcal{O}P$. Restricting further we get

$(2)\qquad e_\chi \mathcal{O}G \downarrow_{\langle g \rangle} \cong \mathcal{O} \langle g \rangle^{\oplus \chi(1)/\mathrm{ord}(g)}.$

From the canonical basis for the right-hand we see that the character of $g$ acting on $e_\chi \mathcal{O}G$ is zero. But, by Wedderburn’s Theorem, $e_\chi KG$ is isomorphic to the direct sum of $\chi(1)$ copies of a simple $KG$-module with character $\chi$. Hence $\chi(1)\chi(g) = 0$.$\qquad\Box$

I find it quite striking that the modular theory gives a short proof of a result stated entirely in the language of ordinary (i.e. characteristic zero) representation theory. Moreover, from (2), we can easily get a stronger result: $\chi(g) = 0$ whenever $g$ has order divisible by $p$.

Outline proof. let $g$ have order $p^a c$ where $c$ is not divisible by $p$, let $g_p = g^c$, and let $g_{p'} = g^{p^a}$. Take a direct sum decomposition of $e_\chi KG$ into eigenspaces for the action of $g_{p'}$. By (2), $g_p$ has trace zero in its action on each eigenspace, hence so does $g$.

Proposition 1 is also a corollary of the following theorem. My first version of the proof did not make it entirely clear that $e_\chi \mathcal{O}G$ is a direct sum of isomorphic $\mathcal{O}G$-modules, each simple as a $KG$-module. To repair the gap, I borrowed the proof of (1) $\implies$ (2) on page 162 of this recent book by Peter Webb.

Theorem 2. Let $\chi$ be an irreducible character of $G$ such that $\chi(1)$ is divisible by the highest power of $p$ dividing $|G|$. There is a projective $\mathcal{O}G$-module $M$ with ordinary character $\chi$ such that $M/\mathfrak{m}M$ is a simple projective $kG$-module.

Proof. Let $M$ be a $\mathcal{O}G$-module such that $M_K = M \otimes_\mathcal{O} K$ is a simple $KG$-module. By Wedderburn’s Theorem, $e_\chi KG$ is isomorphic to $\mathrm{End}_K(M_K)$, thought of as the complete matrix algebra of $\chi(1) \times \chi(1)$ matrices with coefficients in $K$. Let $\rho : KG \rightarrow \mathrm{End}_K(M_K)$ be the corresponding ring homomorphism, with kernel $(1-e_\chi)KG$. The restriction of $\rho$ to $\mathcal{O}G$ has kernel $(1-e_\chi)\mathcal{O}G$. So it is sufficient to prove that the image of $\rho : \mathcal{O}G \rightarrow \mathrm{End}_K(M_K)$ is $\mathrm{End}_O(M)$, thought of as the complete matrix ring of $\chi(1) \times \chi(1)$ matrices with coefficients in $\mathcal{O}$. Since $M$ is a $\mathcal{O}G$-module, the matrix coefficients lie in $\mathcal{O}$.

The surjectivity of the restriction of $\rho$ follows from the following ‘averaging’ formula which expresses an arbitrary $\phi \in \mathrm{End}_\mathcal{O}(M)$ as something in its image:

$\phi = \frac{\chi(1)}{|G|} \sum_{g \in G} \mathrm{Tr}_M\bigl( \rho(g^{-1}) \phi\bigr) \rho(g).$

Subproof. Since the ring homomorphism $\rho: KG \rightarrow \mathrm{End}_K(M)$ is surjective, it is sufficient to prove this when $\phi = \rho(x)$ with $x \in G$; the right-hand side is

$\rho(x) \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1} x) \rho(x^{-1} g) = \rho(x) \rho(e_\chi) = \rho(x),$

as required.

We can now think of the right $\mathcal{O}G$-module $M$ very concretely, as those $d \times d$ matrices with coefficients in $\mathcal{O}$ that are zero outside their first row. Reducing mod $\mathfrak{m}$, we obtain $M/\mathfrak{m}M$; this is the unique simple module (up to isomorphism) for $\mathrm{Mat}_{d \times d}(k)$. Hence $M/\mathfrak{m}M$ is a simple projective $kG$-module, as claimed. $\qquad\Box$