Why is Nagao's theorem the "Module theoretic version of Brauer's second main theorem"?

Question

Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.

Brauer second main theorem states

If $\chi\in Irr(\mathbb{C}G)$ is an ordinary character in the $p$-block $B\in Bl(\mathbb{F}G)$, $u\in G_p$ a $p$-element, and $\phi\in IBr_p(C_G(u))$ an irreducible Brauer character in the $p$-block $b\in Bl(\mathbb{F}C_G(u))$, then $d_{\phi,\chi}^u \neq 0 \implies b^G = B$

Nagao's theorem states

If $e\in Z(\mathbb{F}G)$ is an idempotent, $M \in \mathbb{F}G\mathsf{-mod}$ a module with $eM=M$, $H$ a subgroup with $C_G(D) \leq H \leq N_G(D)$, and $Br_D: Z(\mathbb{F}G) \to Z(\mathbb{F}H)$ the Brauer homomorphism, then the restriction of $M$ to $H$ decomposes as $Res_H^G(M) = Br_D(e)M \oplus M'$ withsome $\mathbb{F}H$-module $M'$ whose indecomposable summands all are relatively projective w.r.t. to some $Q\leq H$ with $D\not\leq Q$.

I have seen in several places, for example in Benson's book, that Nagao's theorem is considered to be "the module theoretic version of Brauer's second main theorem". I struggle to see the similarity. I assume Nagao's theorem (at least) implies the 2nd main theorem, but I am unable to prove it right now. Perhaps I'm missing some crucial insight about the connection between the generalised decomposition numbers and the Brauer homomorphism.

Answer 1

To see the connection, it is easiest to work over a local ring $R$ of characteristic zero with residue field $R/J(R) \cong \mathbb{F}$ (there are some technicalities I am omitting here for the sake of brevity).

The key point is that if $B$ is a $p$-block with defect group $P$ (I avoid $D$ for the name of a defect group since you have used it more generally), and $x$ is an element of $P$, then Nagao's lemma as you state it (but lifted to to the version over $R$) applies to an $RG$-module affording irreducible character $\chi \in B$ with $D = \langle x \rangle$ and with $H = C_{G}(D)$, $e = 1_{B}$ and with $M$ an $RG$-module affording $\chi.$

The block summands of ${\rm Br}_{D}(e)RH$ are just ( the lifts of) the Brauer correspondent blocks of $H$ for $B$.

Nagao's theorem (together with Mackey decomposition) tells us that all indecomposable summands of $N = (e - {\rm Br}_{D}(e))M$ (viewed as $RD$-module) have vertex strictly less than $D$. Then Green's indecomposability theorem tells us that each such indecomposable summand is induced from $\langle x^{p} \rangle$ (using the fact that $D$ is cyclic).

Now take a $p$-regular element $y$ of $H$. Then the primitive idempotents of $R\langle y \rangle$ (there are $|\langle y \rangle |$ of these as $y$ has order prime to $p$) give a decomposition of $N$ as a sum of indecomposable $RD$-modules such that $y$ acts as a scalar on each summand. Each of these still has vertex strictly less than $D$,so $x$ has trace zero on each of them. Since $y$ acts a scalar on each summand, we see that $xy$ acts with trace zero on $N$.

Hence we see that only the Brauer correspondent blocks of $B$ for $H = C_{G}(x)$ need to be considered when calculating $\chi(xy).$ Since $y$ was an arbitrary $p$-regular element of $H = C_{G}(x),$ we do obtain fairly easily the usual statement of Brauer's second main theorem from this.

Thus it is true that (with a little effort) one can deduce Brauer's second main theorem from Nagao's theorem and "standard" theory, and it is reasonable to consider Nagao's theorem as a (strict) generalization of Brauer's second main theorem.